418 Dr. L. Silberstein on the Electron 



4. Full spherical conductor. — Let the conductor have the 

 total charge \pdr — q, and let there be no " external " field. 

 Then p is obviously a function of r alone, r being the distance 

 from the centre of the conducting sphere. Under these 

 circumstances the differential equation (II) becomes 



^p) -pCrp), 



and its most general integral 



p=-(A?'* + Be-*P) (8) 



Both of the arbitrary constants A, B could be determined 

 from the integral equation and from the given q. In the 

 present case, however, the procedure can be simplified by 

 noting that to avoid p = <x> we must have B = —A*. Thus, 



and the single constant A will be determined from the total 

 charge. (Notice that the relation between B and A could, 

 but the value of A. could not, be determined from (i), since, 

 in the present case, $ e = 0, it is a homogeneous equation.) 

 If R be the radius of the sphere, the total charge is 



q = kir\ r 2 pdr, 



Jo 



which on substitution of the above p gives A without 

 trouble. It is convenient to replace A by the density p at 

 the centre of the sphere, which is p = 2A/L. Thus the 

 final solution becomes 



P=po-smh^ 9 (9) 



where p Q = ql^.irL z l-j- cosh-^ — sinhy\. 



In order to bring into evidence the rapid decrease of density 

 below the surface, compare p with the density p R at the 



surface, thus : 



p = R sinh (r/L) 

 p R r sinh {R/L) ' 



* We shall see, in fact, from the next example that the relation 

 B= —A would follow automatically from the integral equation. 



