ELECTRICAL TRANSMISSION ALONG WIRES 13 



Maxwell's equations for a continuous homogeneous medium may be 

 written in the compact form ^ 



curl E= - M, 



curlM= v'E, 



div £ = 0, ^^^ 



div M = 0. 



From this set of equations it is easily shown that each component 

 of the vectors E and M individually satisfies the ivave equation 





5 + ^2 + ^-2- -M/=0 (2) 



or in vector notation 



(V^ - v^")f = 0. 



Here / denotes any vector component ; thus in Cartesian coordinates 

 / may stand for E^, Ey, E^; Mx, My, Mz, all of which separately 

 satisfy (2). 



Given the electrical constants and geometry of the conducting 

 system and dielectric media, the general problem is to find solutions 

 of (1) and (2) which also satisfy the boundary conditions at the surfaces 

 of separation of the difi'erent media. These boundary conditions are 

 that the tangential components of E and // shall be continuous over 

 such surfaces of separation. These boundary conditions, as may be 

 seen from (1), necessitate also the continuity of the normal components 

 of M and {v'lix)E. 



If we introduce a vector potential A{Ax, y, z) and a scalar potential 

 $, it is easily shown that (1) may be replaced by 



M = curM, 



E= - A - grad $, ^"^^ 



with the further relation 



div A -{- v""^ = 0. (4) 



$ and the components of the vector A individually satisfy the wave 

 equation; thus 



(V^ - ^^)$ = 0, 



(V' - i'')A = 0. ^^^ 



In Cartesian coordinates these equations are 



2 Note that in this form the constants of the medium appear explicitly only through 

 the parameter i>^. 



