EARNSHAV/S THEOREM. 47 



If the body is connected with the earth, we shall have to consider 

 the two follow ng states of equilibrium : 



Potentials. Charges. 



C Sphere P U m 



i st. Sphere P' V 



\ Body A x 



C Sphere P V 



2nd. ] Sphere P' U' m 



( Body A x' 



Applying Gauss' theorem to these two conditions, we get 



v;y, 



whence 



V = V. 



If the body A is insulated, its potential is not equal to zero, but 

 its charge is null in both states, and the final result is the same. 



II. If each of the two conductors A and B is successively put in 

 connection with a source which raises it to potential V, the other being 

 in connection with the earth, and therefore at zero potential, the quantity 

 of electricity developed by induction on the latter is the same in both cases. 



We have, in fact, in the first case, 



Potential. Charge. 



A V x 



B -M 



and in the second, 



Potential. Charge. 



A -M' 



B V x 



Applying the preceding theorem, we get 



M'V = MV, 

 whence 



M' = M. 



64. EARNSHAW'S THEOREM. An electrified body cannot be in 

 stable equilibrium in an electrical field. 



Let an electrified body A be placed in a field produced by 

 external masses B, and let us suppose all the masses fixed, 

 including that of A. 



