< -' A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 



Between these twenty quantities we have found twenty equations, viz. 



Three equations of Magnetic Force ................................. (B) 



Electric Currents .............................. (C) 



Electromotive Force ........................... (D) 



Electric Elasticity .............................. (E) 



Electric Kesistance .............................. (F) 



Total Currents .................................. (A) 



One equation of Free Electricity .................................... (G) 



Continuity ......................................... (H) 



These equations are therefore sufficient to determine all the quantities which 

 occur in them, provided we know the conditions of the problem. In many 

 questions, however, only a few of the equations are required. 



Intrinsic Energy of tlie Electromagnetic Field. 



(71) We have seen (33) that the intrinsic energy of any system of currents 

 is found by multiplying half the current in each circuit into its electromagnetic 

 momentum. This is equivalent to finding the integral 



E = &(Fp'+G< I ' + Hr')dV. .......................... (37) 



over all the space occupied by currents, where p, q, r are the components of 

 currents, and F, G, II the components of electromagnetic momentum. 



Substituting the values of p', q', r' from the equations of Currents (C), 

 this becomes 



tdy d$\ (da. dy\ 

 - - 



Integrating by parts, and remembering that a, /8, y vanish at an infinite 

 distance, the expression becomes 



IdH dG\ ldF dll\ fdG 



where the integration is to be extended over all space. Referring to the equa- 

 tions of Magnetic Force (B), p. 556, tliis becomes 



(38), 



