74 ELECTRICAL EQUILIBRIUM. 



86. GENERAL PROBLEM OF THE RECIPROCAL INFLUENCE OF 

 Two INSULATED CONDUCTORS. MURPHY'S METHOD. In order to 

 determine the distribution of electricity on two insulated conductors 

 A and B, charged with the total masses M a and M 6 and only sub- 

 mitted to their reciprocal action, it is sufficient if we know for each 

 of them : 



ist The capacity and the distribution on the surface when it is 

 insulated and not subject to any external induction ; 



2nd. The distribution of the electricity induced on the surface 

 when it is in connection with the earth, and is subject to the in- 

 ductive action of an electrical mass placed at any point outside it. 



Let m be the capacity of the conductor A alone that is to say, 

 the charge which would then produce potential unit. 



Let us fix this mass, the distribution of which is known, and let 

 us place in the desired position the conductor B in connection with 

 the earth. This will be at potential zero, and will become charged 

 with a known mass of the opposite electricity - m'. 



In like manner let us fix the mass m' on B. Let this conductor 

 be insulated, and let the first one be connected with the earth ; this 

 latter will acquire a mass m 1 at potential zero. 



In like manner let the mass +m 1 be fixed on A, an induced 

 layer - m" will be obtained on B, and so forth. 



Continuing in the same manner, we shall successively obtain the 

 masses m t m v m z ... on the former, and m\ m", m'" ... on the 

 latter, each of them tending to verge rapidly towards zero. 



The superposition of all the layers m, m v m 2 . . . on A, and of all 

 the layers m', m", m'" on B will result in a state of equilibrium 

 with zero potential on B, and potential equal to unity on A. In fact, 

 the successive layers m and - m' t m l and - m", . . . taken in pairs, 

 give zero potential on B ; the layers m' and m v m" and m 2 , . . . give, 

 in like manner, zero potential on A. We have only thus to consider 

 the mass m on the first conductor, which produces a potential equal 

 to unity. 



Putting Cm+mm + ..... , 



we see that C a represents the capacity of the insulated conductor A 

 in the presence of the conductor B connected with the earth, and 

 - C' the coefficient of electricity induced on B (69). 



Multiply these two masses by V , the respective charges C a V a 

 and - C' a V' a correspond to a state of equilibrium with zero potential 

 on B, and potential equal to V a on A. 



