8.1 Introductory Remarks

In the following we consider circuits of currents, we define inductances L and consider coils and solenoids, and return to gauge transformations. Again examples are given. An important result will be that the product of self-inductance L and capacity C of a transmission line is equal to the product of dielectric constant and magnetic permeability of the surrounding medium. In Chapter 9 it will be shown that the latter product is, in fact, l/c ², where c is the velocity of light. Thus the quantity represents the velocity of the electromagnetic wave travelling along the transmission line.

8.2 Inductances L _ij

We assume that several current-carrying circuits are given. For each, say the k-th circuit, we have

Here ? _k = ? _{F k} B. dF is the magnetic flux through the k-th conductor with enclosed area F _k, and external or generator voltage V _e. Equation (8.1) represents a combination of the usual Ohm's law with Faraday's law. If now the current I _j in circuit j is varied with time, then the magnetic flux through circuit k varies and hence according to the relation ( V _induced) _k = ? ? ? _k/ ? t also the induced voltage ( V _induced) _k, and we can write (using the superposition principle, i.e. the summability of contributions of all circuits)

where the coefficients of...