146 PARTICULAR CASES OF EQUILIBRIUM. 



For the density to be null at A, the original values of the poten- 

 tial at the two poles of the sphere must be in the ratio of i to 2. 



The density at any other point is negative, and therefore the 

 surface of the sphere is entirely negative as long as V 2 < 2V r 



In the contrary case, the greater part of the surface is still 

 negative, but about the point A there is a more or less extensive 

 zone of positive electricity. 



162. DIELECTRIC SPHERE IN A UNIFORM FIELD. Uniform polari- 

 zation, or electrification by layers of displacement, also represents, 

 on Poisson's theory, the condition of a dielectric in a uniform field. 

 Yet if <f> be the strength of the field, F^ the internal force due to the 

 fictive layer, the resultant force at each point of the interior, instead 

 of being zero, will have a constant value equal to < + F^. 



We can demonstrate that the condition relative to the equilibrium 

 of dielectrics is then satisfied that is to say, that there is a constant 

 ratio over the whole surface between the perpendicular components 

 on the interior and on the exterior. 



This ratio ft, being given by the nature of the dielectric, will 

 enable us to determine the force F^, and consequently the distri- 

 bution of the fictive layer. 



For a point P on the surface in a direction to, the external 

 perpendicular component is < cos to + F n = (< 2F^) cos to, and the 

 internal perpendicular component (</> + F { ) cos to. The ratio of these 

 two forces 



(< - 2F,.) cos to < - 2F,. 

 cos to 



is therefore constant, and we deduce from it 



It thus appears that the problem is completely determinate, and that 

 the state of the sphere is identical with that of a conducting sphere 

 of the same radius situate in a uniform field, the strength of which 



is < ^^ . We deduce from this (159) 



p+2 



3 A*" 1 



