106 BELL SYSTEM TECHNICAL JOURNAL 



very near the open end. The complementary waves are needed only 

 for logical consistency and to satisfy the boundary conditions. 



Thus let us suppose that the field distribution in the coaxial pair 

 is known to a high degree of accuracy. In order to calculate the field 

 outside the coaxial pair and hence obtain the radiated power we can 

 use the Equivalence Principle in two ways. We can fit our surface C 

 smoothly over the outer cylinder and the open end (Fig. 7a) or, re- 

 garding this surface as a perfectly elastic rubber sheet, we can press 

 it through the open end and fit it smoothly over the inner surface of 

 the outer conductor and the outer surface of the inner conductor 

 (Fig. lb). Since by hypothesis the conductors are perfect, the com- 

 ponents of E tangential to the cylinders vanish; hence in the second 

 choice of C the equivalent layer consists of only an electric current 

 sheet. Naturally this current distribution is precisely that which 

 actually exists in the conductors so that this choice of C leads to some- 

 thing that we knew beforehand, namely: if the actual sources, that is, 

 if the electric currents in the structure are known exactly or approxi- 

 mately, the entire field can be calculated exactly or approximately. 



The first choice of C is more important. Over the lateral portion 

 (12, 34) of C the equivalent magnetic current sheet vanishes as in the 

 preceding case on account of the perfect conductivity of the cylinders. 

 The magnetic intensity just outside the coaxial pair is also zero except 

 near the open end where it must be exceedingly small. To see this, 

 we need only recall that the electric currents in the two cylinders are 

 equal and opposite and that except in the neighborhood of the open 

 end the displacement currents are transverse. Thus the equivalent 

 electric current sheet can be ignored altogether. What is left is the 

 magnetic current sheet over the surface of the oj)en end; the density 

 of this sheet is determined by the radial component of the electric 

 intensity and in the final analysis by the voltage existing between the 

 ends of the inner and outer conductors. Presently we shall carry 

 out the actual calculations but just now we shall examine the question 

 of the accuracy of the results. Of course, the results would be exact 

 if we knew the equivalent electric and magnetic sheets accurately; 

 and the above approximations appear to be reasonable. We shall not 

 be able to find out how good these approximations are but we can 

 prove that they are just as good as the approximations usually made 

 in calculating the radiated power from the distribution of electric 

 currents. The only virtue of the Equivalence Principle is to save a 

 certain amount of mathematical work and furnish a further insight 

 into the phenomena of radiation. 



