SOME EQUIVALENCE THEOREMS OF ELECTROMAGNETICS 99 



Let US suppose that we have a continuous distribution of sources 

 on a closed surface C (Fig. 3) and that there are no other sources. 

 We assume that the sources are harmonic of frequency /. The elec- 

 tromagnetic field 5 produced by these sources can be calculated 

 directly from this distribution with the aid of the above mentioned 

 vector potentials. On the other hand, we can reason as follows. 



Fig. 3^A cross-section of a closed surface C. 



There are no sources either inside or outside of C; hence everywhere 

 except on C, we have 



curl E = — ioofxH, curl 11= (g -\- i(>}e)E. (15) 



In the region inside of C we take that solution of (15) which is finite 

 throughout this region and outside of C we select the solution vanishing 

 at infinity. Both solutions will contain constants which can be deter- 

 mined from conditions (13) and (14) across the surface. The field 

 5' obtained in this manner is identical with J because the difierence 

 % — %' is everywhere source-free and thus must vanish. 



Let us now reverse the process and, instead of starting with the 

 known distribution of sources on C, suppose that we know the field 

 and wish to find its sources. Let the known field i^ be source free 

 everywhere except on C. In order to determine these sources 5 we 

 merely calculate the discontinuities in the tangential components of 

 E and H across C. We can utilize this result to establish the major 

 Equivalence Principle. For the outside portion of 5 we can choose 

 the outside portion of the field %' produced by a given system of sources 

 S' situated inside C and for the inside part of ^^ we take any field which 

 is source-free there. The latter may be, for instance, the inside 

 portion of the field 5" produced by some sources S" situated outside C. 

 Thus we arrive at the following Equivalence Principle discovered by 

 Love and Macdonald ^: a distribution of electric and magnetic currents 

 on a given surface C can be found such that outside C it produces the 



* See references 3 and 4 and also H. M. Macdonald, "Electroniagnetism" (1934). 



