2 BELL SYSTEM TECHNICAL JOURNAL 



hence the phenomena of radiation. Again it involves the assumption 

 that the network can be represented by a finite number of coordinates 

 and thus that it constitutes a rigid dynamic .system. The rigorous 

 equations of electromagnetic theory formulate the relations between 

 current and charge densities and the accompanying fields. Circuit 

 theory, on the other hand, expresses approximate relations between 

 total currents and charges and impressed electromotive forces. 



With the rapid development of electro-technics an increasing number 

 of problems is being encountered where the application of classical 

 electric circuit theory is of doubtful validity, or where the conclusions 

 derived from it must be interpreted with great care. Such problems 

 are the result not only of the use of very high frequency in radio- 

 transmission but arise also in connection with the need of a more 

 precise theory of wire transmission. 



In view of the foregoing it seems desirable to examine the founda- 

 tions of circuit theory. This is the problem dealt with in the present 

 paper: — a derivation of the classical circuit theory equations from the 

 standpoint of electromagnetic theory, in the course of which the 

 approximations involved are pointed out. 



A second reason, pedagogic in character, is believed to justify the 

 present study. This is, that, as circuit theory is usually taught to 

 technical students no picture of its true relation to electromagnetic 

 theory is given, and the student comes to regard inductance, resistance, 

 capacity, voltage, etc., as fundamental concepts. 



To start with our problem in a general form, consider a conducting 

 system of any form whatsoever, in which the charge density at any 

 point X, y, z at any time t is denoted by 



p{x, y, 2, /) = p, 



and the vector current density by 



u(x, y, z, t) = u, 



the functional notation indicating that the charge and the current 

 density are functions of space and time. At any point in the system let 



E{x, y, z, t) = E 



denote the vector electric intensity. This we shall suppose to be com- 

 posed of two parts; thus 



E = E° -\- E'. (1) 



In this equation E° is the impressed electric intensity and E' the 

 electric intensity due to the reaction of the currents and charges in the 

 system. Thus E° may be the electric intensity due to a distant system, 

 as in radio transmission, or that due to a generator, battery or other 



