ELECTROMAGNETIC FIELD ANALYSIS 503 



we would not be able to detect any difference between the ''exact" solution 

 and a much simpler approximate solution. 



In the case of rectangular tubes we don't even know how to obtain the 

 ''exact" solution in any form; but good approximate solutions are exceed- 

 ingly simple. The word "exact" is in quotation marks because there can 

 be no really exact solutions of actual physical problems. In the first place 

 the properties of materials are not known exactly; the boundaries between 

 media do not exist in the exact sense of the term; and we just don't know 

 the exact laws of nature. All we really want of any solution is to be ac- 

 curate enough for some particular purpose. And here is where the idea 

 of ideaHzed boundaries helps in the formulation of simplified, clear-cut 

 mathematical problems. The idea lends flesh and blood to ideaHzed 

 mathematical boundary conditions. Perfect conductors have long been 

 mentioned in literature as idealizations of good conductors; but other 

 types of boundaries are of much more recent origin. Perfect conductors 

 are boundaries of zero surface impedance] they support electric currents of 

 finite strength when the tangential electric intensity is zero. At these 

 boundaries the tangential magnetic intensity is different from zero. The 

 natural counterpart is a boundary of infinite impedance at which the tan- 

 gential magnetic intensity vanishes but the tangential electric intensity 

 does not. The further generalization is a boundary with a given finite 

 surface impedance which is defined as the ratio of two mutually perpendic- 

 ular tangential components of the electric and magnetic intensity. The 

 boundary may be isotropic, with its surface impedance the same in all 

 directions; likewise, the boundary may be aelotropic. The surface imped- 

 ance is defined as the ratio of the tangential components of E and H. Since 

 it is necessary to adopt a convention regarding "positive directions" of 

 E and H, these are so chosen that a right-handed screw will advance into 

 the boundary if its handle is turned through 90° from the positive direction 

 of E to coincide with the positive direction of H. In accordance with this 

 convention the positive real part of the surface impedance is associated 

 with an average flow of power into the boundary — that is, with a passive 

 boundary. An active boundary is a boundary with a negative surface re- 

 sistance; such boundaries may be used to represent idealized generators of 

 electromagnetic waves and to eliminate from explicit consideration the 

 internal mechanisms of these generators. 



Field Equations 



Thus far I have tried to present the ideas behind the physical and mathe- 

 matical analysis of electromagnetic transmission phenomena. These are 

 broader than the electromagnetic laws themselves and, with some super- 



