INDUCTANCES AND RESISTANCES 



Mutual inductance 



When two coils of wire are arranged so that some of the flux caused by 

 one threads the windings of the other, the arrangement is known as a 'mutual 

 inductance'. It is found that an e.m.f. appears across the ends of one winding 

 proportional to the rate of change of current in the other. In Figure 4.4, if 



'M 



-> 



J — iJ 



«'2 



i 



/\ 



Figure 4.4 



current changes in the primary winding at an instantaneous rate of one 

 ampere per second, and if the e.m.f. induced into the secondary winding 

 is one volt, then the value of the mutual inductance is one henry 



^2 = M d/'i/d/ (volts, henries, amps per second) 



Mutual inductance is reciprocal, in the sense that, in addition, 



e-^ = M dijdt 



Self inductance 



Since the flux produced by a coil threads the coil itself, it is not surprising 

 that, upon varying the current, an e.m.f. appears across the terminals of the 

 self-same coil. If the current in the winding is varied at the rate of one 

 ampere per second, and if an e.m.f. of one volt appears across the coil ter- 

 minals, then the 'self inductance' of the coil is one henry {Figure 4.5) and 



e-^ = — L dijdt (volts, henries, amps per second) 



The negative sign implies that the polarity of the induced e.m.f. is such as 

 to oppose the flow of current if the latter is increasing, and vice versa ; that 

 is, it is a 'back e.m.f.'. 



The magnitude of the back e.m.f. is proportional to the rate of change of 

 lines of force threading the circuit. By looping the circuit into a coil of N 

 turns, the same flux is made to thread the circuit A^ times. 



Thus 

 but 



e = K^Ndcf>ldt 



so 



and 



d^ 

 d7 



^2 



K, di 



L 



K^l dt 



^TTK J 



Figure 4.5 



Comparing this with e = —L dijdt, it appears that L = {—(/liK^K^)I(K.J)]N^ 

 that is, the self inductance of a coil upon a given core is proportional to the 



56 



