A capacitor is a device consisting of two conducting electrodes separated by an insulator. Equal and opposite charges on the electrodes result in a potential difference between them. The amount of charge per unit potential difference is called the capacitance of the capacitor. Large capacitance is favored by large surface areas on the electrodes and small separation between them. The parallel plate capacitor is the prototype, but the capacitance of other configurations can be computed if the scalar potential associated with the charge on the electrodes can be computed.
A capacitor on which the charge is increasing with time must have a current flowing into one electrode and out of the other. The increasing charge results in a correspondingly increasing scalar potential field between the electrodes. However, the Lorentz condition therefore implies the existence of non-zero vector potential, which means that there is also a magnetic field. Examination of this effect in a parallel plate capacitor leads to Maxwell's extension of Ampere's law, which states that the circulation of the magnetic field around a loop is non-zero if either (1) current passes through the loop, or (2) the electric flux through the loop is changing with time, or both.
Magnetic induction occurs when a current flowing in a loop changes with time. The changing current means that the magnetic flux through the loop is also changing. Thus, by Faraday's law, an EMF around the loop is induced -- hence the word ``induction''. The direction of the EMF is such as to oppose the change in the current, and the inductance of the loop is the EMF divided by the time rate of change of current. The parallel plate inductor is the prototype device. The relationship between time-variable vector potential, magnetic field, and electric field is used to derive the inductance of this device.
Resistors are non-conservative devices in which negative work is done on electrons passing through them. The energy extracted is converted to heat. The resistivity is a property of the material used to construct the resistor. The resistance is a function of the resistivity and the dimensions of the resistor.
Capacitors and inductors are conservative devices, in that work done on them results in the storage of recoverable energy. We think of the energy as being stored in the internal electric and magnetic fields in these devices. The resulting formulae for electric and magnetic energy density are not limited to the fields inside these devices, but are actually valid in general.
Kirchhoff's laws are approximations which are used for computing the properties of electric circuits consisting of discrete resistors, capacitors, inductors, and other devices, connected by wires. The resistance, capacitance, and inductance of the wires must be small compared to the values in the discrete components in order for Kirchhoff's laws to work. Furthermore, the EMFs of changing external magnetic fields must be small relative to the EMFs in the circuit components for Kirchhoff's laws to be valid.
Kirchhoff's first law states that the current going into any wire segment is the same as the current coming out the other end. Likewise, if a wire branches into two wires, the net current going in or out of the branch is zero.
Kirchhoff's second law states that the sum of the works done by each device on a positively charged particle passing around a circuit loop is zero. In order to better understand this, recall that a battery or a generator does positive work on moving charge, whereas a resistor does negative work. An inductor does positive work if the current is decreasing with time, whereas it does negative work if the current is increasing. Charge never actually passes through a capacitor, since the two electrodes are separated by an insulator. However, since charge entering one electrode always corresponds to charge leaving the other electrode, it looks like charge is passing through the capacitor. A capacitor does positive or negative work on charge ``passing through'' it depending on the sign of the charge on the capacitor plate into which current is flowing -- if this charge is positive, the capacitor does negative work, since charge is being imported into the capacitor at high potential and exported at low potential. Positive work occurs in the opposite case.