Impulsive Motion of Electrified Systems. 133 



Thus the energy carried off in the pulse in the case of the 

 volume-charge is six fifths o£ the energy carried off in the 

 case of a surface-charge. 



In exactly the same way, it follows that P for a s[ there 

 with a volume-charge is six fifths of the value for a sphere 

 with an equal surface-charge. 



The electrostatic potential at a point within the sphere at a 

 distance r from its centre, when the sphere is at rest, is easily 

 found to be 



QC3a 2 -r 2 )/2Ka% 



and hence by the formula 



U =iX charge x potential, 



the electric energy is given by 



TjJsQl 



and this is six fifths of the energy when the charge is on the 

 surface. 



It therefore follows, by § 15, that the values of U, T, and 

 M for a sphere with a volume-charge are six fifths as great 

 as for the same sphere with an equal surface-charge*. 



§ 21. Pair of concentric spheres with uniformly distributed 

 complementary charges. — Let the outer sphere have the radius 

 -a and the charge Q, while the inner sphere has the radius h 

 and the charge — Q. Then from ~= — a to z=—l> and also 

 from z — b to z= a, 



dQ/dz = QJ2a, 



but from z = — b to z = b 



dQ/dz=-Q(a-b)/2ab. 



Thus C +a fdQ\* _ Q%a-l>) Q 2 (a-b) - Q s a-b 



J^Adz) 2a 2 + 2a% ~ 2a' b' 



For the outer sphere alone 



r ;(©*'-£■ 



and the value of U for the pair of spheres is Q 2 («— b)/2abK. 

 Thus the values of TV, P, and U for the system are (a — b)jh 



* Dr. M. Abraham has proved similar results for an ellipsoid by a 

 method not depending- on the calculation of "W and P for the pulse. 

 See his Theorie der EleMrizitat, vol. ii. p. 170. 



