A Radically Modern Approach to Introductory Physics Second Edition

Volume 2: Four Forces

David J. Raymond

Copyright © 2012, 2015 David J. Raymond

Second Edition
Original state, 14 December 2015

Content of this book available under the Creative Commons Attribution-Noncommercial-Share-Alike License. See http://creativecommons.org/licenses/by-nc-sa/3.0/ for details.

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OCLC Number: 932065971 (print) – 932066146 (ebook)

Published by The New Mexico Tech Press, a New Mexico nonprofit corporation

This copy printed by CreateSpace, Charleston, SC
Printed in the United States of America

The New Mexico Tech Press
Socorro, New Mexico, USA
http://press.nmt.edu

Contents

Preface

The idea for a “radically modern” introductory physics course arose out of frustration in the physics department at New Mexico Tech with the standard two-semester treatment of the subject. It is basically impossible to incorporate a significant amount of “modern physics” (meaning post-19th century!) in that format. It seemed to us that largely skipping the “interesting stuff” that has transpired since the days of Einstein and Bohr was like teaching biology without any reference to DNA. We felt at the time (and still feel) that an introductory physics course for non-majors should make an attempt to cover the great accomplishments of physics in the 20th century, since they form such an important part of our scientific culture.

It would, of course, be easy to pander to students – teach them superficially about the things they find interesting, while skipping the “hard stuff”. However, I am convinced that they would ultimately find such an approach as unsatisfying as would the educated physicist. What was needed was a unifying vision which allowed the presentation of all of physics from a modern point of view.

The idea for this course came from reading Louis de Broglie’s Nobel Prize address.¹ De Broglie’s work is a masterpiece based on the principles of optics and special relativity, which qualitatively foresees the path taken by Schrödinger and others in the development of quantum mechanics. It thus dawned on me that perhaps optics and waves together with relativity could form a better foundation for all of physics, providing a more interesting way into the subject than classical mechanics.

Whether this is so or not is still a matter of debate, but it is indisputable that such a path is much more fascinating to most college students interested in pursuing studies in physics — especially those who have been through the usual high school treatment of classical mechanics. I am also convinced that the development of physics in these terms, though not historical, is at least as rigorous and coherent as the classical approach.

After 15 years of gradual development, it is clear that the course failed in its original purpose, as a replacement for the standard, one-year introductory physics course with calculus. The material is way too challenging, given the level of interest of the typical non-physics student. However, the course has found a niche at the sophomore level for physics majors (and occasional non-majors with a special interest in physics) to explore some of the ideas that drew them to physics in the first place. It was placed at the sophomore level because we found that having some background in both calculus and introductory college-level physics is advantageous for most students. However, we allow incoming freshmen into the course if they have an appropriate high school background in physics and math.

The course is tightly structured, and it contains little or nothing that can be omitted. However, it is designed to fit into two semesters or three quarters. In broad outline form, the structure is as follows:

• Optics and waves occur first on the menu. The idea of group velocity is central to the entire course, and is introduced in the first chapter. This is a difficult topic, but repeated reviews through the year cause it eventually to sink in. Interference and diffraction are done in a reasonably conventional manner. Geometrical optics is introduced, not only for its practical importance, but also because classical mechanics is later introduced as the geometrical optics limit of quantum mechanics.
• Relativity is treated totally in terms of space-time diagrams – the Lorentz transformations seem to me to be quite confusing to students at this level (“Does gamma go upstairs or downstairs?”), and all desired results can be obtained by using the “space-time Pythagorean theorem” instead, with much better effect.
• Relativity plus waves leads to a dispersion relation for free matter waves. Optics in a region of variable refractive index provides a powerful analogy for the quantum mechanics of a particle subject to potential energy. The group velocity of waves is equated to the particle velocity, leading to the classical limit and Newton’s equations. The basic topics of classical mechanics are then done in a more or less conventional, though abbreviated fashion.
• Gravity is treated conventionally, except that Gauss’s law is introduced for the gravitational field. This is useful in and of itself, but also provides a preview of its deployment in electromagnetism. The repetition is useful pedagogically.
• Electromagnetism is treated in a highly unconventional way, though the endpoint is Maxwell’s equations in their usual integral form. The connection to relativity is exploited rather than buried. In particular, the seemingly simple question of how potential energy can be extended to the relativistic context gives rise to the idea of potential momentum. The potential energy and potential momentum together form a four-vector which is closely related to the scalar and vector potential of electromagnetism. The Aharonov-Bohm effect is easily explained using the idea of potential momentum in one dimension, while extension to three dimensions results in a version of Snell’s law valid for matter waves, from which the Lorentz force law is derived.
• The generation of electromagnetic fields comes from Coulomb’s law plus relativity (I borrowed from my graduate advisor Mel Schwartz’s text on electromagnetism here), with the scalar and vector potential being used to produce a much more straightforward treatment than is possible with electric and magnetic fields. Electromagnetic radiation is a lot simpler in terms of the potential fields as well.
• Resistors, capacitors, and inductors are treated for their practical value, but also because their consideration leads to an understanding of energy in electromagnetic fields.
• At this point the book shifts to a more qualitative (but non-trivial) treatment of atoms, atomic nuclei, the standard model of elementary particles, and techniques for observing the very small. Ideas from optics, waves, and relativity reappear here. The Bohr model of the hydrogen atom is not presented for the simple reason that it gets the angular momentum of the electron wrong!
• The final section of the course deals with heat and statistical mechanics. Only at this point do non-conservative forces appear in the context of classical mechanics. Counting as a way to compute the entropy is introduced, and is applied to the Einstein model of a collection of harmonic oscillators (conceptualized as a “brick”), and in a limited way to an ideal gas. The second law of thermodynamics follows. The book ends with a fairly conventional treatment of heat engines.

A few words about how I have taught the course at New Mexico Tech are in order. As with our standard course, each week contains three lecture hours and a two-hour recitation. The recitation is the key to making the course accessible to the students. I generally have small groups of students working on assigned homework problems during recitation while I wander around giving hints. After all groups have completed their work, a representative from each group explains their problem to the class. The students are then required to write up the problems on their own and hand them in at a later date. The problems are the key to student learning, and associating course credit with the successful solution of these problems insures virtually 100% attendance in recitation.

In addition, chapter reading summaries are required, with the students urged to ask questions about material in the text that gave them difficulties. Significant lecture time is taken up answering these questions. Students tend to do the summaries, as they also count for their grade. The summaries and the questions posed by the students have been quite helpful to me in indicating parts of the text which need clarification.

The writing style of the text is quite terse. This partially reflects its origin in a set of lecture notes, but it also focuses the students’ attention on what is really important. Given this structure, a knowledgeable instructor able to offer one-on-one time with students (as in our recitation sections) is essential for student success. The text is most likely to be useful in a sophomore-level course introducing physics majors to the broad world of physics viewed from a modern perspective.

I freely acknowledge stealing ideas from Edwin Taylor, John Archibald Wheeler, Thomas Moore, Robert Mills, Bruce Sherwood, and many other creative physicists, and I owe a great debt to them. The physics department at New Mexico Tech has been quite supportive of my efforts over the years relative to this course, for which I am exceedingly grateful. Finally, my humble thanks go out to the students who have enthusiastically (or on occasion unenthusiastically) responded to this course. It is much, much better as a result of their input.

My colleagues Alan Blyth, David Westpfahl, Ken Eack, and Sharon Sessions were brave enough to teach this course at various stages of its development, and I welcome the feedback I have received from them. Their experience shows that even seasoned physics teachers require time and effort to come to grips with the content of this textbook!

The reviews of Allan Stavely and Paul Arendt in conjunction with the publication of this book by the New Mexico Tech Press have been enormously helpful, and I am very thankful for their support and enthusiasm. Penny Bencomo and Keegan Livoti taught me a lot about written English with their copy editing.

David J. Raymond

Preface to Second Edition

Aside from numerous corrections, clarifications, and minor enhancements, the main additions to this edition include the following:

• The treatment of the harmonic oscillator is expanded, with complex exponential solutions, damped and forced oscillators, and several new problems.
• The discussion of Faraday’s law is enhanced, with a distinction being made between EMFs generated by changing magnetic fluxes through fixed loops and those generated by magnetic forces on charge moving through translating or deforming loops.
• A section on angular momentum and magnetic moments is added to the chapter on atoms. The Stern-Gerlach experiment is treated in a problem.

As in the first edition, I am greatful for the reviews of Paul Arendt and Allan Stavely, who always manage to catch things that I have overlooked.

David J. Raymond

Physics Department

New Mexico Tech

Socorro, NM, USA

djraymondnm@gmail.com

Chapter 13
Newton’s Law of Gravitation

In this chapter we study the law that governs gravitational forces between massive bodies. We first introduce the law and then explore its consequences. The notion of a test mass and the gravitational field is developed, followed by the idea of gravitational flux. We then learn how to compute the gravitational field from more than one mass, and in particular from extended bodies with spherical symmetry. We finally examine Kepler’s laws and learn how these laws and the conservation laws for energy and angular momentum may be used to solve problems in orbital dynamics.

13.1 The Law of Gravitation

Of Newton’s accomplishments, the discovery of the universal law of gravitation ranks as one of the greatest. Imagine two masses, M₁ and M₂, separated by a distance r. The force has the magnitude

(13.1)

where G = 6.67 × 10^-11 m³ kg^-1 s^-2 is the universal gravitational constant. The gravitational force is always attractive and it acts along the line of centers between the two masses.

13.2 Gravitational Field

The gravitational field at any point is equal to the gravitational force on some test mass placed at that point divided by the mass of the test mass. The dimensions of the gravitational field are length over time squared, which is the same as acceleration. For a single point mass M (other than the test mass), Newton’s law of gravitation tells us that

(13.2)

where r is the position of the test point relative to the mass M. Note that we have written this equation in vector form, reflecting the fact that the gravitational field is a vector. Thus, r = x_test - x_mass, where x_test and x_mass are the position vectors of the test point and the mass M. The vector r points from the mass to the test point. The quantity r = |r| is the distance from the mass to the test point.

If there is more than one mass, then the total gravitational field at a test point is obtained by computing the individual fields produced by each mass at the test point and vectorially adding these fields. This process is schematically illustrated in figure 13.1.

13.3 Gravitational Flux

The next concept we need to discuss is the gravitational flux. Figure 13.2 shows a rectangular area S with a vector S perpendicular to the rectangle. The vector S is defined to have length S, so it is a compact way of representing the size and orientation of a rectangle in three dimensional space. The vector S could point either upward or downward, and the choice of directions turns out to be important. This is why we say that S represents a directed area.

Figure 13.2 also shows a vector g, representing the gravitational field on the surface of the rectangle. Its value is assumed here not to vary with position on the rectangle. The angle θ is the angle between the vectors S and g.

The gravitational flux through the rectangle is defined as

(13.3)

where g_n = g cos θ is the component of g normal to the rectangle. The flux is thus larger for larger areas and for larger gravitational fields. However, only the component of the gravitational field normal to the rectangle (i. e., parallel to S) counts in this calculation. A consequence is that the gravitational flux through area 1, S₁ ⋅ g, in figure 13.3 is the same as the flux through area 2, S₂ ⋅ g.

The significance of the directedness of the area is now clear. If the vector S pointed in the opposite direction, the flux would have the opposite sign. When defining the flux through a rectangle, it is necessary to define which way the flux is going. This is what the direction of S does — a positive flux is defined as going from the side opposite S to the side of S.

An analogy with flowing water may be helpful. Imagine a rectangular channel of cross-sectional area S through which water is flowing at velocity v. The flux of water through the channel, which is defined as the volume of water per unit time passing through the cross-sectional area, is Φ_w = vS. The water velocity takes the place of the gravitational field in this case, and its direction is here assumed to be normal to the rectangular cross-section of the channel. The field thus expresses an intensity (e. g., the velocity of the water or the strength of the gravitational field), while the flux expresses an amount (the volume of water per unit time in the fluid dynamical case). The gravitational flux is thus the amount of some gravitational influence, while the gravitational field is its strength. We now try to more clearly understand to what this amount really refers.

We need to briefly consider the case in which the gravitational field varies from one point to another on the rectangular surface. In this case a proper calculation of the flux through the surface cannot be made using equation (13.3) directly. Instead, we must break the surface into a grid of sub-surfaces. If the grid is sufficiently fine, the gravitational field will be nearly constant over each sub-surface and equation (13.3) can be applied separately to each of these. The total flux is then the sum of all the individual fluxes.

There is actually no need for the area in figure 13.2 to be rectangular. We can calculate the gravitational flux through the surface of a sphere of radius R with a mass M at the center. As illustrated in figure 13.4, the gravitational field points inward toward the mass. It has magnitude g = GM∕R², so if we desire to calculate the gravitational flux out of the sphere, we must introduce a minus sign. Finally, the area of a sphere of radius R is S = 4πR², so the flux is

(13.4)

Notice that this flux doesn’t depend on how big the sphere is — the factor of R² in the area cancels with the factor of 1∕R² in the gravitational field. This is a hint that something profound is going on. The size of the surface enclosing the mass is unimportant, and neither is its shape — the answer is always the same — the gravitational flux outward through any closed surface surrounding a mass M is just Φ_g = -4πGM! This is an example of Gauss’s law applied to gravity.

It is possible to formally prove this result using arguments like those posed in figure 13.3, but perhaps the easiest way to understand this result is via the analogy with the flow of water. If we think of the mass as something which destroys water at a certain rate, then there must be an inward flow of water through the surfaces in the left and center examples in figure 13.5. Furthermore, the volume of water per unit time flowing inward through these surfaces is the same in the two examples, because the rate at which water is being destroyed is the same. In the right case the mass is not contained inside the surface and though water flows into the volume bounded by the surface, it also flows out the other side, resulting in a net outward (or inward) volume flux through the surface of zero.

13.4 Flux from Multiple Masses

Gauss’s law extends trivially to more than one mass. As figure 13.6 shows, the outward flux through a closed surface is just

(13.5)

In other words, all masses inside the closed surface contribute to the flux, while no masses outside the surface contribute. This is the most general statement of Gauss’s law as it applies to gravity.

An important application of Gauss’s law is to show that the gravitational field outside of a spherically symmetric extended mass M is exactly the same as if all the mass were concentrated at a point at the center of the sphere. The proof goes as follows: Imagine a sphere concentric with the center of the extended mass, but with larger radius. The gravitational flux from the mass is just Φ_g = -4πGM as before. However, because of the assumed spherical symmetry, we know that the gravitational field points normally inward at every point on the spherical surface and is equal in magnitude everywhere on the sphere. Thus we can infer that Φ_g = -4πR²g, where R is the radius of the sphere and g is the magnitude of the gravitational field at radius R. From these two equations we immediately infer that the field magnitude is

(13.6)

Expressing this in vector form for arbitrary radius r, and remembering that the gravitational field points inward, we find that

(13.7)

which is precisely the equation for g resulting from a point mass M. Recall that r points from the mass to the test point.

13.5 Effects of Relativity

So far our discussion of gravity has been completely non-relativistic. We will not explore in detail how the theory of gravity changes in a completely relativistic treatment. As we noted earlier in the course, Einstein’s general theory of relativity covers this, and the mathematics are formidable. We confine ourselves to two comments:

• As noted previously, gravity is locally equivalent to being in an accelerated reference frame. However, unlike the simple example which we studied earlier, there is in general no universal frame of reference that is everywhere inertial to which we can transform.
• Space is even more non-Euclidean in general relativity than in special relativity. In particular, there is no such thing as a straight line in the geometry of general relativistic spacetime. This is true because spacetime itself is curved. An example of a curved space is the surface of a sphere. Clearly, a straight line cannot be embedded in this space. The closest equivalent to a straight line in a curved space is a geodesic curve. On a sphere great circles are geodesic curves. In general relativity, objects subject only to the force of gravity move along geodesic curves.

One potentially observable prediction of relativity is the existence of gravitational waves. Imagine two stars revolving around each other. The gravitational field from these stars will change periodically due to this motion. However, this change propagates outward only at the speed of light. As a result, ripples in the field, or gravitational waves, spread outward from the revolving stars. Efforts are currently under way to develop apparatus to detect gravitational waves produced by violent cosmic events such as the explosion of a supernova.

13.6 Kepler’s Laws

Johannes Kepler, using data compiled by Tycho Brahe, inferred three laws governing the motions of planets in the solar system:

1. Planets move in elliptical orbits with the sun at one focus.
2. Equal areas are swept out in equal times by the line connecting the sun and the planet.
3. The square of the period of revolution of the planet around the sun is proportional to the cube of the semi-major axis of the ellipse.

These laws were instrumental in the development of modern mechanics and the universal law of gravitation by Isaac Newton.

Showing that the first law is consistent with Newtonian mechanics is mathematically more difficult than we can undertake in this course. However, the second law turns out to be a simple consequence of the conservation of angular momentum. Figure 13.7 shows an elliptical orbit with the area swept out as a planet moves from position 1 to position 2. We estimate this area as dA = Rdx∕2, where we have ignored the small unshaded part of the area to the right of the shaded triangle. The distance traveled by the planet in time dt is ds, so the magnitude of the velocity is v = ds∕dt. However, in computing the angular momentum, we need the tangential component of the velocity, i. e., the component normal to the radius vector R. This is simply v_t = dx∕dt. The angular momentum is L = mRv_t = mRdx∕dt, where m is the mass of the planet. Combining this with the formula for dA results in

(13.8)

Since gravitation is a central force, angular momentum is conserved, which means that dA∕dt is constant. Thus, we have shown that conservation of angular momentum is equivalent to Kepler’s second law.

Kepler’s third law turns out to be a consequence of the universal law of gravitation. We can prove this for circular orbits. We know that a planet moving in a circular orbit around the sun is accelerating toward the sun with the centripetal acceleration a = v²∕R, where v is the speed of the planet’s motion in its orbit and R is the orbit’s radius. This acceleration is caused by the gravitational force, so we can equate the force divided by the planetary mass to a, resulting in

(13.9)

where M is the mass of the sun. This may be solved for v:

(13.10)

Eliminating v in favor of the period of revolution T = 2πR∕v results in

(13.11)

This agrees with Kepler’s third law since the semi-major axis of a circle is simply the radius R.

13.7 Use of Conservation Laws

The gravitational force is conservative, so two point masses M and m separated by a distance r have a potential energy:

(13.12)

It is easily verified that differentiation recovers the gravitational force.

The conservation of energy and angular momentum in planetary motions can be used to solve many practical problems involving motion under the influence of gravity. For instance, suppose a bullet is shot straight upward from the surface of the moon. One might ask what initial velocity is needed to insure that the bullet will escape from the gravity of the moon. Since total energy E is conserved, the sum of the initial kinetic and potential energies must equal the sum of the final kinetic and potential energies:

(13.13)

For the bullet to escape the moon, its kinetic energy must remain positive no matter how far it gets from the moon. Since the potential energy is always negative, asymptoting to zero at infinite distance (i. e., U_final = 0), the minimum total energy consistent with this condition is zero. For zero total energy we have

(13.14)

where m is the mass of the bullet, M is the mass of the moon, R is the radius of the moon, and v_initial is the minimum initial velocity required for the bullet to escape. Solving for v_initial yields

(13.15)

This is called the escape velocity. Notice that the escape velocity from a given radius is a factor of 2^1∕2 larger than the velocity needed for a circular orbit at that radius (see equation (13.10)).

An object is energetically bound to the sun if its kinetic plus potential energy is less than zero. In this case the object follows an elliptical orbit around the sun as shown by Kepler. However, if the kinetic plus potential energy is zero, the object follows a parabolic orbit, and if it is greater than zero, a hyperbolic orbit results. In the latter two cases the sun also resides at a focus of the parabola or hyperbola. Figure 13.8 shows a typical hyperbolic orbit. The impact parameter, defined in this figure, is the closest the object would have come to the center of the sun if it hadn’t been deflected by gravity.

Sometimes energy and angular momentum conservation can be used together to solve problems. For instance, suppose we know the energy and angular momentum of an asteroid of mass m and we wish to infer the maximum and minimum distances of the asteroid from the sun, the so-called aphelion and perihelion distances. Since the asteroid is gravitationally bound to the sun, it is convenient to characterize the total energy by E_b = -E, the so-called binding energy. If v is the orbital speed of the asteroid and r is its distance from the sun, then the binding energy can be written in terms of the kinetic and potential energies:

(13.16)

The magnitude of the angular momentum of the asteroid is L = mv_tr, where v_t is the tangential component of the asteroid’s velocity. At aphelion and perihelion, the radial part of the velocity of the asteroid is zero and the speed equals the tangential component of the velocity, v = v_t. Thus, at aphelion and perihelion we can eliminate v in favor of the angular momentum:

(13.17)

This can be rearranged into a quadratic equation

(13.18)

which can be solved to yield

(13.19)

The larger of the two solutions yields the aphelion value of the radius while the smaller yields the perihelion.

Equation (13.19) tells us something else interesting. The quantity inside the square root cannot be negative, which means that we must have

(13.20)

In other words, for a given value of the binding energy E_b there is a maximum value for the angular momentum. This maximum value makes the square root zero, which means that the aphelion and the perihelion are the same — i. e., the orbit is circular. Thus, among all orbits with a given binding energy, the circular orbit has the maximum angular momentum.

13.8 Problems

1. Assume a mass M is located at (-2 m, 0 m) and a mass 2M is located at (0 m, 3 m). Find the (vector) gravitational field at the point (1 m, 1 m).
2. If two equal masses M are located at x₁ = (-3 m,-4 m) and x₂ = (-3 m, +4 m), determine where a third mass M must be placed to result in zero gravitational force at the origin.
3. Given the value of g at the Earth’s surface, the radius of the Earth (look it up), and the universal gravitational constant G, determine the mass of the Earth.
4. Given the situation in figure 13.9:
   1. What is the gravitational flux through the illustrated surface?
   2. Explain why you cannot use this information to compute the gravitational field on the surface in this case.

   ---

   

   
   

   Figure 13.9: Various masses inside and outside a Gaussian surface.

   _____________________________________

   ---

5. Suppose mass is distributed uniformly with density μ kg m^-1 in an infinite thin line along the z axis. Try to figure out a way of using Gauss’s law and symmetry arguments to predict the gravitational field resulting from this mass distribution.
6. Assume that the Earth is of uniform density, ρ.
   1. Use Gauss’s law to determine the gravitational field inside the Earth as a function of distance from the center.
   2. Using the results of the above problem, determine the motion of an object moving through an evacuated hole drilled through the center of the Earth. Ignore the Earth’s rotation.
7. Two infinite thin sheets of mass, each with σ mass per unit area, are aligned perpendicular to each other. Determine the gravitational field from this combination. Hint: Compute g from each sheet separately and add vectorially.
8. Suppose that the universal law of gravitation says that the (attractive) gravitational force takes the form F = M₁M₂Gr, where r is the separation between the two masses M₁ and M₂ and G is a constant. Find the relationship between the orbital radius and the period for a circular orbit of a planet around the sun in this case.
9. An alien spaceship enters the solar system at distance D from the sun with speed v₀. (D may be considered to be very far from the sun.) It coasts through the solar system, approaching within a distance d ≪ D of the sun.
   1. Find its speed at the point of closest approach.
   2. Find the angular momentum of the spaceship with respect to the center of the sun.
   3. What was the tangential component of the spaceship’s velocity (i. e., the component normal to the radius vector) when it entered the solar system at distance D?
10. As a result of tidal torques, the spin angular momentum of the Earth is gradually being converted into orbital angular momentum of the moon, which causes the radius of the moon’s (circular) orbit to increase. Hint: Recall that for a solid sphere the moment of inertia is I = 2mr²∕5.
    1. Obtain a relationship between the moon’s orbital velocity and its distance from the Earth, assuming that the orbit is circular.
    2. If the Earth’s rotation rate is cut in half due to this effect, what will the new radius of the moon’s orbit be?

Chapter 14
Forces in Relativity

In this chapter we ask an apparently simple question: How can the idea of potential energy be extended to the relativistic case? The answer to this question is unexpectedly complex, but it leads us to immensely fruitful results. In particular, it prompts us to investigate the idea of potential momentum, which results ultimately in gauge theory, of which electromagnetism is an example.

Along the way we show that conservation of four-momentum has an unexpected consequence — the idea of force at a distance is inconsistent with the theory of relativity. This means that momentum and energy must be carried between interacting particles by another type of particle that we call an intermediary particle. These particles are virtual in the sense that they don’t have their real-world mass when acting in this role.

In relativistic quantum mechanics, we find that particles can take on negative energies. Feynman’s interpretation of this fact is discussed, which leads us to a model for antiparticles.

14.1 Potential Momentum

For a free, non-relativistic particle of mass m, the total energy E equals the kinetic energy K and is related to the momentum Π of the particle by

(14.1)

(Note that we have ignored the contribution of the rest energy to the total energy here.) In the non-relativistic case, the momentum is Π = mv, where v is the particle velocity.

If the particle is not free, but is subject to forces associated with a potential energy U(x,y,z), then equation (14.1) must be modified to account for the contribution of U to the total energy:

(14.2)

The force on the particle is related to the potential energy by

(14.3)

For a free, relativistic particle, we have

(14.4)

The obvious way to add forces to the relativistic case is by rewriting equation (14.4) with a potential energy, in analogy with equation (14.2):

(14.5)

Unfortunately, equation (14.5) is incomplete, because we have subtracted U from the energy E without subtracting a corresponding term from the momentum Π as well. However, Π = (Π,E∕c) is a four-vector, so an equation with something subtracted from just one of the components of this four-vector is not relativistically invariant. In other words, equation (14.5) doesn’t obey the principle of relativity, and therefore cannot be correct!

How can we fix this problem? One way is to define a new four-vector with U∕c being its timelike part and some new vector Q being its spacelike part:

(14.6)

We then subtract Q from the momentum Π. When we do this, equation (14.5) becomes

(14.7)

The quantity Q is called the potential momentum and Q is the potential four-momentum.

Some additional terminology is useful. We define

(14.8)

as the kinetic momentum for reasons discussed below. In order to avoid confusion, we rename Π the total momentum.¹ Thus, the total momentum equals the kinetic plus the potential momentum, in analogy with energy.

So far, we have shown that the introduction of a potential momentum complements the potential energy so as to make the energy-momentum relationship for a particle relativistically invariant. However, we as yet have no idea what causes potential momentum nor what it does to the affected particle. We shall put off answering the former question and address only the latter at this point. A hint comes from the corresponding behavior of energy. The total energy of a particle is related to the quantum mechanical frequency ω of the particle, and the total momentum is related to its wave vector k:

(14.9)

However, the kinetic energy² and the kinetic momentum are related to the particle’s velocity v:

(14.10)

where v = |v|.

The relationship between kinetic momentum and velocity can be proven by dividing equation (14.7) by ℏ to obtain a dispersion relation and then computing the group velocity, which we equate to the particle velocity. However, we will not do this here.

14.2 Aharonov-Bohm Effect

Let us now study a phenomenon that depends on the existence of potential momentum. If the potential energy of a particle is zero and both the kinetic and potential momenta point in the ±x direction, the total energy equation (14.7) for the particle becomes

(14.11)

Since its total energy E is conserved, the magnitude of the kinetic momentum p of the particle doesn’t change according to the above equation. Thus, if a region of non-zero potential momentum is encountered, the total momentum of the particle must change so as to keep the kinetic momentum constant. This results in a change in the wavelength of the matter wave associated with the particle. In particular, if the potential momentum points in the same direction as the kinetic momentum, the total momentum is increased and the wavelength decreases, while a potential momentum pointing in the direction opposite the kinetic momentum results in an increase in wavelength.

Figure 14.1 illustrates what might happen to a particle moving through a channel that splits into two sub-channels for an interval. If we arrange to have non-zero potential momenta pointing in opposite directions in the sub-channels, the wavelength of the particle will be different in the two regions. At the end of the interval, the waves recombine, interfering constructively or destructively, depending on the magnitude of the phase difference between them. If destructive interference occurs, then the particle cannot pass. The potential momentum thus acts as a valve controlling the flow of particles through the channel. This is an example of the Aharonov-Bohm effect.

14.3 Forces from Potential Momentum

In the Aharonov-Bohm effect, the potential momentum didn’t result in any force on the particle — its only manifestation was to change the particle’s wavelength. In such situations the potential momentum’s presence is only revealed by quantum mechanical effects.

The potential momentum has more of an influence on the non-quantum world when the problem is two or three-dimensional or when the potential momentum is changing with time. The total force on a particle due to all possible effects involving the potential energy and the potential momentum is given by

(14.12)

where v is the particle velocity and P is a vector obtained from the potential momentum vector as follows:

(14.13)

This is unexpectedly complicated. However, equation (14.12) consists of three parts. The first part involves derivatives of the potential energy and is exactly the same as in the non-relativistic case. The new effects are confined to the second and third parts, -∂Q∕∂t and v × P. A full derivation of these equations involves rather complex mathematics. However, it is possible to understand the origin of these additional contributions to the force by looking at a couple of simple examples.

14.3.1 Refraction Effect

A matter wave impingent on a discontinuity in potential momentum is refracted, just as it is refracted by a discontinuity in potential energy. Refraction of a matter wave packet means that the velocity of the associated particle changes as it moves across the interface. This means that the particle undergoes an acceleration, implying that it is subject to a force.

As in the case of Snell’s law for optics, the frequency of a matter wave doesn’t change as it crosses such a discontinuity in potential momentum. Furthermore, neither does the component of the wave vector parallel to the discontinuity. These two conditions together ensure phase continuity at the interface.

Figure 14.2 shows an example of what happens when a wave encounters a series of parallel slabs with increasing values of Q. The y component of the wave vector doesn’t change as the wave crosses each of the interfaces between slabs, for reasons discussed above. Hence, Π_y = ℏk_y doesn’t change either, which means that dΠ_y∕dx = 0. The y component of kinetic momentum, p_y = Π_y - Q_y, must therefore decrease as Q_y increases, as illustrated in figure 14.2.

Newton’s second law tells us that the y component of the force on the particle associated with the wave is just the time derivative of the y component of the kinetic momentum:

(14.14)

In the last step of this equation we used the fact that dΠ_y∕dx = 0.

The x component of the force can be obtained by similar reasoning, using the additional information that the speed, and hence the magnitude of the kinetic momentum, p² = p _x² + p _y², doesn’t change under the influence of the potential momentum:

(14.15)

Aside from assuming that p² = constant, we have used the relationships p_x = (p² - p _y²)^1∕2 and p _y∕p_x = u_y∕u_x. Equations (14.14) and (14.15) constitute a special case of equations (14.12) and (14.13) which is valid when Q points in the y direction and is a function only of x.

14.3.2 Time-Varying Potential Momentum

The necessity for the term -∂Q∕∂t in equation (14.12) is easily understood from the following argument, which is illustrated in figure 14.3. The example in the previous section showed that a particle moving in the +x direction with velocity u_x through a field of increasing Q_y (left panel of figure 14.3) experiences a force in the -y direction equal to F_y = -(dQ_y∕dx)u_x. However, viewing this same process from a reference frame in which the particle is stationary (right panel of this figure), we see that the potential momentum at the position of the particle increases with time at the rate dQ_y∕dt. The particle is not moving in this reference frame, so the term v×P = 0. However, the stationary particle must still experience the above force in this reference frame in order to satisfy the principle of relativity.

Noting that dQ_y∕dt = (dQ_y∕dx)u_x, we see that equation (14.12) provides this force via the term -∂Q∕∂t in the reference frame moving with the particle. Thus, the time derivative term in equation (14.12) is needed to maintain the principle of relativity; the same force occurs in the two different reference frames but originates from the term v × P in the original reference frame and the term -∂Q∕∂t in the frame moving with the particle.

Arguments similar to these were actually made by Einstein in his original 1905 paper on relativity.

14.4 Lorenz Condition

It turns out that the four components of the potential four-momentum are not independent, but are subject to the condition

(14.16)

This is called the Lorenz condition. The physical meaning of this condition will become clear when we study electromagnetism.³

14.5 Gauge Theories and Other Theories

The theory of potential momentum is only one of three ways in which the idea of potential energy can be extended to the relativistic case. This theory is called gauge theory for obscure historical reasons. Gauge theory is important because electromagnetism as well as the theories of weak and strong sub-nuclear interactions are all of this type.

Gravity is the only fundamental force that does not take the form of a gauge theory. Instead, gravity takes the form of one of two other possible relativistic extensions of potential energy. This theory is called general relativity. The gravitational force in general relativity can be interpreted geometrically as a consequence of the curvature of spacetime. Mathematically, it is far too difficult to pursue here.

The third relativistic extension of potential energy considers potential energy to be a field which alters the rest energy of particles. High energy physicists believe that the elementary particles gain their mass by this mechanism. The field is called the Higgs field after the English physicist who first proposed this theory, Peter Higgs. The recent discovery of the Higgs boson at CERN’s Large Hadron Collider in Geneva, Switzerland supports this idea.

14.6 Conservation of Four-Momentum

We earlier introduced the ideas of energy and momentum conservation. In other words, if we have a number of particles isolated from the rest of the universe, each with momentum p_i and energy E_i, then particles may be created and destroyed and they may collide with each other.⁴ In these interactions the energy and momentum of each particle may change, but the sum total of all the energy and the sum total of all the momentum remains constant with time:

(14.17)

The expression is simpler in terms of four-momentum:

(14.18)

At this point a statement such as the one above should ring alarm bells. Just what does it mean to say that the total energy and momentum remain constant with time in the context of relativity? Which time? The time in which reference frame?

Figure 14.4 illustrates the problem. Suppose two particles exchange four-momentum remotely at the time indicated by the fat horizontal bar in the left panel of figure 14.4. Conservation of four-momentum implies that

(14.19)

where the subscripted letters correspond to the particle labels in figure 14.4. Primed values refer to the momentum after the exchange while no primes indicates values before the exchange.

Now view the exchange from the reference frame in the right panel of figure 14.4. A problem with four-momentum conservation exists in the region between the thin horizontal lines. In this region particle B has already transferred its four-momentum, but it has yet to be received by particle A. In other words, four-momentum is not conserved in this reference frame!

This problem is so serious that we must eliminate the concept of force at a distance from the repertoire of physics. The only way to have particles interact remotely and still conserve four-momentum in all reference frames is to assume that all remote interactions are mediated by another particle, as indicated in figure 14.5. In other words, momentum and energy are transferred from particle A to particle B in a two step process. First, particle A emits particle C in a manner which conserves the four-momentum. Second, particle C is absorbed by particle B in a similarly conservative interaction. Four-momentum is conserved at all times in all reference frames in this picture.

14.7 Virtual Particles

Another problem is evident from figure 14.5. As drawn, the velocity of the intermediary particle exceeds the speed of light. This is reflected in the fact that different reference frames yield contradictory results as to whether the intermediary particle moves from A to B or B to A. These difficulties turn out to be much less severe than those arising from non-locality. Let us address them in sequence.

For sake of definiteness, let us view the emission of particle C by particle A in a reference frame in which the velocity of particle A is just reversed in the emission process. In this case the four-momentum before the emission is p_A = (p,E∕c), where E = (p²c² + m²c⁴)^1∕2. After the emission we have p′_A = (-p,E∕c). Conservation of four-momentum in the emission process requires that

(14.20)

where q is the four-momentum of particle C. From the above assumptions it is clear that

(14.21)

Suppose that the real, measured mass of particle C is m_C. This conflicts with the apparent or virtual mass of this particle in its flight from A to B, which is

(14.22)

where q ≡|q| is the momentum transfer. Note that the apparent mass is imaginary because the four-momentum is spacelike.

Classically, this discrepancy in the apparent and actual masses of the particle C would simply indicate that the process wasn’t possible. However, recall that the uncertainty principle allows there to be an uncertainty in the mass if it doesn’t persist for too long in terms of the proper time interval along the particle’s world line. The statement of this law is ΔμΔτ ≈ 1. Expressed in terms of mass, this becomes

(14.23)

Let us convert the proper time to an interval since the world line of particle C is horizontal in the reference frame in which we are viewing it. Ignoring the factor of i, Δτ = ΔI∕c. We finally compute the absolute value of the mass discrepancy as follows: |m_C -iq∕c| = [(m_C -iq∕c)(m_C + iq∕c)]^1∕2 = (m _C² + q²∕c²)^1∕2. Solving for I yields the approximate maximum invariant interval that particle C can move from its source point while keeping its erroneous mass hidden by the uncertainty principle:

(14.24)

A particle forced into having an apparent mass different from its actual mass is called a virtual particle. The interaction shown in figure 14.5 can only take place if particles A and B come closer to each other than the distance ΔI. This argument thus produces an estimate for the “range” of an interaction with momentum transfer 2p and intermediary particle mass m_C.

Two distinct possibilities exist. If the intermediary particle is massless (a photon, for instance), then the range of the interaction is inversely related to the momentum transfer: ΔI ≈ℏ∕q. Thus, small momentum transfers can occur at large distances. An interaction of this type is called “long range”. On the other hand, if the intermediary particle has mass, the range is simply ΔI ≈ℏ∕m_Cc when q ≪ m_Cc. The range is thus constant and inversely proportional to the mass of the intermediary particle for low momentum transfers. For large momentum transfer, i. e., when q ≫ m_Cc, the range decreases from this value with increasing momentum transfer, as in the case of a massless intermediary particle.

14.7.1 Virtual Particles and Gauge Theory

According to quantum mechanics, particles are represented by waves. The absolute square of the wave amplitude represents the probability of finding the particle. In gauge theory the potential four-momentum performs this role for the virtual particles intermediary interactions. Thus a larger potential four-momentum at some point means a higher probability of finding the related virtual particles at that point.

14.8 Negative Energies and Antiparticles

Figure 14.5 illustrates another oddity in the role of intermediary particles in collisions. In the unprimed frame, particle C appears to be emitted by particle A and absorbed by particle B. In the primed frame the reverse is true; it appears to be emitted by B and absorbed by A. These judgements are based on the fact that the A vertex occurs earlier than the B vertex in the unprimed frame, while the B vertex occurs earlier in the primed frame. However, since these distinctions are based on time ordering in different reference frames of events separated by a spacelike interval, they are inherently not relativistically invariant. Since the principle of relativity states that physical laws are the same in all inertial reference frames, we have a conceptual problem to overcome.

A related problem has to do with the computation of energy from mass and momentum. The solution of equation E² = p²c² + m²c⁴ for the energy has a sign ambiguity that we have so far ignored:

(14.25)

A natural tendency would be to omit the minus sign and just consider positive energies. However, this would be a mistake — experience with quantum mechanics indicates that both solutions must be considered.

Richard Feynman won the Nobel Prize in physics largely for developing a consistent interpretation of the above negative energy solutions, which we now relate. Notice that the four-momentum points backward in time in a spacetime diagram if the energy is negative. Feynman suggested that a particle with four-momentum p is equivalent to the corresponding antiparticle with four-momentum -p. Thus, we interpret a particle with momentum p and energy E < 0 as an antiparticle with momentum -p and energy -E > 0.

Antiparticles are known to exist for all particles. If a particle and its antiparticle meet, they can annihilate, creating one or more other particles. Correspondingly, if energy is provided in the right form, a particle-antiparticle pair can be created.

Suppose a particular kind of particle, call it an A particle, produces a B particle when it annihilates with its antiparticle A. This is illustrated in the left panel in figure 14.6. In Feynman’s view, this process is equivalent to the scattering of an A particle backward in time by a B particle, the scattering of an A backward in time by a B particle, the creation of an AA pair moving backward in time by a B particle (an antiB), and the emission of a B particle by an A particle moving forward in time.

The statement “moving backward in time” has stimulated generations of physics students to contemplate the possibility that Feynman’s picture makes time travel possible. As far as we know, this is not so. The key phrase is equivalent to. In other words, causality still works forward in time as we have come to expect.

The real utility of the “backward in time” picture is that it makes calculations easier, since processes which are normally thought of as being very different turn out to have the same mathematical form.

Returning to the ambiguity shown in figure 14.5, it turns out that it does not matter whether the picture in the left or right panel is chosen. According to the Feynman view the two processes are equivalent if one small correction is made — if the intermediary particle going from left to right is a C particle, then the intermediary particle going from right to left in the other picture is a C particle, or an antiC. It is immaterial whether the arrow representing either the C or the C points forward or backward in time. The key point is that if an arrow points into a vertex, the four-momentum of that particle contributes to the input side of the momentum-energy budget for that vertex. If an arrow points away from a vertex, then the four-momentum contributes to the output side.

14.9 Problems

1. An alternate way to modify the energy-momentum relation while maintaining relativistic invariance is with a “potential mass”, H(x):

   If |H|≪ m and p² ≪ m²c², show how this equation may be approximated as

   

   and determine the form of “something” in terms of H. Is this theory distinguishable from the theory involving potential energy at nonrelativistic velocities?

2. For a given channel length L and particle speed in figure 14.1, determine the possible values of potential momentum ±Q in the two channels that result in destructive interference between the two parts of the particle wave.
3. Show that equations (14.14) and (14.15) are indeed recovered from equations (14.12) and (14.13) when Q points in the y direction and is a function only of x.
4. Show that the force F = v × P is perpendicular to the velocity v. Does this force do any work on the particle? Is this consistent with the fact that the force doesn’t change the particle’s kinetic energy?
5. Show that the potential momentum illustrated in figure 14.2 satisfies the Lorenz condition, assuming that U = 0. Would the Lorenz condition be satisfied in this case if Q depended only on x and pointed in the x direction?
6. A mass m moves at non-relativistic speed around a circular track of radius R as shown in figure 14.7. The mass is subject to a potential momentum vector of magnitude Q pointing counterclockwise around the track.
   1. If the particle moves at speed v, does it have a longer wavelength when it is moving clockwise or counterclockwise? Explain.
   2. Quantization of angular momentum is obtained by assuming that an integer number of wavelengths n fits into the circumference of the track. For given |n|, determine the speed of the mass (i) if it is moving clockwise (n < 0), and (ii) if it is moving counterclockwise (n > 0).
   3. Determine the kinetic energy of the mass as a function of n.

   ---

   

   
   

   Figure 14.7: The particle is constrained to move along the illustrated track under the influence of a potential momentum Q.

   _____________________________________

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7. Suppose momentum were conserved for action at a distance in a particular reference frame between particles 1 cm apart as in the left panel of figure 14.4 in the text. If you are moving at velocity 2 × 10⁸ m s^-1 relative to this reference frame, for how long a time interval is momentum apparently not conserved? Hint: The 1 cm interval is the invariant distance between the kinks in the world lines.
8. An electron moving to the right at speed v collides with a positron (an antielectron) moving to the left at the same speed as shown in figure 14.8. The two particles annihilate, forming a virtual photon, which then decays into a proton-antiproton pair. The mass of the electron is m and the mass of the proton is M = 1830m.
   1. What is the mass of the virtual photon? Hint: It is not 2m. Why?
   2. What is the maximum possible lifetime of the virtual photon by the uncertainty principle?
   3. What is the minimum v the electron and positron need to have to make this reaction energetically possible? Hint: How much energy must exist in the proton-antiproton pair?

   ---

   

   
   

   Figure 14.8: Electron-positron annihilation leading to proton-antiproton production.

   _____________________________________

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9. A muon (mass m) interacts with a proton as shown in figure 14.9, so that the velocity of the muon before the interaction is v, while after the interaction it is -v∕2, all in the x direction. The interaction is mediated by a single virtual photon. Assume that v ≪ c for simplicity.
   1. What is the momentum of the photon?
   2. What is the energy of the photon?

   ---

   

   
   

   Figure 14.9: Collision of a muon with a proton, mediated by the exchange of a virtual photon.

   _____________________________________

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10. A photon with energy E and momentum E∕c collides with an electron with momentum p = -E∕c in the x direction and mass m. The photon is absorbed, creating a virtual electron. Later the electron emits a photon in the x direction with energy E and momentum -E∕c. (This process is called Compton scattering and is illustrated in figure 14.10.)
    1. Compute the energy of the electron before it absorbs the photon.
    2. Compute the mass of the virtual electron, and hence the maximum proper time it can exist before emitting a photon.
    3. Compute the velocity of the electron before it absorbs the photon.
    4. Using the above result, compute the energies of the incoming and outgoing photons in a frame of reference in which the electron is initially at rest. Hint: Using E_photon = ℏω and the above velocity, use the Doppler shift formulas to get the photon frequencies, and hence energies in the new reference frame.

    ---

    

    
    

    Figure 14.10: Compton scattering.

    _____________________________________

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11. The dispersion relation for a negative energy relativistic particle is

    Compute the group velocity of such a particle. Convert the result into an expression in terms of momentum rather than wavenumber. Compare this to the corresponding expression for a positive energy particle and relate it to Feynman’s explanation of negative energy states.

12. The potential energy of a charged particle in a scalar electromagnetic potential ϕ is the charge times the scalar potential. The total energy of such a particle at rest is therefore

    where q is the charge on the particle and ±mc² is the rest energy, with the ± corresponding to positive and negative energy states. Assume that |qϕ|≪ mc².

    1. Given that a particle with energy E < 0 is equivalent to the corresponding antiparticle with energy equal to -E > 0, what is the potential energy of the antiparticle?
    2. From this, what can you conclude about the charge on the antiparticle?

    Hint: Recall that the total energy is always rest energy plus kinetic energy (zero in this case) plus potential energy.

Chapter 15
Electromagnetic Forces

In this chapter we begin the study of electromagnetism. The forces on charged particles due to electromagnetic fields are introduced and related to the general case of force on a particle by a gauge field. The principles of electric motors and generators are then addressed as an example of such forces in action.

15.1 Electromagnetic Four-Potential

Electromagnetism is a gauge theory. Particles that have a property called electric charge are subject to forces exerted by the gauge fields of electromagnetism. The potential four-momentum Q = (Q,U∕c) of a particle with charge q in the presence of the electromagnetic four-potential a is just

(15.1)

In the simplest case the four-potential represents the amplitude for finding the intermediary particle associated with the electromagnetic gauge field. This particle has zero mass and is called the photon. If more than one photon is present, the interpretation of a becomes more complicated. This issue will be considered later.

The four-potential has space and time components A and ϕ∕c such that a = (A,ϕ∕c). The quantity A is called the vector potential and ϕ is called the scalar potential. The scalar and vector potential are related to the potential energy U and potential momentum Q of a particle of charge q by

(15.2)

The Lorenz condition written in terms of A and ϕ is

(15.3)

15.2 Electric and Magnetic Fields and Forces

Electric and magnetic fields manifest themselves observationally by the forces that they cause. These vector quantities are related to the scalar and vector potentials as follows:

(15.4)

(15.5)

Note that arbitrary scalar and vector constants may be added respectively to ϕ and A without changing either the electric or magnetic fields, since the latter are functions only of space and time derivatives of the former. This is a simple example of the concept of gauge invariance in action. We will see later that not just a constant, but any time-independent vector function A′(x,y,z) may be added to A with similar null results, as long as ∂A_x′∕∂y = ∂A_y′∕∂x, etc. Gauge invariance is an important part of gauge theory, but a full understanding depends on more sophisticated mathematics than currently at our disposal.

By comparison of equations (15.4) and (15.5) with the general expression for force in gauge theory, we find that the electromagnetic force on a particle with charge q is

15.3 Charged Particle Motion

We now explore some examples of the motion of charged particles under the influence of electric and magnetic fields.

15.3.1 Particle in Constant Electric Field

Suppose a particle with charge q is exposed to a constant electric field E_x in the x direction. The x component of the force on the particle is thus F_x = qE_x. From Newton’s second law the acceleration in the x direction is therefore a_x = F_x∕m = qE_x∕m where m is the mass of the particle. The behavior of the particle is the same as if it were exposed to a constant gravitational field equal to qE_x∕m.

15.3.2 Particle in Conservative Electric Field

If ∂A∕∂t = 0, then the electric force on a charged particle is

(15.7)

This force is conservative, with potential energy U = qϕ. Recalling that the total energy, E = K + U, of a particle under the influence of a conservative force remains constant with time, we can infer that the change in the kinetic energy with position of the particle is just minus the change in the potential energy: ΔK = -ΔU. Notice in particular that if the particle returns to its initial position, the change in the potential energy is zero and the kinetic energy recovers its initial value.

If ∂A∕∂t≠0, then there is the possibility that the electric force is not conservative. Recall that the magnetic field is derived from A. Interestingly, a necessary and sufficient criterion for a non-conservative electric force is that the magnetic field be changing with time. This result was first inferred experimentally by the English physicist Michael Faraday in 1831 and at nearly the same time by the American physicist Joseph Henry. It will be further explored later in this chapter.

15.3.3 Torque on an Electric Dipole

Let us now imagine a “dumbbell” consisting of positive and negative charges of equal magnitude q separated by a distance d, as shown in figure 15.1. If there is a uniform electric field E, the positive charge experiences a force qE, while the negative charge experiences a force -qE. The net force on the dumbbell is thus zero.

The torque acting on the dumbbell is not zero. The total torque acting about the origin in figure 15.1 is the sum of the torques acting on the two charges:

(15.8)

The vector d can be thought of as having a length equal to the distance between the two charges and a direction going from the negative to the positive charge.

The quantity p = qd is called the electric dipole moment. (Don’t confuse it with the momentum!) The torque is just

(15.9)

This shows that the torque depends on the dipole moment, or the product of the charge and the separation. Thus, halving the separation and doubling the charge results in the same dipole moment.

The tendency of the torque is to rotate the dipole so that the dipole moment p is parallel to the electric field E. The magnitude of the torque is given by

(15.10)

where the angle θ is defined in figure 15.1 and p = |p| is the magnitude of the electric dipole moment.

The potential energy of the dipole is computed as follows: The scalar potential associated with the electric field is ϕ = -Ez where E is the magnitude of the field, assumed to point in the +z direction. Thus, the potential energy of a single particle with charge q is U = qϕ = -qEz. The total potential energy of the dipole is the sum of the potential energies of the individual charges:

The tendency of the electric field to align the dipole moment with itself is confirmed by the potential energy formula. The potential energy is lowest when the dipole moment is aligned with the field and highest when the two are anti-aligned.

15.3.4 Particle in Constant Magnetic Field

The magnetic force on a particle with charge q moving with velocity v is F_magnetic = qv × B, where B is the magnetic field. The magnetic force is directed perpendicular to both the magnetic field and the particle’s velocity. Because of the latter point, no work is done on the particle by the magnetic field. Thus, by itself the magnetic force cannot change the magnitude of the particle’s velocity, though it can change its direction.

If the magnetic field is constant, the magnitude of the magnetic force on the particle is also constant and has the value F_magnetic = qvB sin(θ) where v = |v|, B = |B|, and θ is the angle between v and B. If the initial velocity is perpendicular to the magnetic field, then sin(θ) = 1 and the force is just F_magnetic = qvB. The particle simply moves in a circle with the magnetic force directed toward the center of the circle. This force divided by the mass m must equal the particle’s centripetal acceleration: v²∕R = a = F _magnetic∕m = qvB∕m in the non-relativistic case, where R is the radius of the circle. Solving for R yields

(15.12)

The angular frequency of revolution is

(15.13)

Notice that this frequency is a constant independent of the radius of the particle’s orbit or its velocity. This is called the cyclotron frequency.

If the initial velocity is not perpendicular to the magnetic field, then the particle still has a circular component of motion in the plane normal to the field, but also drifts at constant speed in the direction of the field. The net result is a spiral motion in the direction of the magnetic field, as illustrated in figure 15.2. The radius of the circle is R = mv_p∕(qB) in this case, where v_p is the component of v perpendicular to the magnetic field.

15.3.5 Crossed Electric and Magnetic Fields

If we have perpendicular electric and magnetic fields as shown in figure 15.3, then it is possible for a charged particle to move such that the electric and magnetic forces simply cancel each other out. From the Lorentz force equation (15.6), the condition for this happening is E + v × B = 0. If E and B are perpendicular, then this equation requires v to point in the direction of E × B (i. e., normal to both vectors) with the magnitude v = |E|∕|B|. This, of course, is not the only possible motion under these circumstances, just the simplest.

It is interesting to consider this situation from the point of view of a reference frame that is moving with the charged particle. In this reference frame the particle is stationary and therefore not subject to the magnetic force. Since the particle is not accelerating, the net force, which in this frame consists only of the electric force, is zero. Hence, the electric field must be zero in the moving reference frame.

This argument shows that the electric field perceived in one reference frame is not necessarily the same as the electric field perceived in another frame. Figure 15.4 shows why this is so. The left panel shows the situation in the reference frame moving to the right, which is the unprimed frame in this picture. The charged particle is stationary in this reference frame. The four-potential is purely spacelike, having no time component ϕ∕c. Assuming that a is constant in time, there is no electric field, and hence no electric force. Since the particle is stationary in this frame, there is also no magnetic force. However, in the primed reference frame, which is moving to the left relative to the unprimed frame and therefore is equivalent to the original reference frame in which the particle is moving to the right, the four-potential has a time component, which means that a scalar potential and hence an electric field is present.

15.4 Forces on Currents in Conductors

So far we have talked mainly about point charges moving in free space. However, many practical applications of electromagnetism have charges moving through a conductor such as copper. A conductor is a material in which electrically charged particles can freely move. An insulator is a material in which charged particles are fixed in place. Practical conductors are often surrounded by insulators in order to confine the motion of charge to particular paths.

The current through a wire is defined as the amount of charge passing through the wire per unit time. When defining current, one needs to decide which direction constitutes a positive current for the problem at hand, i. e., the direction in which the positive charge is moving. If the current consists of particles carrying negative charge, then the direction of the current is opposite the direction of the motion of the particles.

Metals tend to be good conductors, while glass, plastic, and other non-metallic materials are usually insulators. All materials contain both positive and negative charges. In metals, negatively charged electrons can escape from atoms and are free to move about the material. When atoms lose one or more electrons, they become positively charged. Atoms tend to be fixed in place. Since the electron charge is negative, the current in a wire actually has a direction opposite the direction of motion of the electrons, as noted above.

If a conductor is in the form of a wire, we can compute the magnetic force on the wire if we know the number of mobile particles per unit length of wire N, the charge on each particle q, and the speed v with which they are moving down the wire. The total force on a length of wire L is F = qNLvn × B, where n is a unit vector pointing in the direction of motion of the particles through the wire. The quantity i ≡ qNv is called the current in the wire. It equals the amount of charge per unit time flowing down the wire. Written in terms of the current, the force on a length L of the wire is

(15.14)

15.5 Torque on a Magnetic Dipole and Electric Motors

Figure 15.6 shows a rectangular loop of wire mounted on an axle in a magnetic field. A current i exists in the loop as shown. The currents in loop segments 2 and 4 experience a force parallel to the axle. These forces generate no net torque. However, the magnetic forces on loop segments 1 and 3 are each F = idB in magnitude, where B = |B| is the magnitude of the magnetic field. Together these forces generate a counterclockwise torque about the axle equal to τ = 2F(w∕2) sin(θ) = iwdB sin(θ). This can be represented in vector form as

(15.15)

where m is a vector with magnitude iwd and direction normal to the loop as shown in figure 15.6. The vector m is called the magnetic dipole moment.

The loop can actually be any shape, not just rectangular. In the general case the magnitude of the magnetic moment equals the current i times the area S of the loop:

(15.16)

In the above example the area is S = wd. The direction of m is determined by the right hand rule; curl the fingers on your right hand around the loop in the direction of the current and your thumb points in the direction of m.

In analogy with the electric dipole in an electric field, the potential energy of a magnetic dipole in a magnetic field is

(15.17)

Figure 15.6 illustrates the principle of an electric motor. A motor consists of multiple loops of wire on an axle carrying a current in a magnetic field. The torque on the axle turns the loops so that the magnetic moment is parallel to the field. The angular momentum of the loops carries the rotation of the axle through the zero torque region, which occurs when the magnetic moment is either perfectly parallel or perfectly anti-parallel (i. e., pointing in the opposite direction) to the field. At this point either the magnetic field is reversed by some mechanism or the magnetic dipole is reversed by making the current circulate around the loops in the opposite direction. The torque due to the magnetic force then turns the axle through another half-turn, whereupon the field or the magnetic moment is again reversed, and so on.

15.6 Electric Generators and Faraday’s Law

As was shown earlier, the electric field is derived from two different sources, spatial derivatives of the scalar potential and time derivatives of the vector potential:

(15.18)

In time independent situations the vector potential part drops out and we are left with a dependence only on the scalar potential. In this case a particle with charge q has an electrostatic potential energy U = qϕ, which means that the electric force is conservative. However, in the time dependent situation there is no guarantee that the part of the electric field derived from the vector potential will be conservative.

An example of a non-conservative electric field occurs when we have

(15.19)

where C is a constant. In this case the electric and magnetic fields are

(15.20)

The magnetic field points in the -z direction and increases in magnitude with time. The electric field vectors are shown in figure 15.7. Notice that a positively charged particle moving in a counterclockwise circle as shown is continually being accelerated in the direction of motion, and is therefore continually gaining energy. This is impossible with a conservative force.

How much energy is gained by a particle with charge q moving in a complete circle of radius R under the above circumstances? The magnitude of the electric field at this radius is E = CR, so the force on the particle is F = qCR. The circumference of the circle is 2πR, so the total work done by the electric field in one revolution is just ΔW = 2πRF = 2πqCR² = 2qCS, where S = πR² is the area of the circle. Let us define ΔV = ΔW∕q = 2CS. For historical reasons this is called the electromotive force or EMF. This is deceptive terminology, because in fact ΔV doesn’t have the dimensions of force — it is really just the work per unit charge done on a particle making a single loop around the circle in figure 15.7.

Recall that the z component of the magnetic field in this case is B_z = -2Ct. Note that the time derivative of the magnetic field is just ∂B_z∕∂t = -2C. Comparison with the equation for electromotive force shows us that

(15.21)

where the area is brought inside the time derivative since it is constant in time.

Notice that the argument of the time derivative in the above equation is the component of B perpendicular to the plane of the loop. The loop area multiplied by the normal component of B is the magnetic flux through the loop: Φ_B ≡ B_normalS. The generalization of equation (15.21) for any loop fixed in space is expressed as

(15.22)

It is valid for arbitrary loop and magnetic field configurations, not just for the simple case we have been investigating.

The minus sign in equation (15.22) means the following: If the fingers on your right hand curl around the loop in the direction opposite to the direction that causes a positive charge to gain energy, then your thumb points in the direction of the time rate of change of the magnetic flux passing through the loop. This is illustrated in figure 15.8.

If a loop is changing size or shape with time, then the magnetic force also acts on charges in the loop. The work per unit charge done in a segment j of the loop defined by the displacement vector Δl_j is

(15.23)

where v_j is the velocity with which the loop segment is moving and B_j is the magnetic field at the location of the segment. To get the total EMF around the loop due to the magnetic force, the ΔV _j values from each segment must simply be summed up and added to the right side of equation (15.22):

(15.24)

It turns out that the part of ΔV resulting from a moving or deforming loop equals minus the time rate of change of magnetic flux through the loop due solely to loop movement and deformation. Therefore, we can rewrite equation (15.24) in the more compact form

(15.25)

where the time derivative of the magnetic flux now includes both changes in the magnetic field and changes in the position, shape, and orientation of the loop. This is called Faraday’s law, even though it is a hybrid of equation (15.22), which is Faraday’s law as generally defined in advanced physics and the Lorentz force law as expressed in equation (15.23).

The electric generator is perhaps the best known application of Faraday’s law. Figure 15.9 shows a rectangular loop of wire fixed to an axle that rotates at an angular rate Ω. The magnetic flux through the loop thus varies with time according to Φ_B = wdB cos(θ) = wdB cos(Ωt). The EMF around the loop is thus

(15.26)

In a real generator there are many loops forming a coil of wire and the ends of the coil are brought out through the axle so that the resulting current can be tapped for practical use.

15.7 EMF and Scalar Potential

The EMF ΔV has the same units as the scalar potential ϕ. What is the difference between the two quantities? Both are related to the work done per unit charge by the electric field on a particle moving through the field. However, recall that the electric field is composed of two parts:

(15.27)

Δϕ = ϕ₂ - ϕ₁ is minus the work done on the particle in going from point 1 to point 2 by the part of the electric field associated with the scalar potential; moving to a lower potential results in a release of kinetic energy according to the conservation of energy. On the other hand, ΔV is (plus) the work done per unit charge on a particle by the part of the electric field associated with the time derivative of the vector potential.

Aside from the different sign conventions, there is one other fundamental difference between the two quantities: Δϕ is always zero for closed paths, i. e., paths in which the particle returns to its initial point. This is because point 1 is then the same as point 2, so ϕ₁ = ϕ₂. This condition doesn’t necessarily apply to the EMF. ΔV often is non-zero for closed particle paths. The electric generator that we have just discussed is an important case in point. The total work done per unit charge by the electric field on a charged particle moving along some path is thus ΔV - Δϕ. The Δϕ term drops out if the path is closed.

15.8 Problems

1. Given a four-potential a = (Cyt,-Cxt, 0, 0) where C is a constant:
   1. Determine whether this four-potential satisfies the Lorenz condition.
   2. Compute the electric and magnetic fields from this four-potential.
2. Given a₁ = (C₁zt, 0, 0,C₂x), find the electric and magnetic field components. Compare with the fields you get from a₂ = (C₁zt,C₃y,-C₃z,C₂x). C₁, C₂, and C₃ are constants. Can one have more than one four-potential field giving rise to the same electric and magnetic fields?
3. Suppose that in the rest frame we have a four-potential of the form a = (0, 0, 0,Ky) where K is a constant.
   1. Find the electric and magnetic fields in this frame.
   2. Find the components of a in a reference frame moving in the -x direction at speed U. Hint: Draw a spacetime diagram showing the a vector and resolve into components in the moving frame using the spacetime Pythagorean theorem.
   3. Find the electric and magnetic fields in the moving frame.
4. Assume a four-potential of the form a = (A,ϕ∕c), where A = (Ky, 0, 0) and ϕ = 0 in the rest frame, K being a constant.
   1. Compute the electric and magnetic fields in the rest frame.
   2. Find the components of the four-potential in a reference frame moving in the -x direction at speed U.
   3. Compute the electric and magnetic fields in the moving frame using the above results.
5. Using the right-hand rule, show that the electric torque acting on an electric dipole tries to align the dipole so that it is in its state of lowest potential energy.
6. The net electric force on an electric dipole is zero in a uniform electric field. However, if the field varies with position, this is not necessarily true. Consider an electric field that has the form E = E₀(1 + αz)k along the z axis, where E₀ and α are positive constants.
   1. An electric dipole consisting of charges ±q spaced by a distance d is centered at the origin. If the dipole is aligned with the electric field, determine the direction and magnitude of the net force on the dipole.
   2. Determine the force on the dipole if it is anti-aligned with (i. e., pointing in the opposite direction from) the electric field.
7. Suppose that a charged particle is moving under the influence of electric and magnetic fields such that it periodically returns to some point P. If the four-potential is independent of time, will the kinetic energy of the particle be the same or different every time it returns to P? Explain.
8. Given constant electric and magnetic fields E = Ej and B = Bk:
   1. Find the velocity (magnitude and direction) of a charged particle for which the Lorentz force is zero.
   2. Using this result, describe how you would build a setup to select out only those particles in a beam moving at a certain velocity.
9. Determine qualitatively how a charged particle moves in crossed electric and magnetic fields in the general case in which it is not moving at constant velocity. For the sake of definiteness, assume that the magnetic field points in the +z direction and the electric field in the +x direction. Hint: Is there a reference frame in which the electric field vanishes? If there is, describe the motion in this reference frame and then determine how this motion looks in the original reference frame.
10. A horizontal wire of mass per unit length 0.1 kg m^-1 passes through a horizontal magnetic field of strength B = 0.1 T with an orientation of 45^∘ to the field as shown in figure 15.10. What current must the wire carry for the magnetic force on the wire to just balance gravity?

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    Figure 15.10: Horizontal wire with current i in a magnetic field.

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    Figure 15.11: Magnetic dipole (current loop) in an inhomogeneous magnetic field.

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11. Figure 15.11 shows a current loop in a magnetic field. The magnetic field diverges with increasing z, so that its magnitude decreases with height.
    1. Which way does the magnetic dipole vector due to the current loop point?
    2. Is this dipole oriented so as to have maximum or minimum potential energy, or is it somewhere in between?
    3. Is there a net force on the dipole? If so, what direction does it point? Hint: Determine the direction of the v × B force at each point on the current loop. What direction does the sum of all these forces point?

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    Figure 15.12: A moving crossbar on a U-shaped wire in a magnetic field.

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12. A charged particle moving in a circle in a magnetic field constitutes a circular current which forms a magnetic dipole.
    1. Determine whether the dipole moment produced by this current is aligned or anti-aligned with the initial magnetic field.
    2. Do charged particles moving in a non-uniform magnetic field as shown in figure 15.11 tend to accelerate toward regions of stronger or weaker field?
13. Why do electric motors have many turns of wire around the loop that cuts the magnetic field, instead of just one? Hint: Magnetic fields in normal motors are of order 0.1 T and currents are typically a few amps. Estimate the torque on a reasonably sized current loop for these conditions. Compare this to the torque you could expect to exert with your hand acting on a 1 m moment arm.
14. Imagine a stationary U-shaped conductor with a moving conducting bar in contact with the U as shown in figure 15.12. A uniform magnetic field exists normal to the plane of the U and has magnitude B. The bar is moving outward along the U at speed v as shown.
    1. Using the fact that the charged particles in the moving bar are subject to a Lorentz force due to the motion of the bar through a magnetic field, compute the EMF around the closed loop consisting of the bar and the U. Hint: Recall that the EMF is the work done per unit charge on a charged particle moving around the loop.
    2. Compute the EMF around the above loop using Faraday’s law. Is the answer the same as obtained above?

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    Figure 15.13: The charged bead continuously accelerates around the loop due to electromagnetic fields.

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15. A bead on a loop has a positive charge q and accelerates continuously around the loop in the counterclockwise direction, as shown in figure 15.13. Explain qualitatively what this information tells you about
    1. the vector potential in the vicinity of the loop, and
    2. the magnetic flux through the loop.

Chapter 16
Generation of Electromagnetic Fields

In this chapter we investigate how charge produces electric and magnetic fields. We first introduce Coulomb’s law, which is the basis for everything else in the section. We then discuss Gauss’s law for the electric and magnetic field, drawing on what we learned while using it on the gravitational field. Coulomb’s law and the theory of relativity together show that magnetic fields are generated by moving charge. We then use this fact to compute the magnetic fields from some simple charge distributions. We finish with a discussion of electromagnetic waves.

16.1 Coulomb’s Law and the Electric Field

A stationary point electric charge q is known to produce a scalar potential

(16.1)

a distance r from the charge. The constant ϵ₀ = 8.85 × 10^-12 C² N^-1 m^-2 is called the permittivity of free space. The vector potential produced by a stationary charge is zero.

The potential energy between two stationary charges is equal to the scalar potential produced by one charge multiplied by the value of the other charge:

(16.2)

Notice that it doesn’t make any difference whether one multiplies the scalar potential from charge 1 by charge 2 or vice versa – the result is the same.

Since r = (x² + y² + z²)^1∕2, the electric field produced by a charge is

(16.3)

where r = (x,y,z) is the vector from the charge to the point where the electric field is being measured. The magnetic field is zero since the vector potential is zero.

The force between two stationary charges separated by a distance r is the value of one charge multiplied by the electric field produced by the other charge. Thus the magnitude of the force is

(16.4)

with the force being repulsive if the charges are of the same sign, and attractive if the signs are opposite. This is called Coulomb’s law.

Equation (16.4) is the electric equivalent of Newton’s universal law of gravitation. Replacing mass by charge and G by -1∕(4πϵ₀) in the equation for the gravitational force between two point masses gives us equation (16.4). The most important aspect of this result is that both the gravitational and electrostatic forces decrease as the square of the distance between the particles.

16.2 Gauss’s Law for Electricity

The electric flux is defined in analogy to the gravitational flux as

(16.5)

where S is the directed area through which the flux passes. (This is strictly true only for small, flat areas S over which the component of E normal to S can be assumed constant.) Since the electric field obeys an inverse square law, Gauss’s law applies to the electric flux Φ_E just as it applies to the gravitational flux. In particular, since the magnitude of the outward electric field a distance r from a charge q is E = q∕(4πϵ₀r²), the electric flux through a sphere of radius r (and area 4πr²) concentric with the charge is ES = [q∕(4πϵ ₀r²)] × (4πr²) = q∕ϵ ₀. This generalizes to an arbitrary distribution of charge as in the gravitational case:

(16.6)

where Φ_E in this equation is the outward electric flux through a closed surface and q_inside is the net charge inside this surface. This is an expression of Gauss’s law for the electric field. Since Gauss’s law for electricity and for gravitation are so similar, we can use all our insights from studying gravity on the electric field case.

16.2.1 Sheet of Charge

Figure 16.1 shows how to set up the Gaussian surface to obtain the electric field emanating from an infinite sheet of charge. We assume a charge density of σ Coulombs per square meter, which means that the amount of charge inside the box is q_inside = σhd, where the box has height h and depth d into the page. The total electric flux out of the left and right faces of the box is Φ_E = 2Ehd, where E is the magnitude of the electric field on these surfaces. The field is assumed to point away from the charge, and hence out of the box on both faces. Due to the assumed direction of the electric field, there is no electric flux out of any of the other faces of the box.

Applying Gauss’s law, we infer that 2Ehd = σhd∕ϵ₀, which means that the electric field emanating from a sheet of charge with charge density per unit area σ is

(16.7)

The scalar potential associated with this electric field is easily obtained by realizing that equation (16.7) gives the x component of this field — the other components are zero. Using E = -∂ϕ∕∂x, we infer that

(16.8)

The absolute value signs around x take account of the fact that the direction of the electric field for negative x is opposite that for positive x.

16.2.2 Line of Charge

Similar reasoning is used to obtain the electric field due to a line of charge. A sketch of the expected electric field vectors and a Gaussian cylinder coaxial with the line of charge is shown in figure 16.2. If the charge per unit length is λ, the amount of charge inside the cylinder is q_inside = λd, where d is the length of the cylinder. The outward electric flux at radius r is Φ_E = 2πrdE. Gauss’s law therefore tells us that the electric field at radius r is just

(16.9)

In this case E = -∂ϕ∕∂r, so that the scalar potential is

(16.10)

16.3 Gauss’s Law for Magnetism

By analogy with Gauss’s law for the electric field, we could write a Gauss’s law for the magnetic field as follows:

(16.11)

where Φ_B is the outward magnetic flux through a closed surface, C is a constant, and q_{magnetic inside} is the “magnetic charge” inside the closed surface. Extensive searches have been made for magnetic charge, generally called a magnetic monopole. However, none has ever been found. Thus, Gauss’s law for magnetism can be written

(16.12)

This of course doesn’t preclude non-zero values of the magnetic flux through open surfaces, as illustrated in figure 16.3.

16.4 Coulomb’s Law and Relativity

The equation (16.1) for the scalar potential of a point charge is valid only in the reference frame in which the charge q is stationary. By symmetry, the vector potential must be zero. Since ϕ is actually the timelike component of the four-potential, we infer that the four-potential due to a charge is tangent to the world line of the charged particle.

A consequence of the above argument is that a moving charge produces a magnetic field, since the four-potential must have spacelike components in this case.

16.5 Moving Charge and Magnetic Fields

We have shown that electric charge generates both electric and magnetic fields, but the latter result only from moving charge. If we have the scalar potential due to a static configuration of charge, we can use this result to find the magnetic field if this charge is set in motion. Since the four-potential is tangent to the particle’s world line, and hence is parallel to the time axis in the reference frame in which the charged particle is stationary, we know how to resolve the space and time components of the four-potential in the reference frame in which the charge is moving.

Figure 16.4 illustrates this process. For a particle moving in the +x direction at speed v, the slope of the time axis in the primed frame is just c∕v. The four-potential vector has this same slope, which means that the space and time components of the four-potential must now appear as shown in figure 16.4. If the scalar potential in the primed frame is ϕ′, then in the unprimed frame it is ϕ, and the x component of the vector potential is A_x. Using the spacetime Pythagorean theorem, ϕ′²∕c² = ϕ²∕c² - A _x², and relating slope of the ct′ axis to the components of the four-potential, c∕v = (ϕ∕c)∕A_x, it is possible to show that

(16.13)

where

(16.14)

Thus, the principles of special relativity allow us to obtain the full four-potential for a moving configuration of charge if the scalar potential is known for the charge when it is stationary. From this we can derive the electric and magnetic fields for the moving charge.

16.5.1 Moving Line of Charge

As an example of this procedure, let us see if we can determine the magnetic field from a line of charge with linear charge density in its own rest frame of λ′, aligned along the z axis. The line of charge is moving in a direction parallel to itself. From equation (16.10) we see that the scalar potential a distance r from the z axis is

(16.15)

in a reference frame moving with the charge. The z component of the vector potential in the stationary frame is therefore

(16.16)

by equation (16.13), with all other components being zero. This is illustrated in figure 16.5.

We infer that

(16.17)

where we have used r² = x² + y². The resulting field is illustrated in figure 16.6. The field lines circle around the line of moving charge and the magnitude of the magnetic field is

(16.18)

There is an interesting relativistic effect on the charge density λ′, which is defined in the co-moving or primed reference frame. In the unprimed frame the charges are moving at speed v and therefore undergo a Lorentz contraction in the z direction. This decreases the charge spacing by a factor of γ and therefore increases the charge density as perceived in the unprimed frame to a value λ = γλ′.

We also define a new constant μ₀ ≡ 1∕(ϵ₀c²). This is called the permeability of free space. This constant has the assigned value μ₀ = 4π × 10^-7 N s² C^-2. The value of ϵ₀ = 1∕(μ₀c²) is actually derived from this assigned value and the measured value of the speed of light. The reasons for this particular way of dealing with the constants of electromagetism are obscure, but have to do with making it easy to relate the values of constants to the experiments used in determining them.

With the above substitutions, the magnetic field equation becomes

(16.19)

The combination λv is called the current and is symbolized by i. The current is the charge per unit time passing a point and is a fundamental quantity in electric circuits. The magnetic field written in terms of the current flowing along the z axis is

(16.20)

16.5.2 Moving Sheet of Charge

As another example we consider a uniform infinite sheet of charge in the x - y plane with charge density σ′. The charge is moving in the +x direction with speed v. As we showed in the section on Gauss’s law for electricity, the electric field for this sheet of charge in the co-moving reference frame is in the z direction and has the value

(16.21)

where we define

(16.22)

The sgn(z) function is used to indicate that the electric field points upward above the sheet of charge and downward below it (see figure 16.7).

The scalar potential in this frame is

(16.23)

In the stationary reference frame in which the sheet of charge is moving in the x direction, the scalar potential and the x component of the vector potential are

(16.24)

according to equation (16.13), where σ = γσ′ is the charge density in the stationary frame. The other components of the vector potential are zero. We calculate the magnetic field as

(16.25)

where sgn(z) is defined as before. The vector potential and the magnetic field are shown in figure 16.7. Note that the magnetic field points normal to the direction of motion of the charge but parallel to the sheet. It points in opposite directions on opposite sides of the sheet of charge.

16.6 Electromagnetic Radiation

We have found so far that stationary charge produces an electric field while moving charge produces a magnetic field. It turns out that accelerated charge produces electromagnetic radiation. Electromagnetic radiation is nothing more than one or more photons that have zero mass, and are therefore real, not virtual.

Acceleration of a charged particle is needed to produce radiation because of the conservation of energy and momentum. The left panel of figure 16.8 shows why. Since a photon carries off energy and momentum, conservation means that the energy and momentum of the emitting particle change due to the emission of a photon. This corresponds in classical mechanics to an acceleration.

The process in the left panel of figure 16.8 actually cannot occur if particles A and B have the same mass. If the mass of the outgoing particle B is less than the mass of the incoming particle A, then this reaction can and does occur. An example is the decay of an atom from a higher energy state to a lower energy state (and hence lower mass), accompanied by the emission of a photon.

Another type of reaction that can generate radiation occurs when two charged particles (say, electrons) collide, as illustrated in the right panel of figure 16.8. In an elastic collision both electrons are real both before and after the photon transfer. However, it is possible for one of the electrons to have a virtual mass that is greater than the normal electron mass after the collision, which means that it is free to decay to a real electron plus a real photon.

We now try to understand the characteristics of free electromagnetic radiation. In our studies of waves we found it easiest to examine plane waves. We will follow this path here, writing the four-potential for an electromagnetic plane wave moving in the x direction as

(16.26)

where a₀ = (A₀,ϕ₀∕c) is a constant four-vector representing the direction and maximum amplitude of the four-potential, and k_x and ω are the wavenumber and the angular frequency of the wave. Since the real photon is massless, we have ω = k_xc in this case. Virtual photons are not subject to this constraint.

By substituting A and ϕ from equation (16.26) into the Lorenz condition, we find that

(16.27)

Thus, the Lorenz condition requires that the scalar potential ϕ be related to the x or longitudinal component of the vector potential, A_x, i. e., the component pointing in the direction of wave propagation. The transverse components, A_y and A_z, are unconstrained by the Lorenz condition, since they don’t depend on y and z.

Using equations for the electric and magnetic field, as well as equations (16.26) and (16.27), we can now find E and B in an electromagnetic plane wave:

(16.28)

(16.29)

The electric field has a longitudinal or x component proportional to k_xϕ₀ -ωA_0x = -ω(ϕ₀∕c-A_0x). However, comparison with equation (16.27) shows that E_x = 0 as long as ω∕k_x = c, i. e., as long as the photons travel at the speed of light, c. Thus, virtual photons, i. e., those that have a non-zero mass and therefore travel at a speed other than that of light, can have a non-zero longitudinal component of the electric field, but real photons cannot.

The dot product of the electric and magnetic fields in a plane wave is E ⋅ B = 0, as can be verified from equations (16.28) and (16.29). This means that E and B are perpendicular to each other. Furthermore, both E and B are perpendicular to the direction of wave motion for real photons.

Figure 16.9 shows the electric and magnetic fields for real photons in the special case where A_z = 0. The electric field points in the same direction as the transverse part of the vector potential, while the magnetic field points in the other transverse direction. The ratio of the magnitudes of the electric and magnetic fields is easily inferred from equations (16.28) and (16.29):

(16.30)

Notice that the electric and magnetic fields for a wave do not depend on the longitudinal component of the vector potential, A_x. This is because the Lorenz condition forces A_x to cancel with the term containing ϕ in the expression for E_x.

16.7 The Lorenz Condition

We are now in a position to see what the Lorenz condition means. For an isolated stationary charge, the scalar potential is given by equation (16.1) and the vector potential A is zero. The Lorenz condition reduces to

(16.31)

From this we see that the Lorenz condition applied to the four-potential for a point charge is equivalent to the statement that the charge on a point particle is conserved, i. e., it doesn’t change with time. This is extended to any stationary distribution of charge by the superposition principle.

We thus see that the Lorenz condition is closely related to charge conservation for the four-potential of any charge distribution in the reference frame in which the charge is stationary. If we can further show that the Lorenz condition is an equation that is equally valid in all reference frames, then we will have demonstrated that it is true for the four-potential produced by moving charged particles as well.

If the Lorenz condition is valid in one reference frame, it is valid in all frames for the special case of a plane electromagnetic wave. This follows from substituting the four-potential for a plane wave into the Lorenz condition, as was done in equation (16.27) in the previous section. In this case the Lorenz condition reduces to k ⋅ a = 0. Since the dot product of two four-vectors is a relativistic scalar, the Lorenz condition is equally valid in all frames.

16.8 Problems

1. Imagine that an electron actually consists of two point charges, each with charge e∕2, separated by a distance D, where e is the charge on the electron. Compute D such that the potential energy of the two charges equals the rest energy of the electron. Look up the constants and compute a numerical value for D. Finally, compute the force between the two charges and compare to the gravitational force between two masses each equal to half the electron mass separated by this distance.
2. Verify that the equations for the scalar potentials associated with a sheet and a line of charge, (16.8) and (16.10), yield the corresponding electric fields.
3. Two sheets of charge, one with charge density σ, the other with -σ, are aligned as shown in figure 16.10. Compute the electric field in each of the regions A, B, and C.

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   Figure 16.10: Two parallel sheets of charge, one with surface charge density σ, the other with -σ.

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4. Positive charge is distributed uniformly on the upper surface of an infinite conducting plate with charge per unit area σ as shown in figure 16.11. Use Gauss’s law to compute the electric field above the plate. Hint: Is there any electric field inside the plate?

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   Figure 16.11: A charged metal plate.

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5. Suppose a student proposes that a magnetic field can take the form shown in figure 16.12. Is the proposed form of the magnetic field consistent with Gauss’s law for magnetism? Explain.

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   Figure 16.12: Hypothesized magnetic field. Does it satisfy Gauss’s law for magnetism?

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6. The magnetic flux through the sides of the cone illustrated in figure 16.13 is zero. The magnetic field may be assumed to be approximately normal to the ends of the cone and the magnetic flux into the left end is Φ_B. The areas of the left and right ends of the cone are S_a and S_b.

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   Figure 16.13: Converging magnetic field passing through a closed surface.

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   1. What is the magnetic flux out of the right end of the cone?
   2. What is the value of the magnetic field B on the left end of the cone?
   3. What is the value of B on the right end?
7. In the lab frame a wire has negative charge with linear charge density -λ moving at speed -U corresponding to a current i = λU as shown in figure 16.14. Positive charge is stationary, and has charge density λ, so the net charge is zero.¹

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   Figure 16.14: A horizontal wire with current i viewed in two different reference frames.

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   1. What are the electric and magnetic fields produced by the charge in the wire in the stationary frame?
   2. In a reference frame moving at velocity -U in the x direction, such that the negative charge is stationary, what is the apparent linear charge density of (1) the negative charge, and (2) the positive charge? Hint: The Lorentz contraction must be taken into account here.
   3. What is the electric field produced by the charge in the wire in the moving frame? Hint: Do the charge densities from the positive and negative charge cancel in this frame?
   4. What is the current in the wire in the moving frame, and hence, what is the magnetic field around the wire in this frame? Hint: Is the positive or negative charge causing the current in this frame?
   5. Explain why the net force on a separate charged particle some distance from the wire and stationary in the lab frame is zero in both reference frames.
8. The left panel of figure 16.8 shows a real charged particle A emitting a real photon, turning into a possibly different real particle B after the emission. If particle A and particle B have the same mass, show that this process is energetically impossible. Hint: Work in a reference frame in which particle A is stationary.
9. Given the four-potential for an electromagnetic plane wave, show why the longitudinal component of the magnetic field is zero.
10. Referring to figure 16.9, show that the vector E × B points in the direction of propagation of a plane electromagnetic wave.
11. Referring to figure 16.9, what direction and speed must a charged particle move in the presence of a free electromagnetic wave such that the net electromagnetic force on the particle is zero?

Chapter 17
Capacitors, Inductors, and Resistors

Various electronic devices are considered in this chapter. This is useful not only for understanding these devices but also for revealing new aspects of electromagnetism. The capacitor is first discussed and Ampère’s law is introduced. The theory of magnetic inductance is then developed. Ohm’s law and the resistor are discussed. The energy associated with electric and magnetic fields is calculated and Kirchhoff’s laws for electric circuits are briefly discussed.

17.1 The Capacitor and Ampère’s Law

We first discuss a device that is commonly used in electronics, called the capacitor. We then introduce a new mathematical idea called the circulation of a vector field around a loop. Finally, we use this idea to investigate Ampère’s law.

17.1.1 The Capacitor

The capacitor is an electronic device for storing charge. The simplest type is the parallel plate capacitor, illustrated in figure 17.1. This consists of two conducting plates of area S separated by distance d, with the plate separation being much smaller than the plate dimensions. Positive charge q resides on one plate, while negative charge -q resides on the other.

The electric field between the plates is E = σ∕ϵ₀, where the charge per unit area on the inside of the left plate in figure 17.1 is σ = q∕S. The density on the right plate is just -σ. All charge is assumed to reside on the inside surfaces and thus contributes to the electric field crossing the gap between the plates.

The above formula for the electric field comes from applying Gauss’s law to the sheet of charge on the positive plate. The factor of 1∕2 present in the equation for an isolated sheet of charge is absent here because all of the electric flux exits the Gaussian surface on the right side — the left side of the Gaussian box is inside the conductor where the electric field is zero, at least in a static situation.

There is no vector potential in this case, so the electric field is related solely to the scalar potential ϕ. Integrating E_x = -∂ϕ∕∂x across the gap between the conducting plates, we find that the potential difference between the plates is Δϕ = E_xd = qd∕(ϵ₀S), since E_x is known to be constant in this case. This equation indicates that the potential difference Δϕ is proportional to the charge q on the left plate of the capacitor in figure 17.1. The constant of proportionality is d∕(ϵ₀S), and the inverse of this constant is called the capacitance:

(17.1)

The relationship between potential difference, charge, and capacitance is thus

(17.2)

The equation for the capacitance of the illustrated parallel plates contains just a fundamental constant (ϵ₀) and geometrical factors (area of plates, spacing between them), and represents the amount of charge the parallel plate capacitor can store per unit potential difference between the plates. A word about signs: The higher potential is always on the plate of the capacitor that has the positive charge.

Note that equation (17.1) is valid only for a parallel plate capacitor. Capacitors come in many different geometries and the formula for the capacitance of a capacitor with a different geometry will differ from this equation. However, equation (17.2) is valid for any capacitor.

We now show that a capacitor that is charging or discharging has a magnetic field between the plates. Figure 17.2 shows a parallel plate capacitor with a current i flowing into the left plate and out of the right plate. This current is necessarily accompanied by an electric field that is changing with time: E_x = q∕(ϵ₀S) = it∕(ϵ₀S). Such an electric field can be derived from a scalar potential that is a function of time: ϕ = -itx∕(ϵ₀S). However, the Lorenz condition

(17.3)

demands that some component of the vector potential A be non-zero under these circumstances, since ∂ϕ∕∂t is non-zero.

How much can we infer about the vector potential from the geometry of the capacitor and equation (17.3)? Substituting ϕ = -itx∕(ϵ₀S) into this equation results in

(17.4)

which suggests a number of different possibilities for A. For instance, A = (0,ixy∕(ϵ₀c²S), 0) and A = [0, 0,ixz∕(ϵ ₀c²S)] both satisfy equation (17.4). However, neither of these trial choices is satisfactory by itself, as they are not consistent with the cylindrical symmetry of the capacitor about the x axis.

A choice of vector potential that is consistent with the shape of the capacitor and satisfies the Lorenz condition is obtained by combining these two trial solutions:

(17.5)

This vector potential leads to the magnetic field

(17.6)

These fields are illustrated in the right-hand panel of figure 17.2.

17.1.2 Circulation of a Vector Field

We have already seen one example of the circulation¹ of a vector field, though we didn’t label it as such. In chapter 15 we computed the work done on a charge by the electric field as it moves around a closed loop in the context of the electric generator and Faraday’s law. The work done per unit charge, or the EMF, is an example of the circulation of a field, in this case the electric field, Γ_E. Faraday’s law can be restated as

(17.7)

In the simple case of a circular loop with the field directed along the loop, the circulation is just the magnitude of the field multiplied by the circumference of the loop, as illustrated in the left panel of figure 17.3. In more complicated cases in which the field points in a direction other than the direction of the loop, just the component in the direction of traversal around the loop enters the circulation. If this component varies as one progresses around the loop, the calculation must be broken into pieces. The total circulation is then obtained by adding up the contributions from segments of the loop in which the value of the field component parallel to the motion around the loop is constant. An example of this type is the calculation of the EMF around a square loop of wire in an electric generator. Another is illustrated in the right panel of figure 17.3.

17.1.3 Ampère’s Law

The magnetic circulation Γ_B around the periphery of the capacitor in the right panel of figure 17.2 is easily computed by taking the magnitude of B in equation (17.6). The magnitude of the magnetic field on the inside of the capacitor is just B = ir∕(2ϵ₀c²S), since r = (y² + z²)^1∕2 in figure 17.2. Thus, at the periphery of the capacitor, r = R, and B = iR∕(2ϵ₀c²S) there. The area of the capacitor plates is S = πR² and ϵ ₀c² = 1∕μ ₀, as we discussed previously. Thus, the magnetic field is B = μ₀i∕(2πR) at the periphery. If the periphery is traversed in the counter-clockwise direction, the magnetic circulation around the capacitor is Γ_B = 2πRB = μ₀i.

Let us now compute the magnetic circulation around a wire carrying a current. The magnetic field a distance r from a straight wire carrying a current i is B = μ₀i∕(2πr). The magnetic field points in the direction of a circle concentric with the wire. The magnetic circulation around the wire is thus Γ_B = 2πrB = μ₀i.

Notice that the magnetic circulation is found to be the same around the wire and around the periphery of the capacitor. Furthermore, this circulation depends only on the current in the wire and the constant μ₀.

One further item needs to be calculated, namely the electric flux across the gap between the capacitor plates. This is just the electric field E = σ∕ϵ₀ multiplied by the area S, or Φ_E = Sσ∕ϵ₀ = q∕ϵ₀. The current into the capacitor is the time rate of change on the capacitor, so i = dq∕dt = ϵ₀dΦ_E∕dt.

We are now in a position to understand Ampère’s law:

(17.8)

This states that the magnetic circulation around a loop equals the sum of two contributions, (1) μ₀ multiplied by the electric current through the loop and (2) μ₀ϵ₀ multiplied by the time rate of change of the electric flux through the loop. In the above example the first term dominates when the loop is around the wire, while the second term acts when the loop is around the gap between the capacitor plates.

Ampère actually formulated an incomplete version of the law named after him — he included only the first term containing the current. The Scottish physicist James Clerk Maxwell added the second term, based primarily on theoretical reasoning. Maxwell’s additional term solved a serious internal inconsistency in electromagnetic theory — in our terms, the Lorenz condition requires a magnetic field to exist if the scalar potential ϕ is time-dependent. This magnetic field is only predicted by Ampère’s law if Maxwell’s term is included. The quantity ϵ₀dΦ_E∕dt was called the displacement current by Maxwell since it has the dimensions of current and is numerically equal to the current entering the capacitor. However, it isn’t really a current — it is just an electric flux that changes with time!

Gauss’s law for electricity and magnetism, Faraday’s law, and Ampère’s law are collectively called Maxwell’s equations. Together they form the basis for electromagnetism as it developed historically. However, our formulation of electromagnetism in terms of the four-potential, the dispersion relation for free electromagnetic waves, the Lorenz condition, and Coulomb’s law, is precisely equivalent to Maxwell’s equations, and is much closer to the modern approach to electromagnetism.

17.2 Magnetic Induction and Inductors

Induction is the tendency of a current in a conductor to maintain itself in the face of changes in the potential difference driving the current. Figure 17.4 shows a parallel plate inductor in which a current i passes through the two plates in opposite directions.² The vector potential between plates of width w and spacing d is

(17.9)

as long as w ≫ d (see figure 17.4).

Let us try to understand how this vector potential is constructed from what we already know. The vector potential for a single current sheet in the x-y plane at z = 0 moving in the x direction was computed in the previous chapter as A_x = -vσ|z|∕(2ϵ₀c²), with A _y = A_z = 0. The quantity σ is the charge per unit area on the sheet and v is the velocity of the charge sheet in the x direction. We use the relationship 1∕(ϵ₀c²) = μ ₀ and also realize that if each plate has a width w, then the current in each plate is i = vσw, which means that we can rewrite A_x = -μ₀i|z|∕(2w) for a single plate.

To proceed further, we first need to understand that |z| in the above equation is only valid if the charge sheet is at z = 0. If the sheet is located a distance a from the origin, then we must replace |z| by |z - a|. We also need to call on the definition of absolute value to realize that |z - a| = z - a if z > a, and |z - a| = -z + a if z < a. Figure 17.5 shows how the profiles of A_x from each of the charge sheets add together to form a combined profile for the two sheets together.

The resulting magnetic field between the plates can be computed from the vector potential:

(17.10)

Above and below the plates the magnetic field is zero because the vector potential is constant.

Let us now ask what happens when the current through the inductor increases or decreases with time. Assuming initially that no scalar potential exists, the x component of the electric field in the device is

(17.11)

while E_y = E_z = 0. Substituting the z values for each plate, we see that

(17.13)

where l is the length of the plate, as illustrated in figure 17.4.

The minus sign means that the electric field acts so as to oppose a change in the current. This result is called Lenz’s law. Lenz’s law is not an independent law, but arises from the minus sign in the statement of Faraday’s law.

In order for the current i through the inductor to increase with time, an external potential difference Δϕ must be imposed between the input and output wires of the inductor, which just balances the effects of the internally generated electric field:

(17.14)

If this potential difference is positive, i. e., if the input wire of the inductor is at a higher potential then the output wire, then the current through the inductor will increase with time. If it is lower, the current will decrease.

As with capacitors, inductors come in many shapes and forms. The above equation is valid only for a parallel plate inductor, but the relationship

(17.15)

is valid for any inductor, assuming that the inductance L is known. Comparison of the above two equations reveals that the inductance for the parallel plate inductor shown in figure 17.4 is just

(17.16)

17.3 Resistance and Resistors

Normal conducting materials require an electric field to keep an electric current flowing through them. The electric field causes a force on the electrons in the material, which is balanced by the energy loss that occurs when the electrons collide with the atoms forming the material. Most objects exhibit a linear relationship between the current i through them and the potential difference Δϕ applied to them. This relationship is called Ohm’s law,

(17.17)

where the constant of proportionality R is called the resistance. The quantity Δϕ is sometimes called the voltage drop across the resistor.

For certain materials, such as semiconductors, the resistance depends on the current. For such materials, the above equation defines resistance, but since the resistance doesn’t remain constant when the current changes, these materials don’t obey Ohm’s law.

Figure 17.6 illustrates a rectangular resistor. The resistance of such a resistor can be written

(17.18)

where the resistivity ρ is characteristic only of the material and not its shape or size.

Unlike capacitors and inductors, resistors are dissipative devices. The work done on a charge q passing through a resistor is just qΔϕ. This energy is converted to heat. The work done per unit time, which equals the power dissipated by a resistor is therefore

(17.19)

17.4 Energy of Electric and Magnetic Fields

In this section we calculate the energy stored by a capacitor and an inductor. It is most profitable to think of the energy in these cases as being stored in the electric and magnetic fields produced respectively in the capacitor and the inductor. From these calculations we compute the energy per unit volume in electric and magnetic fields. These results turn out to be valid for any electric and magnetic fields — not just those inside parallel plate capacitors and inductors!

Let us first consider a capacitor starting in a discharged state at time t = 0. A constant current i is caused to flow through the capacitor by some device such as a battery or a generator, as shown in the left panel of figure 17.7. As the capacitor charges up, the potential difference across it increases with time:

(17.20)

The EMF supplied by the generator has to increase to match this value.

The generator does work on the positive charges moving around the circuit in the direction indicated by the arrow. We assume that Δϕ equals the EMF or work per unit charge done by the generator V _G, so the work done in time dt by the generator is dW = V _Gdq = V _Gidt. Using the equation for the potential difference across a capacitor, we see that the power input is

(17.21)

Integrating this in time yields the total energy U_E supplied to the capacitor by the generator:

(17.22)

Assuming that we have a parallel plate capacitor, let’s insert the formula for the capacitance of such a device, C = ϵ₀S∕d. Let us further recall that the electric field in a parallel plate capacitor is E = σ∕ϵ₀ = q∕(ϵ₀S), so that q = ϵ₀ES and

(17.23)

The combination Sd is just the volume between the capacitor plates. The energy density in the capacitor is therefore

(17.24)

This formula for the energy density in the electric field is specific to a parallel plate capacitor. However, it turns out to be valid for any electric field.

A similar analysis of a current increasing from zero in an inductor yields the energy density in a magnetic field. Imagine that the generator in the right panel of figure 17.7 produces a constant EMF, V _G, starting at time t = 0 when the current is zero. The work done by the generator in time dt is dW = V _Gdq = V _Gidt so that the power is

(17.25)

We have assumed that the EMF supplied by the generator, V _G, balances the voltage drop across the inductor: V _G = Δϕ = L(di∕dt).

If we integrate the above equation in time, we get the energy added to the inductor as a result of increasing the current through it. Substituting the formula for the inductance of a parallel plate inductor, L = μ₀dl∕w, we arrive at the equation for the energy stored by the inductor:

(17.26)

Finally, using the relationship between the current and the magnetic field in a parallel plate inductor, B = μ₀i∕w, we can eliminate the current i and write

(17.27)

The volume between the inductor plates is just dlw, so again we can write an energy density, this time for the magnetic field:

(17.28)

Though we only proved this equation for the magnetic field inside a parallel plate inductor, it turns out to be true for any magnetic field.

The total energy density is just the sum of the electric and magnetic energy densities:

(17.29)

17.5 Kirchhoff’s Laws

In the above discussion of energy we made two assumptions about electric circuits, which consist of electronic components connected by wires:

• Currents are conserved. Thus, the current entering one end of a wire connecting two devices is equal to the current leaving the other end, and the current out of the generator in the left panel of figure 17.7 is assumed to equal the current into and out of the capacitor. No electric charge is stored in any of the wires connecting components.
• The electric fields outside of components are conservative, and therefore are derived solely from the scalar potential. Thus, the net work done on a charged particle passing completely around a circuit loop is zero and the positive work done on charge passing through the generator in the right panel of figure 17.7 is exactly balanced by the potential difference across the inductor. As a result, the effects of electric fields outside of components can be represented by electrostatic potentials, or “voltages”. Every part of a wire connecting components has the same potential and the effect of each component is to maintain a potential difference between its input and output connections.

These are called Kirchhoff’s laws. They are used extensively in electronic circuit design. Figure 17.8 illustrates a typical circuit with a voltage source VS, which may be a battery or generator, and three circuit components. The voltage source provides a potential difference equal to V _a - V _g, which is equal simply to V _a, since we are assuming that V _g = 0. (Since the scalar potential is insensitive to an arbitrary additive constant, we can always set the potential at one point in the circuit to zero to simplify our calculations.) The voltage drop across component 1 is V _a - V _b and the voltage drop across both components 2 and 3 is V _b - V _g. If the components are resistors, then Ohm’s law can be used to relate the voltage drops across the components to the currents through them. If they are capacitors or inductors, the voltage drop is related respectively to the charge on the capacitor or the time rate of change of current through the inductor. The time rate of change of the charge on the capacitor can be related to the current through the capacitor. The final point in figure 17.8 is that current is conserved at junctions, i. e., i₁ = i₂ + i₃. The methods of algebra (for just resistors) or calculus (if there are capacitors or inductors) can then be used to calculate all currents and voltages.

It is important to realize that Kirchhoff’s laws are only approximations that hold when the currents and potentials in a circuit change slowly with time. For steady currents and constant potentials they are precisely true, since imbalances in charge entering and leaving a junction between devices would result in the indefinite buildup of charge in the junction with time and therefore an increasing electrostatic potential, which would violate the steady state assumption. Furthermore, a non-zero EMF around a closed loop would result in net acceleration of charge around the loop and a constantly increasing current.

If currents and potentials are changing with time, Kirchhoff’s laws are approximately valid only if the capacitance, inductance, and resistance of the wires connecting circuit elements are much smaller than the capacitance, inductance, and resistance of the circuit elements themselves. For very high frequency operation, the effects of these “parasitic” properties are not small and must be included in the design of the circuit.

17.6 Problems

1. Compute the capacitance of an isolated conducting sphere of radius R. Hint: Consider the other electrode to be a spherical shell surrounding the conducting sphere at very large radius.
2. Given a parallel plate capacitor with plate area S, fixed charge ±q on the plates, and the possibly variable plate separation x:
   1. Is the force between the plates attractive or repulsive?
   2. Compute the magnitude of the force of each plate on the other. Hint: You know both the electric field and the charge.
   3. Make an alternate computation of the force as follows: Compute the energy U in the electric field between the plates. The force is F = -dU∕dx.
   4. You probably found that the above two calculations of the force didn’t agree. Which is correct? Explain. Hint: In doing part (b), what part of the electric field acting on (say) the negative charge is due to itself, and what part is due to the positive charge? Only the latter part can exert a net force on the negative charge!
3. Compute the circulation of the vector field around the illustrated circle in the left panel of figure 17.3. Assume that the magnitude of the vector field equals Kr where K is a constant.
4. Compute the circulation of the vector field around the illustrated rectangle in the right panel of figure 17.3. Assume that the x component of the vector field equals Ky where K is a constant.
5. The solar wind consists of a plasma (a gas consisting of charged particles with equal amounts of positive and negative charge) streaming out from the sun. In certain sectors of the solar wind the magnetic field points away from the sun while in other sectors it points toward the sun. What is the magnitude and direction of the current flowing through the loop defined by the dashed rectangle that spans a sector boundary as shown in figure 17.9? Assume that the displacement current is negligible.

   ---

   

   
   

   Figure 17.9: Magnetic field at a solar wind sector boundary.

   _____________________________________

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6. A superconducting parallel plate inductor with plate dimensions 0.1 m by 0.1 m and spacing 0.01 m is held together by connectors with maximum breaking strength 500 N and has the input and the output connected by a superconducting wire. A current i is circulating through the inductor.
   1. Is the force between the plates attractive or repulsive?
   2. What is the maximum magnetic field that the inductor can have between the plates without blowing apart? Hint: Find the energy in the magnetic field as a function of plate separation and compute the force between the plates as for the capacitor. The magnetic flux through the inductor remains constant as the plates move in this case, which means that the current can change.
   3. What is the current corresponding to the above maximum field?

   ---

   

   
   

   Figure 17.10: Resistors in parallel and in series.

   _____________________________________

   ---

7. Use Kirchhoff’s laws to compute the net resistance of
   1. resistors in parallel, and
   2. resistors in series,

   as shown in figure 17.10. Hint: In the first case the voltage drop across the resistors is the same, in the second, the current through the resistors is the same. Recall that Ohm’s law relates the current through a device to the voltage drop across it. (If you already know the answers, derive them; don’t just write them down.)

   ---

   

   
   

   Figure 17.11: Battery in parallel with an inductor.

   _____________________________________

   ---

8. Try to explain in physical terms why doubling the length of a resistor doubles its resistance, while doubling its cross-sectional area halves its resistance. Use this argument to justify equation (17.18).
9. Describe qualitatively what happens when
   1. the switch is closed in the circuit in figure 17.11, and
   2. when it is abruptly opened.

   The battery produces a voltage difference V , but also may be thought of as having a small internal resistance R.

   ---

   

   
   

   Figure 17.12: Circuit consisting of a shorted resistor.

   _____________________________________

   ---

10. Given the circuit shown in figure 17.12:
    1. What do Kirchhoff’s laws tell you about Δϕ across the resistor?
    2. Suppose a time-varying magnetic field B = B₀ sin(ωt) is applied normal to the circuit loop, where B₀ is a constant. What is the (time-dependent) voltage drop Δϕ across the resistor in this situation?
    3. Given the above Δϕ, what is the current through the resistor as a function of time?

    You may ignore the effect of the current in creating an additional magnetic field.

11. In the circuit shown in figure 17.13, the voltage source is switched on at time t = 0, at which the voltage V _A goes from zero to some constant positive value. The capacitor initially has no charge.
    1. Just after the source is switched on what is the voltage V _B? Hint: Can the potential difference across the capacitor change instantaneously with the resistor in the circuit? Explain why or why not.
    2. After a very long time, what is the voltage V _B?
    3. Make a qualitative sketch of V _B as a function of time.
       

       ---

       

       
       

       Figure 17.13: Simple RC circuit.

       _____________________________________

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Chapter 18
Measuring the Very Small

To begin our study of matter we discuss experiments in the late 19th and early 20th centuries that led to proof of the existence of atoms and their constituents. We then introduce a fundamental idea about the scattering of waves using the diffraction of light by small particles as a prototype. The famous Geiger-Marsden experiment that led to the idea of the atomic nucleus is discussed. Finally, we examine some of the crucial experiments done with modern particle accelerators and the physical principles behind them.¹

18.1 Continuous Matter or Atoms?

From the time of the ancient Greeks there have been debates about the ultimate nature of matter. One of these debates is whether matter is infinitely divisible or whether it consists of fundamental building blocks that are themselves indivisible. However, it wasn’t until the late 19th century that real progress began to be made on this question.

Advancements in our understanding of matter have largely been coupled to the development of machines to accelerate atomic and sub-atomic particles. The original accelerator was developed in the 19th century and is called the Crookes tube.

J. J. Thomson measured the charge to mass ratio for both electrons and positive ions in the Crookes tube in the following way: If a potential difference Δϕ is applied between the electrodes, then by energy conservation a particle of charge q starting from rest will acquire a kinetic energy moving from electrode to electrode of K = mv²∕2 = qΔϕ. Solving for v, we find v = (2qΔϕ∕m)^1∕2. If a magnetic field B is then imposed normal to the electron beam after it has passed the positive electrode, the beam bends with a radius of curvature of R = mv∕(qB). Since R and B are known, the charge to mass ratio can be computed by eliminating v and solving for q∕m: q∕m = 2Δϕ∕(BR)². Thomson found that positive ions typically had charge to mass ratios several thousand times smaller than the electrons. Furthermore, the ions were positively charged, while the electrons were negatively charged. If the ions and the electrons have electrical charges equal in magnitude (plausible, since the ions are neutral atoms with at least one electron removed), the ions have to be much more massive than the electrons.

Robert Millikan made the first direct measurement of electric charge. He did this by suspending electrically charged oil drops in a known electric field against gravity. The size of an oil drop is directly measured using a microscope, leading to a calculation of its mass, and hence the gravitational force, mg. This is then balanced against the electric force, qE, leading to q = mg∕E. Occasionally an oil drop loses an electron due to photoelectric emission caused by photons from an ultraviolet lamp. This disrupts the force balance, and causes the oil drop to move up or down. If the electric field is quickly adjusted, this motion can be arrested. The change in the charge can be related to the change in the electric field: Δq = mgΔ(1∕E). If only a single electron is emitted, then Δq is equal to the electronic charge.

Between the work of Thomson and Millikan, the masses and the charges of sub-atomic particles were accurately measured for the first time. Ironically, this work also showed that the “atom”, which means “indivisible” in Greek, in fact isn’t. Atoms consist of positive charges with large mass, or protons, in conjunction with low mass electrons of negative charge. Electrons and protons have opposite charges, so they attract each other to form atoms in this picture.

Geiger and Marsden did an experiment that strongly suggested that atoms consist of very small, positively charged atomic nuclei, surrounded by a cloud of circling, negatively charged electrons. This is called the Rutherford model of the atom after Ernest Rutherford.

Chadwick completed our picture of the atom with the discovery of a neutral particle of mass comparable to the proton, called the neutron. The neutron is a constituent of the atomic nucleus along with the proton. The number of protons in a nucleus is denoted Z while the number of neutrons is N. We define A = Z + N to be the total number of nucleons (protons plus neutrons). The parameter Z is often called the atomic number while A is called the atomic mass number.

Marie and Pierre Curie and Henri Becquerel were the first to discover a more fundamental divisibility of atoms in the form of the radioactive decay, though the implications of their results did not become clear until much later. Radioactive decay of atomic nuclei comes in three common forms: alpha, beta, and gamma decay. Alpha decay is the spontaneous emission of a helium-4 nucleus, called an alpha particle by a heavy nucleus such as uranium or radium. The alpha particle consists of two protons and two neutrons, so the emission decreases both Z and N by 2. Beta decay is the emission of an electron or its antiparticle, the positron, by a nucleus, with an accompanying change in the electric charge of the nucleus. For electron emission Z increases by 1 while N decreases by 1. The opposite occurs for positron emission. Gamma decay is the emission of a high energy photon by a nucleus. The values of Z and N remain unchanged. The energy released by these decays is typically of order a few million electron volts.

Of the three forms of decay, beta decay is the most interesting, since it involves the transformation of one sub-atomic particle into another. In the case of neutron decay, a neutron is converted into a proton, an electron, and an antineutrino. For proton decay, a proton becomes a neutron, a positron, and a neutrino. (Only the neutron form occurs for an isolated particle. However, the energetics inside atomic nuclei can result in either form, depending on the nucleus in question.) The neutrino is one of the great theoretical predictions of modern physics. Careful studies of beta decay, which at the time was thought to result only in the emission of a proton and an electron for the neutron form of the reaction, showed apparent non-conservation of energy and angular momentum. Rather than accept this rather unpalatable conclusion, Wolfgang Pauli proposed that a third particle named a neutrino, or little neutral particle, is emitted in the decay, thus accounting for the missing energy and angular momentum. The presumed electrical neutrality of the particle explained the difficulty of detecting it. Over 25 years passed before Frederick Reines and Clyde Cowan from Los Alamos observed this elusive particle.

The three forms of radioactive decay are associated with three of the four known fundamental forces of nature. Gamma decay is electromagnetic in nature, while alpha decay involves the breaking of bonds produced by the nuclear or strong force. Beta decay is a manifestation of the so-called weak force. (The fourth force is gravity, which plays a negligible role on the sub-atomic scale, as far as we know.)

Beta decay gives us a strong hint that even particles such as protons and neutrons, which make up atomic nuclei, are not “atomic” in the sense of the original Greek, since neutrons can change into protons in beta decay and vice versa. We now have excellent evidence that protons, neutrons, and many other sub-nuclear particles are made up of particles called quarks. Quarks and electrons are currently thought to be fundamental in that they are supposedly indivisible, and are hence the true “atoms” of the universe. However, who knows, perhaps someday we will discover that they too are composed of even more fundamental constituents!

18.2 The Ring Around the Moon

Sometimes at night one sees a diffuse disk of light around the moon if it happens to be shining through a thin layer of cloud. This disk consists of light diffracted by the water or ice particles in the cloud. The diameter of the disk contains information about the size of the cloud particles doing the diffraction. In particular, if the particles have diameter d and the light has wavelength λ, then the diffraction half-angle shown in figure 18.2 is approximately

(18.1)

This equation comes from the problem of passage of light through a hole or slit of diameter or width d. This problem was treated in the chapter on waves, and the above formula was concluded to hold in that case. One can think of the diffraction of light by a particle to be the linear superposition of a plane wave minus the diffraction of light by a hole in a mask, as illustrated in figure 18.2. The angular spread of the diffracted light is the same in both cases.

The interesting point about equation (18.1) is that the opening angle of the diffraction cone is inversely proportional to the diameter of the diffracting particles. Thus, for a given wavelength, smaller particles cause diffraction through a wider angle.

Note that when the wavelength exceeds the diameter of the particle by a significant amount, equation (18.1) fails, since scattering through an angle greater than π doesn’t make physical sense. In this case the diffracted photons tend to be isotropic, i. e., they are scattered with equal probability into any direction.

If one wishes to measure the size of an object by observing the diffraction of a wave around the object, the lesson is clear; the wavelength of the wave must be less than or equal to the dimensions of the object — otherwise the scattering of the wave by the object is largely isotropic and equation (18.1) yields no information. Since wavelength is inversely related to momentum by the de Broglie relationship, this condition implies that the momentum must satisfy

(18.2)

in order that the size of an object of diameter d be resolved.

18.3 The Geiger-Marsden Experiment

In 1908 Hans Geiger and Ernest Marsden, working with Ernest Rutherford of the Physical Laboratories at the University of Manchester, measured the angular distribution of alpha particles scattered from a thin gold foil in an experiment illustrated in figure 18.3. In order to understand this experiment, we need to compute the de Broglie wavelength of alpha particles resulting from radioactive decay. Typical alpha particle kinetic energies are of order 5 MeV = 8 × 10^-13 J. Since the alpha particle consists of two protons and two neutrons, its mass is about M_α = 6.7 × 10^-27 kg. This implies a velocity of about v = 1.1 × 10⁷ m s^-1, a momentum of about p = mv = 7.4 × 10^-20 N s, and a de Broglie wavelength of about λ = h∕p = 9.0 × 10^-15 m.

Other evidence indicates that atoms have dimensions of order 10^-10 m, so the de Broglie wavelength of an alpha particle is about a factor of 10⁴ smaller than a typical atomic dimension. Thus, the typical diffraction scattering angle of alpha particles off of atoms ought to be very small, of order α = λ∕(2d) ≈ 10^-4 radian ≈ 0.01^∘.

Imagine the surprise of Geiger and Marsden when they found that while most alpha particles suffered only small deflections when passing through the gold foil, a small fraction of the incident particles scattered through large angles, some in excess of 90^∘!

Ernest Rutherford calculated the probability for an alpha particle, considered to be a positive point charge, to be scattered through various angles by a stationary atomic nucleus, assumed also to be a positive point charge. The calculation was done classically, though interestingly enough a quantum mechanical calculation gives the same answer. The relative probability for scattering with a momentum transfer to the alpha particle of q is proportional to |q|^-4 ≡ q^-4 according to Rutherford’s calculation. (Do not confuse this q with charge!) As figure 18.4 indicates, a larger momentum transfer corresponds to a larger scattering angle. The maximum momentum transfer for an incident alpha particle with momentum p is 2|p|, or just twice the initial momentum. This corresponds to a head-on collision between the alpha particle and the nucleus followed by a recoil of the alpha particle directly backwards. Since this collision is elastic, the kinetic energy of the alpha particle after the collision is approximately the same as before, as long as the nucleus is much more massive than the alpha particle.

Rutherford’s calculation agreed quite closely with the experimental results of Geiger and Marsden. Though the probability for scattering through a large angle is small even in the Rutherford theory, it is still much larger than would be expected if there were no small scale atomic nucleus.

18.4 Cosmic Rays and Accelerators

18.4.1 Early Cosmic Ray Results

Earlier we indicated that particles interacted with each other via the exchange of a virtual intermediary particle that interchanges energy, momentum, and other physical properties between the interacting particles. This idea originated with the Japanese physicist Hideki Yukawa in 1935 in an effort to understand the forces between nucleons. Yukawa hypothesized that the force that holds nucleons together is associated with the exchange of a boson, i. e., a particle with integer spin, with rest energy mc² ≈ 100 MeV. The range of this force at low momentum transfers is I ≈ℏ∕(mc) ≈ 2 × 10^-15 m, or comparable to the observed size of an atomic nucleus.

In 1947 two new particles were discovered in cosmic rays, the negatively charged muon with a rest energy of 106 MeV, and the pion, which comes in three varieties, the π⁺, the π^-, and the π⁰, which respectively have positive, negative, and zero charge. The rest energies of the π⁺ and π^- are 140 MeV while that of the π⁰ is 135 MeV. All of these particles are unstable in that they decay into other, more stable particles in a tiny fraction of a second. In particular, the negative pion decays into a muon and an antineutrino, while the neutral pion decays into two gamma rays, or high energy photons. The antineutrino that results from pion decay is actually distinct from the antineutrino emitted in nuclear beta decay; it is called the mu antineutrino since it is associated with the muon in the same way that the antineutrino in beta decay is associated with the electron. To further distinguish between the two, the latter is called the electron antineutrino. The muon itself decays into an electron, a mu neutrino, and an electron antineutrino.

The muon and its associated neutrino are rather peculiar. In all respects except mass, the muon appears to be identical to the electron. The physicist I. I. Rabi is reputed to have responded “Who ordered that?” upon learning of the properties of the muon. Furthermore, the electron neutrino only interacts with the electron and the muon neutrino only interacts with the muon. This is the first hint that elementary particles occur in families that appear to be replicated at higher energies.

Since the muon is a fermion with spin 1∕2, it can’t be Yukawa’s intermediary particle since all intermediary particles are bosons with integral spin. Furthermore, as with the electron, it is not subject to the nuclear force. The pions are more promising candidates for being intermediary particles of the nuclear force, since they are bosons with spin 0. However, as we shall see, the situation is more complex than Yukawa imagined, and the force between nucleons cannot be so simply treated. However, Yukawa’s idea of intermediary particle exchange lives on in today’s theories of sub-nuclear particles.

18.4.2 Particle Accelerators

Soon after the discovery of muons and pions in cosmic rays, a whole plethora of unstable particles was uncovered. Central to these discoveries was the particle accelerator. In these devices, charged particles, typically electrons or protons, are accelerated to high energy and then smashed into a target. Detectors of various sorts are used to examine the particles created by the collisions of the accelerated particles and the atomic nuclei with which they collide. Sometimes an elastic collision occurs, in which the accelerated particle simply “bounces off” of the target particle, transferring a good bit of its momentum to this sparticle. However, under many circumstances the collision results in the production of new particles that didn’t exist before the collision. This is referred to as an inelastic collision.

The simplest type of target is liquid hydrogen since the nucleus consists of a single proton. The orbital electrons of the target atoms are so light that they are generally just “brushed aside” without greatly affecting the trajectories of the accelerated particles. However, a variety of targets are used under different circumstances.

18.4.3 Size and Structure of the Nucleus

In the late 1950s and early 1960s Robert Hofstadter of Stanford University extended the Geiger-Marsden experiment to much shorter de Broglie wavelengths using high energy electrons from an accelerator, rather than alpha particles, as the probe. The type of results obtained by Hofstadter are shown in figure 18.5. After accounting for some effects having to do with the electron spin, these experiments should agree with the Rutherford formula if the nucleus is truly a point particle. This is shown by the solid line in figure 18.5. However, the actual results (dashed line) show probabilities that drop off more rapidly with increasing momentum transfer q than is predicted by the Rutherford model.

These results are related to the fact that the nucleus is actually of finite size. The diffraction effects discussed in the section on the scattering of moonlight come into play here, in that little scattering takes place for scattering angles larger than roughly λ∕(2d), where λ is the de Broglie wavelength of the probing particle and d is the diameter of the target. For a small scattering angle (which we now call θ), it is clear from figure 18.4 that

(18.3)

where p is the momentum of the incident electron and q is the momentum transfer. If q_max is the maximum momentum transfer for which there is significant scattering, then we can write

(18.4)

where the factor of 2 in the denominator on the right side has been dropped since this is an approximate analysis. However, since λ = h∕p, we find that

(18.5)

Thus, the momentum transfer for which the dashed line in figure 18.5 becomes small compared to the solid line gives us an immediate estimate of the diameter of an atomic nucleus: d ≈ h∕q_max. The results obtained by Hofstadter show that nuclear diameters are typically a few times 10^-15 m.

More than just size information can be extracted from this analysis. Hofstadter’s experiments also led to a great deal of information about the internal structure of atomic nuclei.

18.4.4 Deep Inelastic Scattering of Electrons from Protons

The construction of the Stanford Linear Accelerator Center (SLAC), which accelerates electrons up to 40 GeV, allowed experiments like Hofstadter’s to be carried out at much higher energies. At these energies, many of the collisions between electrons and protons and neutrons are inelastic — generally a great mess of short-lived particles is spewed out, and are very difficult to interpret. However, the so-called deep inelastic collisions, where the electron scatters through a large angle and therefore transfers a large momentum, q, to the proton, yield very interesting results. In particular, these collisions occur essentially with a probability proportional to q^-4 — just as in the Geiger-Marsden experiment!

The electron is a point particle as far as we know. However, previous experiments showed the proton to have a finite size, of order 10^-15 m. Therefore, the scattering probability should drop off more rapidly with increasing momentum transfer q than with q^-4, as in the earlier Hofstadter experiments.

James Bjorken and Richard Feynman showed a way out of this dilemma. They proposed that the proton actually consists of a small number of point particles bound together by weakly attractive forces. A sufficiently energetic photon is able to knock a single one of these particles out of the proton, as illustrated in the right panel of figure 18.6. This leads to a subsequent set of reactions that produce the profusion of particles seen in the left panel of this figure. Feynman called the particles that make up the proton partons. However, we now know that they are actually quarks, spin 1/2 particles with fractional electronic charge that are thought to be the fundamental building blocks of matter, and gluons, the massless spin 1 intermediary particles that carry the strong force.

18.4.5 Storage Rings and Colliders

An alternate way to create interesting collisions is to crash particles and antiparticles of the same energy into each other. This is done via a storage ring, as shown in figure 18.7. A set of magnets forces particles and antiparticles (which have opposite charges) to move in opposing circles within a high vacuum. The circles are slightly offset so that the beams cross at only two points. Collisions occur at these points and are observed by various types of experimental equipment.

An alternate type of collider has two storage rings that intersect at only one point. This type of system can be used to collide particles of the same type together, e. g., protons colliding with protons.

18.4.6 Proton-Antiproton Collisions

If collisions occur by the exchange of a single intermediary particle of zero mass between point particles, the q^-4 dependence of the collision probability on momentum transfer will occur in proton-antiproton collisions as in the Geiger-Marsden experiment. However, if the colliding particles are not point particles, a form factor that decreases for increasing momentum transfer will occur as with the Hofstadter experiments.

When collisions between protons and antiprotons of a few hundred GeV are arranged, certain types of events called two-jet events are recorded. In these events, two jets, each containing many particles, are emitted in opposite directions at wide angles (i. e., with large momentum transfer) from the colliding beams. Furthermore, these jets show a probability distribution as a function of momentum transfer very close to q^-4. This indicates that the colliding particles are point-like, at least down to the minimum spatial resolutions available to today’s accelerators.

According to the Bjorken-Feynman parton model of the proton, the collision between highly energetic protons and antiprotons should operate as shown in figure 18.8. The actual collision is between individual partons. Figure 18.8 illustrates the collision between a quark in the proton and an antiquark in the antiproton. The result of this interaction is the scattering of these particles out of the incident particles, resulting ultimately in a two jet event as described above.

18.4.7 Electron-Positron Collisions

Two-jet events can also be created by the collision of high energy electrons and positrons. Figure 18.9 shows how this process is thought to work. The annihilation of the electron and positron results in a virtual photon, which in turn decays into a quark-antiquark pair. The quarks then produce the jets. These results suggest that quarks can indeed occur outside of protons, at least if they occur in quark-antiquark pairs.

18.5 Commentary

We have examined a selected set of experiments performed over the last 100 years. Though complicated in detail, we have seen that they can be understood in their essence using one idea, namely the uncertainty principle. This principle underlies the diffraction angle formula and also turns out (in an argument that we have not made) to be central to the q^-4 dependence of scattering probability for point particles. For momentum transfers of order 1000 GeV/c, we are able to probe spatial scales of order 10^-17 m, or a factor of 100-500 less than the scale of the atomic nucleus. Even on this scale it appears that both the electron and the quark act like point particles. They thus appear to be the ultimate “atoms” of matter in the original sense of the word. However, it is possible that experiments at even higher momentum transfers would show the electron or the quark to have some kind of internal structure. Perhaps this heirarchy of structure, of which we have noted the atom, the atomic nucleus, nucleons, and quarks, goes on forever.

18.6 Problems

1. If possible, observe the moon through a thin cloud layer and estimate the angular size of the disk of scattered light around the moon. From this, estimate the size of the particles doing the scattering.
2. Which particle can be used to investigate smaller scales, a proton or an electron, at
   1. the same velocity, and
   2. at the same kinetic energy? (Work non-relativistically in both cases.)
   3. Now consider ultra-relativistic protons and electrons with the same total energy. Is there a significant difference between their ability to investigate very small scales?
3. Electron microscope:
   1. What kinetic energy (in electron volts) must electrons in an electron microscope have to match the resolution of an optical microscope? (The resolutions match when the wavelengths of the electrons and the light are the same.)
   2. If the electrons have kinetic energy 50 KeV, how much better resolution does the electron microscope have than the best optical microscope?

   Hint: Use the non-relativistic kinetic energy and check whether this assumption is valid in retrospect.

4. Integrated circuits are made by a system in which the circuit pattern is engraved on a silicon wafer using a photochemical process working with an optical imaging device that projects the circuit image on the wafer.
   1. Assuming visible light is used, estimate the size of the smallest feature that could be produced on the silicon by this system.
   2. Do the same for 1 KeV X-rays.

   Hint: Recall that the smallest feature resolvable by a wave is approximately the wavelength of that wave.

5. The rest energy of two colliding particles is just c² multiplied by the mass of the single particle created by the colliding particles sticking together.
   1. Compute the rest energy (in GeV) of a particle resulting from a 100 GeV energy proton colliding with a stationary proton.
   2. Compute the rest energy of the particle resulting from two 50 GeV protons colliding head-on.

   Hint: These calculations are relativistic, since the rest energy of the proton is about 0.9 GeV.

6. Relativistic charged particle in magnetic field: Assume that a relativistic particle of mass m and charge e is moving in a circle under the influence of the magnetic field B = (0, 0,-B). The position of the particle as a function of time is given by x = [R cos(ωt),R sin(ωt), 0].
   1. Compute the (vector) velocity of the particle and show that its speed is v = ωR.
   2. Compute the (relativistic) momentum (again in vector form) of the particle using the above results.
   3. Compute the magnetic force F on the particle.
   4. Using the relativistic version of Newton’s second law, F = dp∕dt, determine how the rotational frequency ω depends on the speed of the particle, the magnetic field B, and the particle’s charge and mass. Examine particularly the limits where v ≪ c and v ≈ c.
   5. Eliminate ω between the above result and the speed formula to get an equation for the radius R of the circle. Show that this takes the particularly simple form R = p∕(eB) when written in terms of the magnitude of the momentum p = mvγ.
7. A 30 GeV electron is scattered by a virtual photon through an angle of 60^∘ without changing its energy.
   1. Compute its momentum vector before and after the scattering.
   2. Compute the momentum transfer to the electron by the photon in the scattering event.
   3. Compute the wavelength of the virtual photon.
   4. What is the virtual photon’s energy?
   5. What is the virtual photon’s mass?
8. Find α, β, and γ such that ℏ^αc^βG^γ has the units of length. (G is the universal gravitational constant.) Compute the numerical value of this length, which is called the Planck length. Compare this value to the resolution available today in the highest energy accelerators.

Chapter 19
Atoms

In this chapter we investigate the structure of atoms. However, before we can understand these, we first need to review some facts about angular momentum in quantum mechanics.

19.1 Fermions and Bosons

19.1.1 Review of Angular Momentum in Quantum Mechanics

As we learned earlier, angular momentum is quantized in quantum mechanics. We can simultaneously measure only the magnitude of the angular momentum vector and one component, usually taken to be the z component. Measurement of the other two components simultaneously with the z component is forbidden by the uncertainty principle.

The magnitude of the orbital angular momentum of an object can take on the values |L| = [l(l + 1)]^1∕2ℏ where l = 0, 1, 2,…. The z component can likewise equal L_z = mℏ where m = -l,-l + 1,…,l.

Particles can have an intrinsic spin angular momentum as well as an orbital angular momentum. The possible values for the magnitude of the spin angular momentum are |S| = [s(s + 1)]^1∕2ℏ and the z component of the spin angular momentum S_z = m_sℏ where m_s = -s,-s + 1,…,s. Spin differs from orbital angular momentum in that the spin can take on half-integer as well as integer values: s = 0, 1∕2, 1, 3∕2,… are possible spin quantum numbers.

Spin is an intrinsic, unchangeable quantity for an elementary particle. Particles with half-integer spins, s = 1∕2, 3∕2, 5∕2,…, are called fermions, while particles with integer spins, s = 0, 1, 2,… are called bosons. Fermions can only be created or destroyed in particle-antiparticle pairs, whereas bosons can be created or destroyed singly.

19.1.2 Two-Particle Wave Functions

We learned in quantum mechanics that a particle is represented by a wave, ψ(x,y,z,t), the absolute square of which gives the relative probability of finding the particle at some point in spacetime. If we have two particles, then we must ask a more complicated question: What is the relative probability of finding particle 1 at point x₁ and particle 2 at point x₂? This probability can be represented as the absolute square of a joint wave function ψ(x₁,x₂), i. e., a single wave function that represents both particles. If the particles are not identical (say, one is a proton and the other is a neutron) and if they are not interacting with each other via some force, then the above wave function can be broken into the product of the wave functions for the individual particles:

(19.1)

In this case the probability of finding particle 1 at x₁ and particle 2 at x₂ is just the absolute square of the joint wave amplitude: P(x₁,x₂) = P₁(x₁)P₂(x₂). This is consistent with classical probability theory.

The situation in quantum mechanics when the two particles are identical is quite different. If P(x₁,x₂) is, say, the probability of finding one electron at x₁ and another electron at x₂, then since we can’t tell the difference between one electron and another, the probability distribution cannot change if we switch the electrons. In other words, we must have P(x₁,x₂) = P(x₂,x₁). There are two obvious ways to make this happen: Either ψ(x₁,x₂) = ψ(x₂,x₁) or ψ(x₁,x₂) = -ψ(x₂,x₁).

It turns out that the wave function for two identical fermions is antisymmetric to the exchange of particles whereas for two identical bosons it is symmetric. In the special case of two non-interacting particles, we can construct the joint wave function with the correct symmetry from the wave functions for the individual particles as follows:

(19.2)

for fermions and

(19.3)

for bosons.

Figure 19.1 shows the joint probability distribution for two particles in different energy states in an infinite square well: P(x₁,x₂) = |ψ(x₁,x₂)|². Three different cases are shown: non-identical particles, identical fermions, and identical bosons. Notice that the probability of finding two fermions at the same point in space, i. e., along the diagonal dotted line in the center panel of figure 19.1, is zero. This follows immediately from equation (19.2), which shows that ψ(x₁,x₂) = 0 for fermions if x₁ = x₂. Notice also that if two fermions are in the same energy level (say, the ground state of the one-dimensional box) so that ψ₁(x) = ψ₂(x), then ψ(x₁,x₂) = 0 everywhere. This demonstrates that the two fermions cannot occupy the same state. This result is called the Pauli exclusion principle.

On the other hand, bosons tend to cluster together. Figure 19.1 shows that the highest probability in the joint distribution occurs along the line x₁ = x₂, i. e., when the particles are colocated. This tendency is accentuated when more particles are added to the system. When there are a large number of bosons, this tendency creates what is called a Bose-Einstein condensate in which most or all of the particles are in the ground state. Bose-Einstein condensation is responsible for such phenomena as superconductivity in metals and superfluidity in liquid helium at low temperatures.

19.2 Angular momentum and magnetic moment

The Rutherford model of the atom has electrons moving in orbit around a small, positively charged nucleus. Thus, in general electrons have both angular momentum and magnetic dipole moments. These magnetic moments are important for understanding the magnetic properties of matter. Even though the Rutherford model has been superseded by quantum mechanics, the classically derived relationship between angular momentum and magnetic moment remains useful.

Imagine a particle with mass M and charge q moving in a circle of radius R with speed v. Treating the motion of this particle classically, with speed much less than the speed of light, the orbital angular momentum of the particle is L = MvR. The time for the particle to make one revolution is T = 2πR∕v, so the circular current represented by this moving charge is i = q∕T = qv∕(2πR). The magnetic moment generated by this circular current is the current times the area of the circle, or m = iA = iπR². The ratio of the magnetic moment to the angular momentum of the particle

(19.4)

is called the gyromagnetic ratio.

This equation turns out to be generally true, even if particles are not moving in circular orbits. However, elementary particles have an intrinsic angular momentum that has nothing to do with orbital motion. For the electron in particular, the gyromagnetic ratio is (to within a tenth of a percent)

(19.5)

where e is the charge on the electron and M_e is its mass, or twice the gyromagnetic ratio of the electron in orbital motion. (Technically, the gyromagnetic ratio of particles with negative charge is negative, though this sign is often ignored.) Since the spin angular momentum of the electron is ℏ∕2, the intrinsic magnetic moment associated with the electron spin is

(19.6)

This is sometimes called the Bohr magneton, hence the subscripted B. The factor of 2 missing from equation (19.5) in comparison to equation (19.4) results from relativistic factors, but the derivation is rather complex, so we will not present it here.

Magnetic dipoles from orbital and spin angular momenta have important practical consequences; they are central to the phenomena of paramagnetism and ferromagnetism. Paramagnetism occurs when atoms have an odd number of electrons, which means that the angular momentum of these electrons does not add up to zero (see next chapter). In this case, there is a net magnetic dipole moment for each atom. If a magnetic field is imposed, this dipole tends to align with the magnetic field, which in turn reinforces the imposed field, resulting in a stronger total field. If the imposed magnetic field is taken away, the atoms reorient themselves randomly, resulting in zero field.

A few materials, such as iron, exhibit a more complex phenomenon called ferromagnetism. In this case the imposition of an external magnetic field causes very strong dipole alignment effects to the point where interactions between the newly aligned dipoles prevent the reversion to random orientation when the imposed field is removed. “Permanent” magnets may be created this way. As the temperature increases, the thermal jiggling of atoms increases in proportion. If this jiggling becomes strong enough, the tendency of neighboring dipoles to maintain a common orientation is overcome and the material becomes paramagnetic. The temperature at which this occurs for a particular material is called the Curie temperature.

If there are no unpaired electrons in a material, then there are no pre-existing dipoles to align. However, the magnetic field still distorts the motions of electrons via the Lorentz force. The result is the production of induced magnetic dipoles that anti-align with the field, as happens with the electrons in a plasma. The resulting induced magnetic field opposes the original field, resulting in a total field that is less than originally imposed. Such a material is called diamagnetic.

19.3 The Hydrogen Atom

The hydrogen atom consists of an electron and a proton bound together by the attractive electrostatic force between the negative and positive charges of these particles. Our experience with the one-dimensional particle in a box shows that a spatially restricted particle takes on only discrete values of the total energy. This conclusion carries over to arbitrary attractive potentials and three dimensions.

The energy of the ground state can be qualitatively understood in terms of the uncertainty principle. A particle restricted to a region of size a by an attractive force will have a momentum equal at least to the uncertainty in the momentum predicted by the uncertainty principle: p ≈ℏ∕a. This corresponds to a kinetic energy K = mv²∕2 = p²∕(2m) ≈ℏ²∕(2ma²). For the particle in a box there is no potential energy, so the kinetic energy equals the total energy. Comparison of this estimate with the computed ground state energy of a particle in a box of length a, E₁ = ℏ²π²∕(2ma²), shows that the estimate differs from the exact value by only a numerical factor π².

We can make an estimate of the ground state energy of the hydrogen atom using the same technique if we can somehow take into account the potential energy of this atom. Classically, an electron with charge -e moving in a circular orbit of radius a around a proton with charge e at speed v must have the centripetal acceleration multiplied by the mass equal to the attractive electrostatic force, mv²∕a = e²∕(4πϵ ₀a²), where m is the electron mass. (The proton is so much more massive than the electron that we can assume it to be stationary.) Multiplication of this equation by a∕2 results in

(19.7)

where U is the (negative) potential energy of the electron and K is its kinetic energy. Solving for U, we find that U = -2K. The total energy E is therefore related to the kinetic energy by

(19.8)

Since the total energy is negative in this case, and since U = 0 when the electron is infinitely far from the proton, we can define a binding energy that is equal to minus the total energy:

(19.9)

The binding energy is the minimum additional energy that needs to be added to the electron to make the total energy zero, and thus to remove it to infinity. Equation (19.9) is called the virial theorem, and it is even true for non-circular orbits if the energies are properly averaged over the entire trajectory.

Proceeding as before, we assume that the momentum of the electron is p ≈ℏ∕a and substitute this into equation (19.7). Solving this for a ≡ a₀ yields an estimate of the radius of the hydrogen atom:

(19.10)

This result was first obtained by the Danish physicist Niels Bohr, using another method, in an early attempt to understand the quantum nature of matter.

The grouping of terms by the large parentheses in equation (19.10) is significant. The dimensionless quantity

(19.11)

is called the fine structure constant for historical reasons. However, it is actually a fundamental measure of the strength of the electromagnetic interaction. The Bohr radius can be written in terms of the fine structure constant as

(19.12)

The binding energy predicted by equations (19.7) and (19.9) is

(19.13)

The binding energy between the electron and the proton is thus proportional to the electron rest energy multiplied by the square of the fine structure constant.

The above estimated binding energy turns out to be precisely the ground state binding energy of the hydrogen atom. The energy levels of the hydrogen atom turn out to be

(19.14)

where n is called the principal quantum number of the hydrogen atom.

19.4 Complex Atoms

The energy levels of the hydrogen atom whose energies are given by equation (19.14) are actually degenerate, in that each energy has more than one state associated with it. Three extra degrees of freedom are associated with angular momentum, expressed by the quantum numbers l, m, and m_s. For energy level n, the orbital angular momentum quantum number can take on the values l = 0, 1, 2,…,n - 1. Thus, for the ground state, n = 1, the only possible value of l is zero. For a given value of l, there are 2l + 1 possible values of the orbital z component quantum number, m = -l,-l + 1,…,l. Finally, there are two possible values of the spin orientation quantum number, m_s. Thus, for the nth energy level there are

(19.15)

states. In particular, for n = 1, 2, 3,…, we have N_n = 2, 8, 18,…. This is summarized in figure 19.2.

These results have implications for the character of atoms with more than one proton in the nucleus. Let us imagine how such atoms might be built. The binding energy of a single electron in the ground state of a nucleus with Z protons is Z² multiplied by the binding energy of the electron in the ground state of a hydrogen atom. If the force between electrons can be ignored compared to the force between an electron and the nucleus (a very poor but initially useful assumption that we will discuss below), then we could construct an atom by dropping Z electrons one by one into the potential well of the nucleus. The Pauli exclusion principle prevents all of these electrons from falling into the ground state. Instead, the available states will fill in order of lowest energy first until all Z electrons are added and the atom becomes electrically neutral. From figure 19.2 we see that Z = 2 fills the n = 1 levels with two electrons, one spin up and one spin down, both with zero orbital angular momentum. For Z = 10 the n = 2 levels fill such that two electrons have l = 0 and six have l = 1.

As electrons are added to an atom, previous electrons tend to shield subsequent electrons from the nucleus, since their negative charge partially compensates for the nuclear positive charge. Thus, binding energies are considerably less than would be expected on the basis of the non-interacting electron model. Furthermore, the binding energies for states with higher orbital angular momentum are smaller than those with lower values, since electrons in these states tend to be more effectively shielded from the nucleus by other electrons. This effect becomes sufficiently important at higher Z to disrupt the sequence in which states are filled by electrons — sometimes level n + 1 states with low l start to fill before all the level n states with large l are full. Accurate calculations of atomic properties in which electron-electron interactions are taken into account are possible, but are computationally expensive.

19.5 Atomic Spectra

The best evidence for atomic energy levels comes from the emission of light by atoms in a gas at low pressure. If the atoms are put in an excited state by some mechanism, say, collisions with energetic electrons accelerated by a potential difference between electrodes, then light is emitted at particular frequencies called spectral lines. These frequencies can be separated by a device called a spectroscope. Spectroscopes use either a prism or a diffraction grating and ancillary optics to make the separation visible to the eye.

The frequency of a spectral line is equal to the energy difference between two states divided by Planck’s constant. This is a consequence of the conservation of energy — the energy released when an atom undergoes a transition from a state with energy E₂ to a state with energy E₁ is just the difference between these energies. The frequency of the emitted photon is then derived from the Planck formula. In terms of the angular frequency,

(19.16)

Figure 19.3 shows the possible transitions between the lowest four energy levels of hydrogen plus the ionized state in which the electron is initially a large distance from the hydrogen nucleus. Transitions from any state to the ground state form a series called the Lyman series, while transitions to the first excited state are called the Balmer series, transitions to the second excited state are called the Paschen series, and so on. Within each series, increasing frequencies are labeled using the Greek alphabet, so the transition from n = 2 to n = 1 is called the Lyman-α spectral line, etc.

Atoms can absorb as well as emit radiation. For instance, if hydrogen atoms in the ground state are bombarded with photons of energy equal to the energy difference between the ground state and some excited state, some of the atoms will absorb these photons and undergo transitions to the excited state. If white light (i. e., many photons with a continuous distribution of frequencies) irradiates such atoms, just those photons with the right energies will be absorbed. Examination of the light with a spectroscope after it passes through a gas of atoms will show absorption lines where the photons with the critical energies have been removed. This is one of the main ways in which astrophysicists learn about the elemental constitution of stars and interstellar gases.

Atoms in excited states emit photons spontaneously. However, a process called stimulated emission is also possible. This occurs when a photon with energy equal to the difference between two atomic energy levels interacts with an atom in the higher energy state. The amplitude for this process is equal to the spontaneous emission amplitude times n + 1, where n is the number of incident photons with energy equal to the energy of the photon which would be spontaneously emitted. If a beam of photons with the right energy shines on atoms in an excited state, the beam will gain energy at a rate that is proportional to the initial intensity of the beam. For intense beams, this stimulated emission process overwhelms spontaneous emission and a large amount of energy can be rapidly extracted from the excited atoms. This is how a laser works.

19.6 Problems

1. The wave function for three non-identical particles in a box of unit length with one particle in the ground state, the second in the first excited state, and the third in the second excited state is
   1. From this write down the wave function for three identical bosons in the above mentioned states.
   2. Do the same for three identical fermions.

   Hint: In each case there are six terms corresponding to the six permutations of x₁, x₂, and x₃. Exchanging any two particles leaves ψ unchanged for bosons but changes the sign for fermions.

2. Two identical particles with equal energies collide nearly head-on, so that they are both deflected through an angle θ, as shown in figure 19.4. A physicist calculates the amplitude ψ as a function of θ for this deflection to take place (using very advanced theory!), resulting in the solid curve shown in figure 19.4. However, measurements show that the actual amplitude as a function of θ (not probability!) is given by the dashed curve.
   1. What did the physicist forget to take into account? Explain.
   2. Are the particles fermions or bosons? Explain.

   Hint: If the outgoing particles (but not the incoming particles) are interchanged, how does the apparent deflection angle change?

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   Figure 19.4: Incorrectly calculated and observed scattering amplitude for a collision between two identical particles.

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3. Consider the hydrogen atom, which has one electron circling the nucleus.
   1. Suppose that the electron moves in a circle of radius R with velocity v, as shown in figure 19.5. (This can only occur in an excited state of the atom.) Use Ampére’s law to estimate the magnetic field produced at the nucleus. Hint: Derive an equation for the average magnetic field over a magnetic circulation loop of radius R that circles the trajectory of the electron. Assume that the field at the nucleus is somewhat greater than this average value. (Pick a number.)
   2. The radius of the hydrogen atom in an excited state is of order 10^-10 m. Compute the orbital velocity of the electron for a circular orbit, given the electron mass and the charges on the electron and the proton nucleus.
   3. From the above information compute an approximate value for the magnetic field at the nucleus caused by the electron’s orbital motion.

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   Figure 19.5: Charge of mass M and charge q moving in a circle of radius R at speed v.

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4. The Stern-Gerlach experiment demonstrated the quantization of angular momentum by passing a beam of neutral silver atoms (approximately 108 times the mass of a proton) through a magnetic field with a strong gradient. (See figure 19.6.) Since silver has one unpaired electron, the magnetic dipole moment is non-zero. Assuming a vertical gradient in the vertical component of the magnetic field, the silver atoms will be deflected either up or down, depending on the orientation of the dipole moment.
   1. For a given value of ∂B_z∕∂z, derive an expression for the z component of the force on a silver atom, assuming that the dipole moment equals one Bohr magneton aligned (1) in the +z and (2) in the -z directions.
   2. If the silver atom has mass M and initial speed v, compute the angular deflection up or down of the trajectory of the silver atom as it passes through the field of the magnet, assumed to have length L.
   3. Assuming ∂B_z∕∂z = 10² T∕m, v = 300 m∕s, and L = 10 cm, compute the above deflection angle.
   4. Explain why the classical result is different from the quantum result as illustrated in figure 19.6.

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   Figure 19.6: Stern-Gerlach experiment. A neutral silver atom moves through a non-uniform magnetic field and is deflected according to the orientation of its magnetic moment vector.

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5. Following the analysis made for the hydrogen atom, compute the “Bohr radius” and the ground state binding energy for an “atom” consisting of Z protons in the nucleus and one electron.
6. Upper and lower bounds on the binding energy of the last (outermost) electron in the sodium atom (Z = 11) may be obtained by assuming (a) that the other electrons have no effect, or (b) that the other electrons neutralize all but one proton in the nucleus. Compute the binding energy of the last electron in sodium in these two limits. (The actual binding energy of the last electron in sodium is 5.139 eV.)
7. A uranium atom (Z = 92) has all its electrons stripped off except the first one.
   1. What is the first electron’s binding energy in electron volts?
   2. What is the ground state radius of the electron orbit in this case?
8. The energy levels of a particle in a box are given by E_n = E₀n² = E ₀, 4E₀, 9E₀,… where E₀ is the ground state energy for the particle. Find the lowest possible total energy of a group of particles, expressed as a multiple of E₀, for the following particles in the box:
   1. 5 identical spin 0 particles.
   2. 5 identical spin 1/2 particles.
   3. 5 identical spin 1 particles.
   4. 5 identical spin 3/2 particles.
9. A charged particle in a 1-D box has energy levels at E_n = E₀n² = E ₀, 4E₀, 9E₀, 16E₀, 25E₀,…, where E₀ is the ground state energy of the particle. If the particle can absorb a photon with any of the energies 5E₀, 12E₀, 21E₀,…, what can you infer about the initial energy of the particle? Explain.
10. The X-rays in your dentist’s office are produced when an energetic beam of free electrons knocks the most tightly bound electrons (n = 1) completely out of the target atoms. Electrons from the next level up (n = 2) then drop into the n = 1 level.
    1. Estimate the energy in electron volts of the resulting photons for a copper target (Z = 29). Hint: For the inner electrons, you may ignore the effects of the other electrons to reasonable accuracy.
    2. What minimum energy must the electron beam have in this case?
11. What is the shortest ultraviolet wavelength usable in astronomy? Hint: UV photons more energetic than the binding energy of the electron in hydrogen are strongly absorbed by this gas.
12. In the naive periodic table model, the first three closed shells occur for Z = 2, 10, 28. However, the first three noble gases have Z = 2, 10, 18. Explain why this is so.

Chapter 20
The Standard Model

In this chapter we learn about the most fundamental known particles of the universe, and how they act as building blocks for everything that we know. The theory describing this scheme is called the standard model. Speculations exist about possible, more fundamental structures in the universe, such as the constructs of string theory. However, with the standard model we have reached the frontier of what is known with any degree of certainty.

20.1 Hadrons and Leptons

The standard model of hadrons and leptons is a united set of quantum mechanical theories encompassing electromagnetism; the weak force, which is responsible for beta decay; and the strong force, which holds atomic nuclei together. Before investigating the standard model, we need to describe the state of affairs previous to its development. The creation of high energy particle accelerators led to the discovery of a plethora of particles in addition to those already known. These particles fall into the following categories:

• Leptons are spin 1/2 particles that do not interact via the strong force. The electron, muon, and the electron and muon neutrinos are examples.
• Hadrons are particles that interact via the strong force. They are divided into two sub-categories depending on their spin:
  • Baryons are hadrons with half-integral spin, mainly 1/2 and 3/2. The proton and neutron are well known examples. The neutral lambda particle is another.
  • Mesons are hadrons with integral spin, mainly 0 and 1. Examples are the pions and kaons.
• Strange particles are baryons and mesons that are unstable, but have much longer half-lives than other particles of similar mass and spin. This is interpreted to mean that such particles possess a property called strangeness that is conserved by strong processes, thus making strange particles stable against strong decay into non-strange particles. However, strangeness is not conserved by weak processes, allowing strange particles to decay via the weak interaction, which indeed is much weaker than the strong interaction at low energies. This explains their anomalously long half-lives. Strange particles are always created in pairs by strong processes in such a way that the total strangeness remains zero. For instance, if one particle has strangeness +1 then the other must have strangeness -1. An example of strange particle production is when a negative pion collides with proton, giving rise to a neutral lambda particle and a neutral kaon.
• Intermediary particles are those that transfer energy, momentum, charge, and other properties from one particle to another in association with one of the four fundamental forces.
  • Photons transmit the electromagnetic force and have zero mass and spin 1.
  • Gravitons are thought to transmit the gravitational force, though they have not been directly observed. The graviton is postulated to have zero mass and spin 2.

  We will discover additional intermediary particles in our discussion of the standard model.

• Antiparticles exist for all particles. These have the same mass and spin but opposite values of the electric charge and various other quantum numbers such as lepton number or baryon number. The lepton number is the number of leptons minus the number of antileptons, with a similar definition for baryon number. Thus, a lepton has lepton number 1 and a baryon has baryon number 1. Their antiparticles have lepton number -1 and baryon number -1. As far as we know, baryon number and lepton number are absolutely conserved, which means that baryons and leptons can only be created or destroyed in particle-antiparticle pairs.¹ Antiparticles are represented by the symbol of the particle with an overbar.

20.2 Quantum Chromodynamics

The standard model postulates that all known particles are either fundamental point particles or are composed of fundamental point particles according to a remarkably small set of rules. Just as atoms are bound states of atomic nuclei and electrons, atomic nuclei are bound states of protons and neutrons. Atomic nuclei are discussed in the next section. In this section we delve one step deeper in the heirarchy of the universe. We now believe that all hadrons are actually bound states of fundamental spin 1/2 particles called quarks. Whereas all other known charged particles have an electric charge equal to an integer multiple of ±e where e is the proton charge, quarks have electric charges equal to either -e∕3 or +2e∕3. Leptons themselves are considered to be fundamental, so the leptons and the quarks form the basic building blocks of all matter in the universe.

Quarks are subject to electromagnetic forces via their charge, but interact most strongly via the so-called strong force. The strong force is carried by massless, uncharged, spin 1 bosons called gluons.

When Murray Gell-Mann and George Zweig first proposed the quark model in 1963, they needed to postulate only three types or flavors of quarks: up, down, and strange. These were sufficient to explain the constitution of all hadrons known at the time. We currently know of six different flavors of quarks. Their properties are listed in table 20.1. The properties charm, topness, and bottomness are analogous to strangeness — these properties are conserved in strong interactions. Weak interactions, discussed in the next section, can turn quarks of one flavor into another flavor. However, the strong and electromagnetic forces cannot do this.