150 The State of the Medium in the Electrostatic Field [CH. vi 



Let us now examine whether the total energy of any field can be regarded 

 as arising from a contribution of this amount per unit volume. The energy 

 contained in a single tube of force, with the notation already used, will be 



S P to r mds < 



KR 



or, since - - = P, where P is the polarisation, this energy 





-/; 



Q 



Rds 

 p 



so that the total energy is ^eV, as before. Thus a distribution of energy of 



TC A? 2 

 amount - per unit volume will account for the energy of any field. 



O7T 



Crystalline dielectrics. 



170. We have seen ( 152) that in a crystalline dielectric, the com- 

 ponents of polarisation and of electric intensity will be connected by equations 

 of the form 



*irf=K n X + KY + K*Z\ 



..................... (89). 



The energy of any distribution of electricity, no matter what the dielectric 

 may be, will be ^EV. If Tf, V, are the potentials at the two ends of 

 a unit tube, the part of this sum which is contributed by the charges at the 

 ends of this tube will be J(K JQ. If d/ds denote differentiation along the 



[dV fdV 



tube, this may be written / -5 ds, or again J I Pa) ds, where P is the 



./ vS J OS 



polarisation, and o> the cross section of the tube. Thus the energy may be 

 supposed to be distributed at the rate of ^ -=- P per unit volume. If e is the 



C/S 



angle between the direction of the polarisation and that of the electric 

 intensity, we have -~- = R cos e, so that the energy per unit volume 



= iRPcos6 = (fX + gY+hZ) .................. (90). 



In a slight increase to the electric charges, the change in the energy of 

 the system is, by 109, equal to 2V&E, so that the change in the energy per 

 unit volume of the medium is 





