Self-induction of Wires. 341 



if V is the line-integral of the radial electric force across the 

 dielectric. Or put non-conducting and non-dielectric plates 

 there similarly. This will make C = 0. Or, which is the fifth 

 case, let the inner and the outer conductors be closed upon 

 themselves. In any of these cases, the electric force will 

 remain longitudinal during the subsidence, which will take 

 place similarly all along the line. By (14), the equation of H 

 will be 



- — rH = 47r&uH + ucfl ; 



dr r dr 



and it is clear that the normal functions are quite definite, so 

 that the expansion of the initial state of E and H can be truly 

 effected. In the already given normal functions take ra = 0. 



But if we were to join the conductors at one end of the line 

 through a resistance, we should, to some extent, upset this 

 regular subsidence everywhere alike. For energy would leave 

 the line ; this would cause radial displacement, first at the end 

 where the resistance was attached, and later all along the 

 line. (By " the line M is meant, for brevity, the system of tubes 

 extending from z = to z=l.) 



Now in short-wire problems the electric energy is of insig- 

 nificant importance, as compared with the magnetic. It is 

 usual to ignore it altogether. This w T e can do by assuming 

 c = 0. This necessitates equality of wire and return current, 

 for one thing ; but, more importantly, it prevents current 

 leaving the conductors, so that C and H and T the current- 

 density, are independent of z. There will be no radial electric 

 force in the conductors, in which therefore the energy-current 

 will be radial. But there will be radial force in the dielectric, 

 and therefore longitudinal energy-current. Since the radial 

 electric force and also the magnetic force in the dielectric 

 vary inversely as the distance from the axis, the longitudinal 

 energy-current density will vary inversely as the square of 

 the distance. But, on account of symmetry, we are only 

 concerned with its total amount over the complete section of 

 the dielectric. This is 



^p^.E r .2 OT *=VC, . . . (118) 



if Y is the line-integral of E r the radial force, and C the wire- 

 current. It is clear, then, that we can now allow terminal 

 connections of the form Y/C = Z before used, and still have 

 correct expansions of the initial magnetic field, giving correct 

 subsidence solutions. 



But it is simpler to ignore Y altogether. For the equation 



