166-169] Energy in the Medium 149 



ENERGY IN THE MEDIUM. 



168. In setting up the system of stresses in a medium originally un- 

 stressed, work must be done, analogous to the work done in compressing 

 a gas. This work must represent the energy of the stressed medium, and 

 this in turn must represent the energy of the electrostatic field. Clearly, 

 from the form of the stresses, the energy per unit volume of the medium 

 at any point must be a function of R only. To determine the form of this 

 function, we may examine the simple case of a parallel plate condenser, 



R* 



arid we find at once that the function must be ^ . 



07T 



We have now to examine whether the energy of any electrostatic field 



7? 2 



can be regarded as made up of a contribution of amount ^ per unit volume 



from every part of the field. 



In fig. 51, let PQ be a tube of force of strength e, passing from P at 

 potential Vp to Q at potential VQ. The ether inside this tube of force 



7? 2 

 being supposed to possess energy ^ per unit volume, | 



the total energy enclosed by the tube will be 



^Q 



where &> is the cross section at any point, and the 

 integration is along the tube. Since Rco = 4>7re, 



this expression 



rQ 



= \e Rds 

 J 



-*/ 



p 



v -^- ds 



p US 



= e(V P -V Q ). 



This, however, is exactly the contribution made by the charges e at 

 P, Q to the expression 2eF. Thus on summing over all tubes of force, we 

 find that the total energy of the field \^eV may be obtained exactly, by 



D2 



assigning energy to the ether at the rate of per unit volume. 



Energy in a Dielectric. 



169. By imagining the parallel plate condenser of 168 filled with 

 dielectric of inductive capacity K, and calculating the energy when charged, 



KR* 



we -find that the energy, if spread through the dielectric, must be -^- 



per unit volume. 



