180 Mr. 0. Heaviside on the Resistance 



resistance of an electromagnet should be such or such a frac- 

 tion of the external resistance ; for the result will be different, 

 not only for every speed, but for every different construction 

 of the electromagnet. 



4. Approximate results are, however, easily obtainable in 

 the case of a solenoidal electromagnet. Let its length be I, 

 external radius x, internal radius y, with an iron core of radius z. 

 Its electromagnetic capacity is 



L = |7r 2 Zw 4 (^-?/) 2 (^ + 2^ + 3 < y 2 + 247r^ 2 ) . . (6) 



(* Maxwell,' vol. ii. p. 283), where k is the coefficient of mag- 

 netization of the core. Its resistance is 



R=VV-</ 2 ), 0) 



where p is the resistance of unit of length of wire of unit dia- 

 meter. Therefore 



^=16^5^=2^ (8) 



approximately, by leaving out x 2 + 2ay + Sy 2 in (6) as small in 

 comparison with 24:7tkz 2 , which is a large number, unless the 

 core is very small. Let /e=32 ; also, if the specific resistance 

 of copper be taken at 1*7 microhm = 1700 c. g. s., then 



p=1700x -, and 



IT 



^- = 2-33^— ^z 2 seconds. ..... (9) 



K x-\-y v y 



5. To determine =— ^ for the line wire, Maxwell (vol. ii. 



1 . 

 p. 282) gives the coefficient of self-induction of a straight 



wire, the circuit being completed by a parallel wire. The 



same method of calculation is applicable to any number of 



parallel straight wires, by finding the integral 



T = ^^B.wdxdydz, 



where T is the kinetic energy of the system, and H, iv are 

 the vector-potential and the current at a point, both parallel 

 to the axes of the wires. Thus, for n parallel straight cylin- 

 drical wires of length /, conveying currents Ci, C 2 , . . . , of 

 radii a 1; a 2 , . . . , specific magnetic capacities ^, /a 2 , . . . , repre- 

 senting the distance between the centre of two wires m and n 

 by b mn we shall have 



; 2T 



y=«MiC? + M 2 C* + ...) 



-2/, { (C>g« 1 + C*loga 2 +...) 



- 4/a (C 1 C 2 log ^ 2 + CA log &! 3 + C 2 C 3 log b 2 3 + /.. . ) 



