POTENTIAL AND CAPACITY 85 



the distance d apart. Now if one cylinder only were charged with 

 surface density or, the corresponding negative charge being infinitely 



distant, the intensity due to it at distance r would be ^TTO- by the 

 preceding investigation. Then the intensity at one cylinder due 

 to the other is practically ^Trcr-j and is very small compared with 



the intensity due to its own charge if -5 is very small. Hence the 



uniformity of distribution on each cylinder will hardly be disturbed 

 by the presence of the other, and we shall obtain a nearly correct 

 result by supposing the charges uniformly distributed on the two 

 cylinders with densities cr. Let us take the potential of one 

 cylinder as and that of the other as V, and let us calculate V by 

 the work done in going straight from one cylinder to the other. 

 At a distance r from the axis of the negative cylinder the intensity is 



r d r 



= d a r- r -\d a 



and V = - /Edr = * ir(Ta lo S JZ 



r a 



log -- - = 87ro-a log - nearly. 



The capacity per unit length is 



d a 



5- 



nearly. 



4 loff - 

 5 a 



This case occurs in practice in two telegraph wires running 

 parallel to each other at a distance from the ground, and equally 

 charged respectively positively and negatively. 



Long thin cylinder parallel to a conducting plane. 

 We may at once deduce the case of a single cylinder running 

 parallel to an indefinitely extended conducting plane a case 

 occurring in practice in a single telegraph wire at a constant height 

 above the ground.* 



If we draw a plane midway between the two cylinders con- 

 sidered above, the lines of force, from symmetry, everywhere cut 



* For a more exact investigation see Heaviside's Electrical Papers, vol. i. 

 p. 42. 



