ENERGY OF THE DIELECTRIC MEDIUM. 93 



The total energy of a system is (91) 



1 if 



W = - V m V, or W = - Vpdv. 



2 2j 



Replacing the density by its value deduced from Poisson's 

 equation 



AV + 47175 = 0, 

 we get 



W= - 



Hitherto the energy has been determined as a function of the 

 electrical masses themselves. In order to vary the signification, we 

 may apply Green's formula (33) 



fvAWz> = fv^^S- (^dv 

 J *n 



to the volume bounded by a sphere of very large radius r which 

 includes the electrical system we are considering. The first term 

 of the second member should be extended to the surface of this 

 sphere. The potential V, as we recede, tends to become inversely 



<)V 



as r; the factor represents the perpendicular component of the 

 on 



force, and becomes inversely as r 2 . As the surface itself is pro- 

 portional to r 2 , this integral is inversely as r, and tends towards 

 zero. 



The second member reduces then to the second term, and we 

 have, for the expression of the energy, 



S7T 



It appears from this that the energy of the system is the same as 

 if each volume element of the medium had a quantity of energy 



F 2 </z>. The energy w for unit volume is accordingly 



OTT 



