SOME EQUIVALENCE THEOREMS OF ELECTROMAGNETICS 101 



outside C must vanish at infinity but it need not be the same as E, II; 

 let it be £ + £',// + //'. The field E', 11' must be source-free out- 

 side C. Inside C the field must be source-free; we shall designate it 

 by £", H". The field E', W is called the reflected field and E" , 11" the 

 refracted field. The boundary conditions are such that the components 

 of the electric and the magnetic intensities tangential to C must be 

 continuous. Thus over the surface C, we have 



Et + E/ = Et", H, + H/ = Ht". (18) 



The bar over the letters is used to designate the values of the corre- 

 sponding quantities on C. From (18) we obtain 



Et" - Et' = Et, Ht" - Ht' = Hi. (19) 



(E',H') 

 S(E,H) ^ 



Fig. 4 — The closed surface C is the boundary between two homogeneous regions in 

 space. (£, H) designates the field produced by some system of sources S\ (£', H') is 

 the field reflected by the body C; and (£", H") is the field in the body. 



Hence the reflected and the refracted fields together constitute an electro- 

 magnetic field in the entire space; this field is source-free everywhere 

 except on C and the distribution of sources on C is calculable from the 

 given sources S. This Induction Theorem is a generalization of the 

 well-known theorem used in calculating the response of an electric 

 circuit to an impressed field. Since the wires constituting the circuit 

 are very thin, only the tangential components of E in the direction 

 of the wires need be considered. 



It may be noted that if the medium inside C is identical with that 

 outside C, the "reflected field" must be absent and the "refracted 

 field" must be identical with the field E,H due to the sources S. 

 Thus the Induction Theorem leads to the Equivalence Principle. 



The Equivalence Principle is evidently an extension of Kirchhoff's 

 theorem. The latter deals with a single wave function instead of two 

 vectors. Kirchhoff derived a formula for computing the wave function 

 in the source-free region from its values and the values of its normal 

 derivative over a closed surface separating the source-free region from 



