ELECTROMAGNETIC THEORY 3 



source of energy. In the following we shall suppose that E° Is specified 

 and we shall keep carefully in mind the fact that E° denotes the 

 electric intensity not due to the reaction of the system itself. This 

 distinction is extremely important. 



We have now to take up the problem of specifying the electric 

 intensity E' In terms of the currents and charges of the system. The 

 necessary relation Is furnished by the Lorentz or retarded potentials 



^ = ^ B^LzJM dv, (scalar) (2) 



A = j "^^ ~ ''""^ dv, (vector) . (3) 



Interpreting equation (2), $ Is equal to the volume integral of the 

 retarded charge density divided by the distance between the point 

 at which $ is evaluated and the location of the charge. The retarded 

 charge density means that at time t we take the value of the charge at 

 the earlier time / — r/c, where c is the velocity of light. It is to be 

 understood that p and u are the true charge and current density, and 

 displacement currents are not Included. Their effect appears in the 

 retardation only, c also Is the true velocity of propagation in vacuo. 

 The potential $ is therefore a generalization of the electrostatic 

 potential Into which it degenerates in an unvarying system. 



Similarly the vector potential A of equation (3) is gotten by a volume 

 integral of the retarded vector current density divided by distance r. 

 As the name indicates it is a vector quantity and in Cartesian co- 

 ordinates has three components Ax, Ay, Ag. 



By means of the equation 



E' = -grsid^-^^A, (4) 



c ot 



the electric intensity due to the reaction of the system is expressed 

 in terms of the charge and current densities. 



Equations (2), (3), (4) and the additional equations 



B' = curl A, (5) 



(where B' Is the magnetic induction due to the currents In the system) 

 are the complete equivalent of Maxwell's equations from which they 

 are Immediately derivable. 



