Self-induction of Wires. 275 



if Ei and E 3 are the longitudinal electric forces "of the field. " 

 Therefore 



e 1 -e 3 = e=p i T 1 -p 3 T 3 -(E l -V 3 ), . . . (54) 



where e is the impressed force per unit length in the circuit 

 at the place considered ; the positive direction in the circuit 

 being along the wire in the direction of increasing z, and 

 oppositely in the return. 



If in (51) we lake r=a 1} and r = a 2 in (52), and use them 

 in (54), then, since C, becomes C, the wire current, and 3 

 becomes the same phis the longitudinal dielectric current, if we 

 agree to ignore the latter, and can put E x — E 3 in terms of 0, 

 (54) will become an equation between e and C. r=a 2 



To obtain the required E x — E 3 , consider 

 a rectangular circuit in a plane through 

 the axis, two of whose sides are of unit 

 length parallel to z at distances a x and a 2 

 from the axis, and the other two sides 

 parallel to r, and calculate the E.M.F. of 

 the field in this circuit in the direction of 

 the circular arrow. If z be positive from — *- z 



left to right, the positive direction of the magnetic force 

 through the circuit is upward through the paper. Therefore, 

 if V be the line integral of the radial electric force from 7* = % 

 to r = a 2 , so that dY/dz is the part of the E.M.F. in the rect- 

 angular circuit due to the radial force, we shall have 



dY T"2 . 



Ei — E 3 + — = - \ fi 2 B. 2 dr, 



«J a-, 



by the Faraday law, or equation (7) ; H 2 being the magnetic 

 force in the dielectric. This being 2G/r, on account of our 

 neglect of T 2 , we get, on performing the integration, — L C, 

 on the right side, where L is the previously used inductance 

 of the dielectric per unit length. This brings (54) to 



e _dV =L c+ PiSj Jofa^-fJi/KQfa^Kofaa!) 

 dz ° 277% J 1 . . . — K x . . . 



0,(55) 



_ Ps^s Jq(s 3 — (Ji/K 1 )(g 3 q 3 )K (g 3 q 2 ) 



27ra 2 Ji ... — Kj . . . 



which, for brevity, write thus, 



dY 

 e-^=L ^C+R/C+B/C ; . . . (56) 



where R/ and R 2 " define themselves in (55). They are gene- 

 ralized resistances of wire and return respectively, per unit 

 length. But of their structure, later. Equation (56) is what 



