156 ELEMENTS OF ELECTRICITY AND MAGNETISM. 



pendence of inductance upon the shape of a coil is much too 

 complicated to permit of its general discussion in this text. The 

 ratio of the inductances of two coils of exactly the same shape de- 

 pends, however, in a very simple way upon the sizes of the coils 

 and upon the relative number of turns of wire in them, as follows : 



The inductance of a coil of wire wound on a given spool is pro- 

 portional to the square of the number of turns of wire. Thus, a 

 given spool wound full of number 16 wire has 500 turns, and an 

 inductance of 0.025 henry; the same spool wound full of num- 

 ber 28 wire has ten times as many turns, and its inductance is one 

 hundred times as great, or 2.5 henrys. 



The inductance of a coil of given shape, the number of turns of 

 wire being unchanged, is proportional to its linear dimensions. 

 Thus, if a given spool of wire be imagined to be increased in 

 dimensions in every detail in the ratio of 1:10, size of wire being 

 increased in the same ratio so that the number of turns will be 

 unchanged, then the inductance of the spool would be increased 

 ten times. 



84. Kinetic energy associated with independent currents in two circuits. 

 Definition of mutual inductance. Consider two adjacent circuits one of which 

 may be called the primary circuit and the other the secondary circuit to distinguish 

 them. Let 7 t be the current in the primary circuit and 7 2 in the secondary circuit. 

 The total kinetic energy associated with these two currents consists of three parts : 

 (a) A part which is proportional to 7j squared, (3) a part which is proportional to 



7, squared, and (c) a part which is propor- 

 tional to 7j7 2 . Therefore we may write 



in which W is the total kinetic energy of 

 the two currents, and (JZ, ), (JZ 2 ), and 

 M are the proportionality factors. The 

 quantities Zj and Z 2 are the inductances 

 of the respective circuits inasmuch as equa- 

 tion (i) reduces to equation (48) when either current is zero. The quantity M is 

 called the mutual inductance of the two circuits. It may be either positive or nega- 

 tive. Mutual inductance is expressed in terms of the same units as inductance. 



Proof of equation (i). Consider a point p in the neighborhood of the two cir- 

 cuits. Let h v Fig. 101, be the intensity at p of the magnetic field due to 7j alone, 

 and let A t be the intensity at / of the magnetic field due to 7, alone. The result- 

 ant magnetic field at / is A, as shown in Fig. 101, and we have 



