Some Equivalence Theorems of Electromagnetics and Their 

 Application to Radiation Problems 



By S. A. SCHELKUNOFF 



After a review of the general aspects of the classical electromagnetic 

 theory several "equivalence" theorems are established and illustrated with 

 a number of examples from the diffraction theory. Then follows a discus- 

 sion of possible applications of these theorems to radiation problems. The 

 latter part of the paper is dev^oted to the calculation of the power radiated 

 from an open end of a coaxial pair. ^ 



THE usual methods of calculating the power radiated by an electric 

 circuit depend upon a determination of the electromagnetic field 

 from the electric current distribution in the circuit. The best known 

 of these methods consists in integrating the Poynting vector over the 

 surface of an infinite sphere surrounding the circuit. This method has 

 been used exclusively until recent years; to facilitate its application, 

 John R. Carson obtained a compact general formula for the radiated 

 power.^ Another method ^ consists in calculating the work done 

 against the forces of the field in supporting a given current distribution 

 in the circuit. Theoretically either of the two methods is sufficient 

 for solving any radiation problem. Practically, aside from inherent 

 difficulties involved in the calculation of the electric current distri- 

 bution in the first place, the preliminary integration for determining 

 the field components E and H may be rather complex. Thus in 

 obtaining the power radiated by a semi-infinite pair of perfectly 

 conducting coaxial cylinders this preliminary integration has to be 

 extended over the infinite surfaces of the two conductors. And yet by 

 the Maxwell-Poynting theory, no energy can flow through the walls of 

 the outer cylinders since the electric intensity E and hence the Poynting 

 vector vanish there. Any energy which is radiated away must pass 

 through the open end and it is natural to expect that there must be a 

 method for calculating this energy from the conditions at the open end. 

 The integration involved in this method would extend only over a 

 comparatively small area of the open end. It is in search of a method 

 of this type for calculating the radiated power that I was led some time 

 ago to certain "equivalence theorems." Subsequently I learned that 



* John R. Carson, "Electromagnetic Theory and the Foundations of Electric 

 Circuit Theory," The Bell System Technical Journal, pp. 1-17, January 1927. 



' A. A. Pistolkors, "The Radiation Resistance of Beam Antennas," Proc. I. R. E., 

 Vol. 17, No. 3 (1929). R. E. Bechmann, "On the Calculation of Radiation Re- 

 sistance of Antennas and Antenna Combinations," Proc. I. R. E., Vol. 19, p. 1471 

 (1931). 



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