ELECTROMAGNETIC THEORY 9 



Derivation of the Familiar Circuit Theory Relations 



In the foregoing we have tacitly assumed that the distribution of 

 currents and charges in the systems is known. We now take up the 

 more difficult problem of determining the distribution in terms of the 

 impressed field and the geometry and electrical constants of the system. 

 This will introduce us to circuit theory and the enormous complexity 

 of the general rigorous expressions will show its important role in 

 physics and engineering. In fact without the beautiful simplifications 

 of circuit theory very few problems of this type could be solved. 



In taking up this problem there are two possible modes of approach. 

 In accordance with one we start with Maxwell's differential equations 

 and try to find a solution which satisfies the geometry of the system 

 and the boundary conditions. For conducting systems of simple 

 geometrical shapes solutions in this way are possible. For compli- 

 cated networks, however, this mode of approach is quite hopeless. 



The other mode of approach is to start with the equation 



-u = E° — grad <l> — iwA 



(8) 

 = £■> - grad /5^^,i. - i.j''±^d., 



which, together with the relation 



iwp = — div u, 



is an integral equation which completely determines the distribution 

 of currents and charges in the system provided g and E° are specified. 



For general purposes of calculation it is quite hopeless as it stands. 

 It has, however, several advantages. First, it is a direct and complete 

 statement of the physical relations which obtain everywhere. Second, 

 it uniquely determines the distribution and does not, like the differ- 

 ential equations, involve the determination of integration constants 

 from the boundary conditions. Third, it leads, through appropriate 

 approximations, to the philosophy and equations of circuit theory. 



To start with a simple case, the solution of which can be extended 

 without difficulty to the general network, consider a conductor forming 

 a closed circuit. We suppose that it is exposed at every point to an 

 impressed electric force £°, and we suppose that the surrounding di- 

 electric is perfectly non-conducting. It is now our problem to derive, 

 for this simple circuit, the circuit equations, in terms of total currents 

 and charges, from the rigorous integral equation for the current and 

 charge densities. 



