SOME EQUIVALENCE THEOREMS OF ELECTRO MAGNETICS 97 

 If the medium is dissipative, we have 



A=^-f^, H = cur\A. E= -.W^+i'i^i^. (8) 

 47rr '^ g -\- ioi€ ^ ' 



The quantity a is the intrinsic propagation constant of the medium and 

 is defined by 



a = >lio)/j.(g + iue). (9) 



In this case the action of the source at some point is not only delayed 

 by the time needed for the disturbance to travel the intervening 

 distance but also exponentially attenuated. 



If instead of an electric current element, we are dealing with a 

 magnetic element, the field components can be expressed in terms of an 

 auxiliary electric vector potential. This vector F is given by 



(10) 



where the moment P of the element is the product of the magnetic 

 current density and the volume of the element. The field components 

 are then given by 



£=-curIi^, H^ -(g+i^,)/. + g^^ddivF ^ 



In the periodic case the general mathematical solution for the vector 

 potential of an element is found to be a linear combination of any two 

 of the following functions 



e~'"' e'"' cosh ar sinh ar , 



— ' y ~r ' ~^~ ' ^"■' 



All of these except the first become exponentially infinite at an infinite 

 distance from the source and cannot be taken to represent the vector 

 potential of a physical source. The last function is finite in any finite 

 region; conceivably it can represent an electromagnetic field in the 

 finite region free from physical sources. If the medium is non- 

 dissipative it is impossible to exclude any of the solutions given by (12) 

 on the grounds of their behavior at infinity— they all vanish there. 

 But we may regard the non-dissipative case as the limit of the dissi- 

 pative one and in this way establish a rule for finding the proper unique 

 solution. 



In the presence of a current sheet, equations (2) are valid on either 

 side of it but not on it. Let us consider a cross-section of an electric 

 current sheet, perpendicular to the lines of flow, and a curvilinear 



