ELECTROMAGNETIC THEORY 5 



Remembering that u is expressed in elm. units, this becomes 



I.D =~W - f (u-grad^)dv - t (u--^ A^dv 

 or 



W ^ D -\- c i (u-grad^)dv + c j (u-~^^A\dv, 



where W is the work done per unit time by the impressed electric 

 field, and D is the dissipation per unit time in the system ; i.e., the rate 

 at which electrical energy is converted into heat. By means of general 

 theorems in vector analysis, the integrals can be transformed and the 

 equation reduced to the form 



W 



^ + al8^/ (^^ + ^^^^^ + 4^ / C^-^^"^^' 



the last integral being taken over any closed surface which includes 

 the system. Translating this equation into words, it states that: — 



The work done per unit time by the impressed forces is equal to the 

 rate of dissipation per unit time plus the rate of increase of the field 

 energy plus the rate at which energy is radiated from the system. The 

 vector (c/47r) [£ • ff] is the radiation vector and gives the density and 

 direction of energy flow per unit time;"* it will be denoted by S. 



We now shall briefly consider the field due to the currents and 

 charges in the system. 



If the current density u and charge density p are everywhere 

 specified, the retarded potentials are uniquely and completely deter- 

 mined by the formulas 



A = j ^ ^ dv, (vector) 



$= {P^l^^dv. (scalar). 



The functional notation u{t — rjc) and p{t — rjc) indicating that u 

 and p are to be evaluated at time / — r/c may profitably be replaced 

 by ue-fp/'-''- and pe-^p''^\ so that 



/■llg—ivlc)r 

 — :: — dv, 



-s 



r 

 pe- 



-(p/c)r 



$ = I ^ dv. 



r 



* This is known as Poynting's theorem. 



