CONDITIONS OF. THE EQUILIBRIUM OF CONDUCTORS. 51 



CHAPTER IV. 

 ELECTRICAL EQUILIBRIUM. 



65. CONDITIONS OF THE EQUILIBRIUM OF CONDUCTORS. The 

 general problem of electrical equilibrium, when restricted to the case 

 of conductors, may be thus enunciated : 



With conductors of given shape and position, one set insulated, and 

 the other in connection with the earth, the former being charged with a 

 definite quantity of electricity, it is required to determine the potential at 

 each point. 



In other words, the problem amounts to determining a function 

 V of co-ordinates, which satisfies the following conditions : 



i st. The function must be zero at an infinite distance, and must 

 have a constant value on each of the conductors, this value being 

 zero on all conductors in communication with the earth. 



2nd. The sum of the three second partial differentials must be 

 zero over the whole surface of the dielectrics and in the interior of 

 conductors, for the electrical density is zero at all these points. 



3rd. At every point of the surface of the conductors, the density 

 is determined by the equation 



i </V 

 4ir dn 



so that the total charge upon one of the conductors is expressed by 



M= fov/S=- f <JS. 

 J 4^J dn 



These three conditions are sufficient, for they determine a state of 

 equilibrium which satisfies the data of the question, and only one 

 state of equilibrium is possible for the system. 



This problem frequently presents great difficulties from the 

 mathematical point of view, and no general solution is known. It 

 has only been completely solved for a few special cases, the most 



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