SOME EQUIVALENCE THEOREMS OF ELECTROMAGNETICS 107 



If a progressive wave is advancing from left to right in a semi- 

 infinite coaxial pair (Fig. 7) and if the generator is at infinity, we can 

 assume it to be the principal wave. At the open end this wave is 

 reflected. It is usually assumed that the reflected wave is also the 

 principal wave but moving in the opposite direction. In other words, 

 it is assumed that the total field is such that the electric lines are 

 radial and the magnetic lines are circular. Since the electric lines 

 are radial, there is no longitudinal displacement current; and since 

 the conduction current at the open end must be zero, the magnetic 

 intensity is zero over the entire open end. This is what follows if we 

 neglect the complementary waves. 



These approximate results correspond to the exact results in the 

 following hypothetical situation. If a hypothetical perfect magnetic 

 conductor is fitted over the open end of the coaxial pair so that it 

 closes it entirely, then the reflection is complete and there are no 

 complementary waves. Perfect magnetic conductors are defined by 

 analogy with perfect electric conductors — the tangential component 

 of the magnetic intensity vanishes at the surface of the former just 

 as the tangential component of the electric intensity vanishes at the 

 surface of the latter. Magnetic conductors support magnetic current 

 sheets just as electric conductors support electric current sheets. The 

 densities of the sheets are given by the discontinuities of the tangential 

 components of E in the former case and H in the latter. 



In the hypothetical case in which the open end is closed with a 

 perfect magnetic conductor, no energy can flow outside the coaxial 

 pair. This is because the flow of energy is given by \E X H and either 

 one or the other factor vanishes over the outer boundary of the struc- 

 ture. The field outside the coaxial pair must now be identically zero. 

 Our sources are the electric current in the walls of the coaxial pair and 

 the magnetic currents in the cap. If one field is designated by S and 

 the other by S' , then 5 -f 5' = and S' = - S. Thus the field 

 produced by the electric currents in the conductors on a supposition 

 that principal waves alone exist, is the same as the field produced by a 

 hypothetical distribution of magnetic currents over the surface of the 

 open end. 



Let us examine another case. It is usual to assume that the electric 

 current in a thin wire (Fig. 8) in free space is distributed sinusoidally. 

 Experimental evidence shows that the radiated power calculated on 

 this assumption is very nearly correct. On the other hand the 

 sinusoidal distribution of the electric current in the wire corresponds to 

 a hypothetical case in which a perfect magnetic conductor is introduced 



