Mutual inductance
We now want to consider a situation in which there are fixed coils of wire but changing magnetic fields. When we described the production of magnetic fields by currents, we considered only the case of steady currents. But so long as the currents are changed slowly, the magnetic field will at each instant be nearly the same as the magnetic field of a steady current. We will assume in the discussion of this section that the currents are always varying sufficiently slowly that this is true. In Fig. 17–8 is shown an arrangement of two coils which demonstrates the basic effects responsible for the operation of a transformer. Coil $1$ consists of a conducting wire wound in the form of a long solenoid. Around this coil—and insulated from it—is wound coil $2$, consisting of a few turns of wire. If now a current is passed through coil $1$, we know that a magnetic field will appear inside it. This magnetic field also passes through coil $2$. As the current in coil $1$ is varied, the magnetic flux will also vary, and there will be an induced emf in coil $2$. We will now calculate this induced emf. We have seen in Section 13–5 that the magnetic field inside a long solenoid is uniform and has the magnitude \begin{equation} \label{Eq:II:17:23} B=\frac{1}{\epsO c^2}\,\frac{N_1I_1}{l}, \end{equation}  where $N_1$ is the number of turns in coil $1$, $I_1$ is the current through it, and $l$ is its length. Let’s say that the cross-sectional area of coil $1$ is $S$; then the flux of $\FLPB$ is its magnitude times $S$. If coil $2$ has $N_2$ turns, this flux links the coil $N_2$ times. Therefore the emf in coil $2$ is given by \begin{equation} \label{Eq:II:17:24} \emf_2=-N_2S\,\ddt{B}{t}. \end{equation}  The only quantity in Eq. (17.23) which varies with time is $I_1$. The emf is therefore given by \begin{equation} \label{Eq:II:17:25} \emf_2=-\frac{N_1N_2S}{\epsO c^2l}\,\ddt{I_1}{t}. \end{equation}  We see that the emf in coil $2$ is proportional to the rate of change of the current in coil $1$. The constant of proportionality, which is basically a geometric factor of the two coils, is called the mutual inductance, and is usually designated $\mutualInd_{21}$. Equation (17.25) is then written \begin{equation} \label{Eq:II:17:26} \emf_2=\mutualInd_{21}\,\ddt{I_1}{t}. \end{equation}  Suppose now that we were to pass a current through coil $2$ and ask about the emf in coil $1$. We would compute the magnetic field, which is everywhere proportional to the current $I_2$. The flux linkage through coil $1$ would depend on the geometry, but would be proportional to the current $I_2$. The emf in coil $1$ would, therefore, again be proportional to $dI_2/dt$: We can write \begin{equation} \label{Eq:II:17:27} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}. \end{equation}  The computation of $\mutualInd_{12}$ would be more difficult than the computation we have just done for $\mutualInd_{21}$. We will not carry through that computation now, because we will show later in this chapter that $\mutualInd_{12}$ is necessarily equal to $\mutualInd_{21}$. Since for any coil its field is proportional to its current, the same kind of result would be obtained for any two coils of wire. The equations (17.26) and (17.27) would have the same form; only the constants $\mutualInd_{21}$ and $\mutualInd_{12}$ would be different. Their values would depend on the shapes of the coils and their relative positions. Suppose that we wish to find the mutual inductance between any two arbitrary coils—for example, those shown in Fig. 17–9. We know that the general expression for the emf in coil $1$ can be written as \begin{equation*} \emf_1=-\ddt{}{t}\int_{(1)}\FLPB\cdot\FLPn\,da, \end{equation*}  where $\FLPB$ is the magnetic field and the integral is to be taken over a surface bounded by circuit $1$. We have seen in Section 14–1 that such a surface integral of $\FLPB$ can be related to a line integral of the vector potential. In particular, \begin{equation*} \int_{(1)}\FLPB\cdot\FLPn\,da=\oint_{(1)}\FLPA\cdot d\FLPs_1, \end{equation*}  where $\FLPA$ represents the vector potential and $d\FLPs_1$ is an element of circuit $1$. The line integral is to be taken around circuit $1$. The emf in coil $1$ can therefore be written as \begin{equation} \label{Eq:II:17:28} \emf_1=-\ddt{}{t}\oint_{(1)}\FLPA\cdot d\FLPs_1. \end{equation}  Now let’s assume that the vector potential at circuit $1$ comes from currents in circuit $2$. Then it can be written as a line integral around circuit $2$: \begin{equation} \label{Eq:II:17:29} \FLPA=\frac{1}{4\pi\epsO c^2}\oint_{(2)}\frac{I_2\,d\FLPs_2}{r_{12}}, \end{equation}  where $I_2$ is the current in circuit $2$, and $r_{12}$ is the distance from the element of the circuit $d\FLPs_2$ to the point on circuit $1$ at which we are evaluating the vector potential. (See Fig. 17–9.) Combining Eqs. (17.28) and (17.29), we can express the emf in circuit $1$ as a double line integral: \begin{equation*} \emf_1=-\frac{1}{4\pi\epsO c^2}\,\ddt{}{t}\oint_{(1)}\oint_{(2)} \frac{I_2\,d\FLPs_2}{r_{12}}\cdot d\FLPs_1. \end{equation*}  In this equation the integrals are all taken with respect to stationary circuits. The only variable quantity is the current $I_2$, which does not depend on the variables of integration. We may therefore take it out of the integrals. The emf can then be written as \begin{equation*} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}, \end{equation*}  where the coefficient $\mutualInd_{12}$ is \begin{equation} \label{Eq:II:17:30} \mutualInd_{12}=-\frac{1}{4\pi\epsO c^2}\oint_{(1)}\oint_{(2)} \frac{d\FLPs_2\cdot d\FLPs_1}{r_{12}}. \end{equation}  We see from this integral that $\mutualInd_{12}$ depends only on the circuit geometry. It depends on a kind of average separation of the two circuits, with the average weighted most for parallel segments of the two coils. Our equation can be used for calculating the mutual inductance of any two circuits of arbitrary shape. Also, it shows that the integral for $\mutualInd_{12}$ is identical to the integral for $\mutualInd_{21}$. We have therefore shown that the two coefficients are identical. For a system with only two coils, the coefficients $\mutualInd_{12}$ and $\mutualInd_{21}$ are often represented by the symbol $\mutualInd$ without subscripts, called simply the mutual inductance: \begin{equation*} \mutualInd_{12}=\mutualInd_{21}=\mutualInd. \end{equation*} 