366 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



189. Point- Charge in Sphere. Suppose we have a point- 

 charge e at the center of a sphere of radius R of homogeneous 

 substance of inductivity /4 l5 surrounded by an infinite homogeneous 

 medium of inductivity ytt 2 . Then Laplace's equation being satisfied 

 in either medium, we may use the solution of 142, and put F in 

 either medium equal to a linear function of I/r 



The condition at the surface r = R gives, since 



so that 



rr e v e 



F! = + C, F 2 = , 



since the integral of g n = - p ^- over any surface inclosing e 

 must be 4-7T0. 



The apparent surface density and charge of the sphere are 



'- JLJ^E 1 8F 4_ JL_iP_jL 



4?r 1 tin* dn* \ 4?r R 2 (Lin IL 



The real charge e at the center acts, by 183 (20'), like an 

 apparent charge e/yu. l5 and the apparent charge of the sphere e acts 

 at outside points as if concentrated at the center. Accordingly 

 the whole force in the medium 2 is 



efh + e' e 



which is the same as found from 3 V. 2 /dr. 



190. Unit of Electricity or Magnetism.. If the charge 

 be situated in a medium of inductivity /u, extending to infinity, the 

 force of the field is, by the above, equal to e/pr 2 and the action of 

 e on a charge of ^ units is ^ times as large, or e^/pr 3 . Now the 

 unit charge has been defined as the charge which repels an equal 

 charge placed at the unit of distance from itself with a unit of 

 force. We accordingly see that the magnitude of the unit will 



