Exceptions to the “flux rule”
We will now give some examples, due in part to Faraday, which show the importance of keeping clearly in mind the distinction between the two effects responsible for induced emf’s. Our examples involve situations to which the “flux rule” cannot be applied—either because there is no wire at all or because the path taken by induced currents moves about within an extended volume of a conductor. We begin by making an important point: The part of the emf that comes from the $\FLPE$-field does not depend on the existence of a physical wire (as does the $\FLPv\times\FLPB$ part). The $\FLPE$-field can exist in free space, and its line integral around any imaginary line fixed in space is the rate of change of the flux of $\FLPB$ through that line. (Note that this is quite unlike the $\FLPE$-field produced by static charges, for in that case the line integral of $\FLPE$ around a closed loop is always zero.) Now we will describe a situation in which the flux through a circuit does not change, but there is nevertheless an emf. Figure 17–2 shows a conducting disc which can be rotated on a fixed axis in the presence of a magnetic field. One contact is made to the shaft and another rubs on the outer periphery of the disc. A circuit is completed through a galvanometer. As the disc rotates, the “circuit,” in the sense of the place in space where the currents are, is always the same. But the part of the “circuit” in the disc is in material which is moving. Although the flux through the “circuit” is constant, there is still an emf, as can be observed by the deflection of the galvanometer. Clearly, here is a case where the $\FLPv\times\FLPB$ force in the moving disc gives rise to an emf which cannot be equated to a change of flux. Now we consider, as an opposite example, a somewhat unusual situation in which the flux through a “circuit” (again in the sense of the place where the current is) changes but where there is no emf. Imagine two metal plates with slightly curved edges, as shown in Fig. 17–3, placed in a uniform magnetic field perpendicular to their surfaces. Each plate is connected to one of the terminals of a galvanometer, as shown. The plates make contact at one point $P$, so there is a complete circuit. If the plates are now rocked through a small angle, the point of contact will move to $P'$. If we imagine the “circuit” to be completed through the plates on the dotted line shown in the figure, the magnetic flux through this circuit changes by a large amount as the plates are rocked back and forth. Yet the rocking can be done with small motions, so that $\FLPv\times\FLPB$ is very small and there is practically no emf. The “flux rule” does not work in this case. It must be applied to circuits in which the material of the circuit remains the same. When the material of the circuit is changing, we must return to the basic laws. The correct physics is always given by the two basic laws \begin{align*} &\FLPF=q(\FLPE+\FLPv\times\FLPB),\\[1ex] &\FLPcurl{\FLPE}=-\ddp{\FLPB}{t}. \end{align*} 