Physics 222 -- Notes on Chapter 15

The main themes of this chapter are (1) to show how electromagnetism is a particular example of a gauge theory, and (2) how electromagnetic forces work.

Electromagnetic forces work on particles which have electric charge. The potential energy of a charged particle equals the charge times the scalar potential. The potential momentum of a charged particle equals the charge times the vector potential. The four-potential is a four-vector with the vector potential as the spacelike part and the scalar potential divided by c as the timelike part.

Electric and magnetic fields are defined by rather messy formulas in terms of space and time derivatives of the scalar and vector potentials. The definitions of these fields are historical and not strictly needed -- we could do everything in terms of the scalar and vector potential without worrying about electric and magnetic fields. However, nobody else would know what we were talking about if we did this!

We write the force on charged particles in terms of the electric and magnetic fields. However, since the components of the four-potential differ in different reference frames, so do the components of the electric and magnetic fields. Thus, whether the electric or magnetic force, or some mixture of the two, is operative in a particular case is a reference frame-dependent question. However, the total force will be the same in different reference frames, at least as long as the different frames are moving at non-relativistic speeds with respect to each other.

Most of the rest of the chapter is really F = ma mechanics applied to charged particles subject to electric and magnetic fields. Certain of these situations are worthy of special mention:

• Moving charged particles tend to spiral around magnetic field lines.
• Crossed electric and magnetic fields can result in a charged particle moving at a constant velocity such that the total electromagnetic force is zero. This is called ``E cross B drift''.
• The current in a wire is the charge passing through the wire per unit time. It can be related to the density of moving charge in the wire times the velocity of this charge.
• The electric dipole consists of equal positive and negative charges spaced by a fixed distance. The electric dipole moment is a vector which points from negative to positive charge and has a magnitude equal to the positive charge times separation distance.
• The magnetic dipole consists of a loop of current. The magnetic dipole moment points normal to the plane of the current in a direction defined by the right-hand rule. The magnitude is the current times the area of the current loop.
• Electric and magnetic dipoles tend to line up with the electric and magnetic fields respectively, and tend to be drawn toward stronger fields when so aligned.
• An electric motor works via the torque which tends to align a magnetic dipole with the magnetic field.

An electric generator does work on charges passing through it via the non-conservative part of the electric field derived from a time-varying vector potential. This time variability leads to a time-varying magnetic field as well. Thus, time-varying magnetic fields and non-conservative electric fields go together. This is formalized in Faraday's law.

Get the quantitative details from the book.