77-79] Spherical Condenser 71 



from P to P' the forces exerted by regions near Q decrease in efficiency, 

 while those exerted by more remote regions gain. The result that the 

 total resultant intensity is the same at P' as at P, shews that the 

 decrease of the one just balances the gain of the other. 



If we replace the infinite plane by a sphere, we find that the force at 

 a near point P is as before contributed 

 almost entirely by the charges in the 

 neighbourhood of Q. On moving from P 

 to P', these forces are diminished just as 

 before, but the number of distant elements 

 of area which now add contributions to 

 the intensity at P' is much less than 

 before. Thus the gain in the contribu- FIG. 30. 



tions from these elements does not suffice 



to balance the diminution in the contributions from the regions near Q, so 

 that the resultant intensity falls off on withdrawing from P to P'. 



The case of a cylinder is of course intermediate between that of a plane 

 and that of a sphere. 



CONDENSERS. 



Spherical Condenser. 



79. Suppose that we enclose the spherical conductor of radius a dis- 

 cussed in 74, inside a second spherical conductor of internal radius 6, the 

 two conductors being placed so as to be concentric and insulated from one 

 another. 



It again appears from symmetry that the intensity at every point must 

 be in a direction passing through the common centre of the two spheres, and 

 must be the same in amount at every point of any sphere concentric with 

 the two conducting spheres. Let us imagine a concentric sphere of radius r 

 drawn between the two conductors, and when the charge on the inner sphere 

 is e, let the intensity at every point of the imaginary sphere of radius r be 

 R. Then, as before, Gauss' Theorem, applied to the sphere of radius r, 

 gives the relation 



so that R - . 



This only holds for values of r intermediate between a and 6, so that to 

 obtain the potential we cannot integrate from infinity, but must use the 

 differential equation. This is 



SVe 



