and Inductance of a Concentric Main. 539 



III. The Formula for the effective Resistance and 

 Inductance of a Concentric Main with a Solid 

 Inner Conductor. 



In order to simplify the problem, we shall first suppose 

 that the inner conductor is a solid metal cylinder o£ radius a, 

 and that the outer conductor is a coaxial hollow cylinder of 

 inner and outer radii b and c respectively. Let fi be the 

 value of the permeability of the metals forming the con- 

 ductors, and let fJ be the permeability of the insulating 

 material separating them. Let p be the volume resistivity of 

 the conducting metal. We shall suppose that /jl, /jl\ and p 

 are constants and, for the present, that both the capacity and 

 leakage currents in the dielectric can be neglected. We can 

 assume, therefore, that the flow of current in the conductors 

 is parallel to their common axis, and hence, that the equi- 

 potential surfaces in each conductor are planes perpendicular 

 to this axis. 



Let us now consider the current in a cylindrical tube of 

 unit length in the inner conductor, whose inner and outer 

 radii are r and r-\-dr respectively. If e Y be the potential 

 difference between the ends of this tube, the equation to 

 determine the current-density i in it is, by Ohm's law and 

 Faraday's law, 



e Y = (p/2«r3r)(t-2iirBr) + 3^/a< 



= pi+Wdt, (59) 



where <£ is the number of magnetic lines linked with the 

 current in this cylindrical tube. 



By hypothesis, the equipotential surfaces in the inner 

 conductor are planes perpendicular to the axis. Hence e x is 

 independent of the value of r, and thus 



°-'! + !S ^ 



From the symmetry of a concentric main, we see that the 

 intensity of the current is the same at all points equidistant 

 from the axis. Hence, since the magnetic force outside an 

 infinite cylindrical tube, carrying a current flowing parallel 

 to its axis, is the same as if all the current were concentrated 

 at this axis, we have, at all points of the inner conductor, 



= 4" ~} 3.v + 2/1 log I +S*Q^£&, (6-1) 



where J z is the algebraical sum of the currents flowing 

 through the cross section of a coaxial cylinder whose radius 



