1909-] BETWEEN PARALLEL CONDUCTING CYLINDERS. 



143 



plane Z'OZ. Let the space above the plane unoccupied by the 

 cylinder be filled by an indefinitely extending medium of uniform 

 resistivity p absohm-cm. Then the linear resistance between the 

 plane and the cylinder, i. e., the resistance of the medium between 

 them, as comprised between a pair of infinite parallel planes per- 

 pendicular to the cylinder and i cm. apart, will be 



r = — cosh" 



P 217 



■0 



absohm-cms. or C.G.S. magnetic 

 units of resistance in a linear cm. 



(0 



If the conducting surface EDF of the cylinder were unrolled 

 into a flat conducting ribbon 27ro- cm. in breadth, and the ribbon 

 were supported parallel to the plane Z'OZ at a uniform distance 

 L^=(j coslr^ (c?/o-) cm. above it, as indicated in Fig. 2, with ver- 

 tical insulating side walls, Ez' and Fs, to limit the flow of current 

 through the medium to the parallel distribution shown ; then the 

 rectangular slab of medium EFzz' of Fig. 2, would be the equi- 

 valent in electric resistance to the indefinitely extending plane and 

 cylinder system of Fig. i. 



In Fig. 2 the depth, or distance across the slab, following 

 the lines of current flow, is L = (t coshr'^ {d / cr) cm., and the 



I 



D 



2Tra-- 



4 



Fig. 2. 



Equivalent slab section corresponding to infinite plane and parallel 

 cylinder of Fig. i. 



surface area of each face of the slab, per linear cm. of its length, 

 is 5^ = 27ro- cm. -/cm. so that the linear resistance of the whole is 



^\ = ~F P 

 p ^ r 



(T cosh ' {d\(T) p 

 i ^-^ — - = - - cosh" 



277(7 27r 



■to 



absohm-cm. (2) 



Since the linear resistance of the plane cylinder system of Fig. i, 

 or of the slab in Fig. 2, does not depend upon its absolute dimen- 

 sions, the scale of linear dimensions in the diagram may be chosen 



