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precisely, Faraday s law states that a time-varying ux causes an induced electromotive force, or emf, e, as follows: d (165) e= dt A little discussion is necessary at this point to explain the meaning of the minus sign in equation 165 Consider the one-turn coil of Figure 164, which forms a circular cross-sectional area, in the presence of a magnetic eld with ux density B oriented in a direction perpendicular to the plane of the coil If the magnetic eld, and therefore the ux within the coil, is constant, no voltage will exist across terminals a and b; if, however, the ux were increasing and terminals a and b were connected for example, by means of a resistor, as indicated in Figure 164(b) current would ow in the coil in such a way that the magnetic ux generated by the current would oppose the increasing ux Thus, the ux induced by such a current would be in the direction opposite to that of the original ux density vector, B This principle is known as Lenz s law The reaction ux would then point downward in Figure 164(a), or into the page in Figure 164(b) Now, by virtue of the right-hand rule, this reaction ux would induce a current owing clockwise in Figure 164(b), that is, a current that ows out of terminal b and into terminal a The resulting voltage across the hypothetical resistor R would then be negative If, on the other hand, the original ux were decreasing, current would be induced in the coil so as to reestablish the initial ux; but this would mean that the current would have to generate a ux in the upward direction in Figure 164(a) (or out of the page in Figure 164(b)) Thus, the resulting voltage would change sign The polarity of the induced voltage can usually be determined from physical considerations; therefore the minus sign in equation 165 is usually left out We will use this convention throughout the chapter In practical applications, the size of the voltages induced by the changing magnetic eld can be signi cantly increased if the conducting wire is coiled many times around, so as to multiply the area crossed by the magnetic ux lines many times over For an N-turn coil with cross-sectional area A, for example, we have the emf d (166) e=N dt Figure 165 shows an N -turn coil linking a certain amount of magnetic ux; you can see that if N is very large and the coil is tightly wound (as is usually the case in the construction of practical devices), it is not unreasonable to presume that each turn of the coil links the same ux It is convenient, in practice, to de ne the ux linkage, , as = N so that d dt (167).

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(168)

Note that equation 168, relating the derivative of the ux linkage to the induced emf, is analogous to the equation describing current as the derivative of charge: dq i= (169) dt

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In other words, ux linkage can be viewed as the dual of charge in a circuit analysis sense, provided that we are aware of the simplifying assumptions just stated in the preceding paragraphs, namely, a uniform magnetic eld perpendicular to the area delimited by a tightly wound coil These assumptions are not at all unreasonable when applied to the inductor coils commonly employed in electric circuits What, then, are the physical mechanisms that can cause magnetic ux to change, and therefore to induce an electromotive force Two such mechanisms are possible The rst consists of physically moving a permanent magnet in the vicinity of a coil for example, so as to create a time-varying ux The second requires that we rst produce a magnetic eld by means of an electric current (how this can be accomplished is discussed later in this section) and then vary the current, thus varying the associated magnetic eld The latter method is more practical in many circumstances, since it does not require the use of permanent magnets and allows variation of eld strength by varying the applied current; however, the former method is conceptually simpler to visualize The voltages induced by a moving magnetic eld are called motional voltages; those generated by a timevarying magnetic eld are termed transformer voltages We shall be interested in both in this chapter, for different applications In the analysis of linear circuits in 4, we implicitly assumed that the relationship between ux linkage and current was a linear one: = Li (1610)

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so that the effect of a time-varying current was to induce a transformer voltage across an inductor coil, according to the expression v=L di dt (1611)

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