CONDUCTING SPHERE ,IN A UNIFORM FIELD. 145 



161. UNINSULATED CONDUCTING SPHERE IN A UNIFORM 

 FIELD. It -is easy to pass from the case we have been treating 

 to that of a sphere situate in a uniform field, and in connection with 

 the earth. For this, the internal potential which had the constant 

 value V must be null; this condition is fulfilled by superposing 

 on the preceding condition a uniform layer capable of producing in 

 the interior a potential equal and of opposite sign to V . Let - M' 

 be the mass of this layer and - a-' its density, we shall have 



V - 



whence 



47TO- 



The resultant density at any point will be 



1 V 



(T = (T COS (0 CT' = - < COS 0) -- , 



4?r 471-0 



47ro- = 3$ cos a) -- - . 

 a 



For the density to be zero at A, we must have 



at the pole of the sphere. 



If Vj and V 2 are the original potentials at A' and A, the strength 

 of the field is 



and the potential at the centre 



The preceding condition reduces to 



whence 



