SOME EQUIVALENCE THEOREMS OF ELECTROMAGNETICS 93 



one of these theorems was discovered long ago, first by A. E. H. Love ^ 

 and then by H. M. MacDonald ^ and proved by the latter ^ for the case 

 of non-dissipative media in 1911. Another proof of this theorem, 

 believed to be helpful from the physical point of view and extended so 

 as to include the dissipative media, is given in this paper. After a 

 brief review of some fundamental concepts we shall prove these 

 equivalence theorems, discuss their significance, and solve one or two 

 simple examples for illustrative purposes. 



The physical sources of electromagnetic fields are electric and mag- 

 netic charges in motion, that is electric and magnetic currents. The 

 radio engineer has never been interested in shaking magnets for the 

 purpose of radiating energy and has settled into a habit of ignoring 

 magnetic currents altogether as if they were non-existent. It is true 

 that there are no magnetic conductors and no magnetic conduction 

 currents in the same sense as there are electric conductors and electric 

 conduction currents but magnetic convection currents are just as real 

 as electric convection currents, although the former exist only in 

 doublets of oppositely directed currents since magnetic charges 

 themselves are observable only in doublets. And, of course, the 

 magnetic displacement current, defined as the time-rate of change of 

 the magnetic flux, is exactly on the same footing as the electric displace- 

 ment current defined by Maxwell as the time-rate of change of the 

 electric displacement. We shall find it convenient, at least for 

 analytical purposes, to employ the concept of magnetic current on a 

 par with the concept of electric current. 



The two fundamental electromagnetic laws can now be stated in a 

 symmetric form. Ampere's law as amended by Maxwell is: An 

 electric current is surrounded by a magnetic field of force; the induced 

 magnetomotive force in a closed curve is equal to the electric current passing 

 through any surface hounded hy the curve. In its original form, the 

 "electric current" meant only the conduction current so that the law 

 was applicable only to closed conduction currents. Maxwell's amend- 

 ment consisted in including the displacement currents, thereby making 

 the law applicable to open conduction currents. The second law is due 

 to Faraday: A magnetic current is surrounded by an electric field of force; 

 the induced electromotive force in a closed curve is equal to the negative of 

 the magnetic current passing through any surface bounded by the curve. 

 The rule for algebraic signs is as follows: choose some direction of the 

 closed curve as positive and have an observer placed in such a way that 



*A. E. H. Love, "The Integration of the Equations of Propagation of Electric 

 Waves," Phil. Trans. A, Vol. 197, pp. 1-45 (1901). 



^H. M. MacDonald, "Electric Waves," p. 16 (1902). 



' H. M. MacDonald, "The Integration of the Equations of Propagation of Electric 

 Waves," Proc. London Mathematical Society, Series E, Vol. 10, pp. 91-95 (1911). 



