272 ELECTROSTATICS. [PT. II. CH. VI. 



Example. In the case of a single conductor, the coefficients 

 q and p reduce in each case to a single one, the capacity and its 

 reciprocal respectively, and 



e = qV, 



V=pe, 



If the conductor is a sphere of radius r, we have 





The only geometrical parameter <f> is here r, and since W e 

 tends to decrease, W v to increase, r tends to increase. 



If the sphere is elastic, as in the case of a soap-bubble, the 

 electrical forces tending to enlarge the sphere may be held in 

 equilibrium by a greater pressure of the air on the outside than on 

 the inside, or by the surface tension of the film. If P denote this 

 excess of pressure, that is, the force acting normally on a unit 

 of surface, the work done by the whole surface 8 in increasing the 

 radius by dr is PSdr. If T is the surface tension of the film, or 

 the elastic force tangential to the surface exerted normally across 

 a curve on the surface per unit of length, in increasing the surface 

 by dS we must do work TdS. Hence we have 



= PSdr + TdS = -dW e = dW V) 



S = *irr\ dS = Sirrdr, W e = 



47r {Pr 2 + 22V} dr = | ^ dr = \ 

 e* = 8-Trr 3 {Pr + 2T\, 



If the soap-bubble be blown on a tube connected with a mano- 

 meter, the difference in pressure P may be observed. T may be 

 determined by an observation when the bubble is unelectrified. 

 Calling r the radius under these circumstances, P the pressure, 



T = -- p , Po being negative, 



