The physics of induction
In the last chapter we described many phenomena which show that the effects of induction are quite complicated and interesting. Now we want to discuss the fundamental principles which govern these effects. We have already defined the emf in a conducting circuit as the total accumulated force on the charges throughout the length of the loop. More specifically, it is the tangential component of the force per unit charge, integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels once around the circuit. We have also given the “flux rule,” which says that the emf is equal to the rate at which the magnetic flux through such a conducting circuit is changing. Let’s see if we can understand why that might be. First, we’ll consider a case in which the flux changes because a circuit is moved in a steady field. In Fig. 17–1 we show a simple loop of wire whose dimensions can be changed. The loop has two parts, a fixed U-shaped part (a) and a movable crossbar (b) that can slide along the two legs of the U. There is always a complete circuit, but its area is variable. Suppose we now place the loop in a uniform magnetic field with the plane of the U perpendicular to the field. According to the rule, when the crossbar is moved there should be in the loop an emf that is proportional to the rate of change of the flux through the loop. This emf will cause a current in the loop. We will assume that there is enough resistance in the wire that the currents are small. Then we can neglect any magnetic field from this current. The flux through the loop is $wLB$, so the “flux rule” would give for the emf—which we write as $\emf$— \begin{equation*} \emf=wB\,\ddt{L}{t}=wBv, \end{equation*}  where $v$ is the speed of translation of the crossbar. Now we should be able to understand this result from the magnetic $\FLPv\times\FLPB$ forces on the charges in the moving crossbar. These charges will feel a force, tangential to the wire, equal to $vB$ per unit charge. It is constant along the length $w$ of the crossbar and zero elsewhere, so the integral is \begin{equation*} \emf=wvB, \end{equation*}  which is the same result we got from the rate of change of the flux. The argument just given can be extended to any case where there is a fixed magnetic field and the wires are moved. One can prove, in general, that for any circuit whose parts move in a fixed magnetic field the emf is the time derivative of the flux, regardless of the shape of the circuit. On the other hand, what happens if the loop is stationary and the magnetic field is changed? We cannot deduce the answer to this question from the same argument. It was Faraday’s discovery—from experiment—that the “flux rule” is still correct no matter why the flux changes. The force on electric charges is given in complete generality by $\FLPF=q(\FLPE+\FLPv\times\FLPB)$; there are no new special “forces due to changing magnetic fields.” Any forces on charges at rest in a stationary wire come from the $\FLPE$ term. Faraday’s observations led to the discovery that electric and magnetic fields are related by a new law: in a region where the magnetic field is changing with time, electric fields are generated. It is this electric field which drives the electrons around the wire—and so is responsible for the emf in a stationary circuit when there is a changing magnetic flux. The general law for the electric field associated with a changing magnetic field is \begin{equation} \label{Eq:II:17:1} \FLPcurl{\FLPE}=-\ddp{\FLPB}{t}. \end{equation}  We will call this Faraday’s law. It was discovered by Faraday but was first written in differential form by Maxwell, as one of his equations. Let’s see how this equation gives the “flux rule” for circuits.  Using Stokes’ theorem, this law can be written in integral form as  \begin{equation} \label{Eq:II:17:2} \oint_\Gamma\FLPE\cdot d\FLPs= \int_S(\FLPcurl{\FLPE})\cdot\FLPn\,da= -\int_S\ddp{\FLPB}{t}\cdot\FLPn\,da, \end{equation} 
 \begin{equation} \begin{aligned} \oint_\Gamma\FLPE\cdot d\FLPs &= \int_S(\FLPcurl{\FLPE})\cdot\FLPn\,da\\[1ex] &= -\int_S\ddp{\FLPB}{t}\cdot\FLPn\,da, \end{aligned} \label{Eq:II:17:2} \end{equation}  where, as usual, $\Gamma$ is any closed curve and $S$ is any surface bounded by it. Here, remember, $\Gamma$ is a mathematical curve fixed in space, and $S$ is a fixed surface. Then the time derivative can be taken outside the integral and we have \begin{equation} \begin{aligned} \oint_\Gamma\FLPE\cdot d\FLPs& =-\ddt{}{t}\int_S\FLPB\cdot\FLPn\,da \\[1ex] &=-\ddt{}{t}(\text{flux through $S$}). \end{aligned} \label{Eq:II:17:3} \end{equation}  Applying this relation to a curve $\Gamma$ that follows a fixed circuit of conductor, we get the “flux rule” once again. The integral on the left is the emf, and that on the right is the negative rate of change of the flux linked by the circuit. So Eq. (17.1) applied to a fixed circuit is equivalent to the “flux rule.” So the “flux rule”—that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit—applies whether the flux changes because the field changes or because the circuit moves (or both). The two possibilities—“circuit moves” or “field changes”—are not distinguished in the statement of the rule. Yet in our explanation of the rule we have used two completely distinct laws for the two cases—$\FLPv\times\FLPB$ for “circuit moves” and $\FLPcurl{\FLPE}=-\ddpl{\FLPB}{t}$ for “field changes.” We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication. We have to understand the “rule” as the combined effects of two quite separate phenomena.  We must look at the “flux rule” in the following way. In general, the force per unit charge is $\FLPF/q=\FLPE+\FLPv\times\FLPB$. In moving wires there is the force from the second term. Also, there is an $\FLPE$-field if there is somewhere a changing magnetic field. They are independent effects, but the emf around the loop of wire is always equal to the rate of change of magnetic flux through it.