SOME EQUIVALENCE THEOREMS OF ELECTROMAGNETICS 95 



sources in the space-time. If they are zero everywhere and at all 

 times, the only physically significant solution of (2) must be E = II 

 = throughout the entire space and at all times. ^ If there are other 

 solutions of (2), they are extraneous and some rule must be found for 

 excluding them. Such extraneous solutions often find their way into 

 mathematical equations because it is usually impossible to express all 

 physical conditions by an equation or a system of equations. Naturally 

 these remarks do not apply to a limited region of space or a finite 

 interval of time. In fact, in many physical problems these "extrane- 

 ous in the large" solutions of (2) can be advantageously used for 

 expressing the general character of electromagnetic phenomena in a 

 limited region and then obtaining, with the aid of the boundary and the 

 initial conditions, the complete answer. But the philosophy of 

 causality demands the dictum "no sources, no field" when considering 

 the whole space-time. It may seem unnecessary to dwell at length on 

 such obvious matters but they happen to be essential in the subsequent 

 discussion if the arguments are to be taken as positive proofs rather 

 than as plausible justifications. 



Equations (2) are linear and the principle of superposition is 

 applicable. This is in accordance with physical intuition which tells us 

 that we can subdivide the impressed currents into elementary cells of 

 volume dv, calculate the field due to a typical element, and obtain the 

 total field by integration. For the typical element (2) becomes 



curl E = - M -^ . curl H = gE -\- e — , (3) 



everywhere except in the infinitely small volume occupied by the 



element. The product of the current density and the volume of the 



element is called the moment of the element. 



At times the impressed currents are confined to sheets so thin that 



their thickness can be disregarded without introducing a serious error 



in the result. This leads to a hypothetical infinitely thin current sheet. 



We pass from a real current sheet to an ideal one by assuming that the 



thickness of the former decreases and the current density increases in 



such a way that their product remains constant. This product is 



called the linear density of the sheet and it represents the current per 



unit length perpendicular to the lines of flow. The moment of a 



current element is now the product of the density of the sheet and the 



area of the element. Finally if the impressed current is confined to a 



' We assume that all the electric and magnetic charges were originally in the 

 neutral state, in which case their separation could be effected only through their 

 motion. The argument could be extended so as to include purely static fields that 

 may constitute an integral part of the universe but it is of no particular interest to us. 



