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A coil containing N turns with magnetic ux linking each turn has total magnetic ux linkage N . By Faraday s law, the induced emf (voltage) in the coil is e d =dt N d =dt . A negative sign is frequently included in this equation to signal that the voltage polarity is established according to Lenz s law. By de nition of self-inductance this voltage is also given by L di=dt ; hence, L di d N dt dt or L N d di 5a

Fig. 4-20

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The unit of being the weber, where 1 Wb 1 V s, it follows from the above relation that 1 H 1 Wb=A. Throughout this book it has been assumed that and i are proportional to each other, making L N constant i 5b

Fig. 4-21

(b) Let x1 be the required maximum grade and z1 its equivalent in standard units. From Fig. 4-21 the area to the left of z1 is 10% 0.10; hence, Area between z1 and 0 and z1 1.28 (very closely). Then z1 (x1 6.7)> 1.2 1.28 and x1 0.40

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In Fig. 14-3, the total ux 1 resulting from current i1 through the turns N1 consists of leakage ux, 11 , and coupling or linking ux, 12 . The induced emf in the coupled coil is given by N2 d 12 =dt). This same voltage can be written using the mutual inductance M: e M di1 d 12 N2 dt dt or M N2 d 12 di1 6

(c) Let x2 be the required minimum grade and z2 the same grade in standard units. From (b), by symmetry, z2 1.28. Then (x2 6.7)> 1.2 1.28, and x2 8.2 or 8 to the nearest integer.

Fig. 14-3

4.59. A Geiger counter is used to count the arrivals of radioactive particles. Find the probability that in time t no particles will be counted.

Let Fig. 4-22 represent the time axis with O as the origin. The probability that a particle is counted in a small time t is proportional to t and so can be written as t. Therefore, the probability of no count in time t is 1 t. More precisely, there will be additional terms involving ( t)2 and higher orders, but these are negligible if t is small.

where 0 k 1. Taking the product of (6) and (7) and assuming that k depends only on the geometry of the system, d 12 d 21 d k 1 d k 2 d 1 d 2 M 2 N2 N1 N2 N1 k2 N1 N2 k2 L1 L2 di1 di2 di1 di2 di1 di2 p p or XM k X1 X2 (8) from which M k L1 L2 p Note that (8) implies that M L1 L2 , a bound that may be independently derived by an energy argument. If all of the ux links the coils without any leakage ux, then k 1. On the other extreme, the coil axes may be oriented such that no ux from one can induce a voltage in the other, which results in k 0. The term close coupling is used to describe the case where most of the ux links the coils, either by way of a magnetic core to contain the ux or by interleaving the turns of the coils directly over one another. Coils placed side-by-side without a core are loosely coupled and have correspondingly low values of k.

Fig. 2-7.

Fig. 4-22

14.3 ANALYSIS OF COUPLED COILS Polarities in Close Coupling In Fig. 14-4, two coils are shown on a common core which channels the magnetic ux . This arrangement results in close coupling, which was mentioned in Section 14.2. To determine the proper signs on the voltages of mutual inductance, apply the right-hand rule to each coil: If the ngers wrap

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