ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 537 



fields, all of whose components vary exponentially in the direction of 

 the axis of symmetry. Whether any of these fields can be produced 

 individually by some simple physical means is impossible to decide on 

 theoretical grounds alone. It may happen, of course, that the field 

 due to any practically realizable source is always a combination of 

 several simple exponential fields. In any case, however, we want to 

 know the properties of pure exponential solutions. 



It is convenient to make the exponential character of the quantities 

 Ep, Ez and H^ explicit and write them respectively in the form Epe~^^, 

 Eze~^^ and H^e~^^. The new quantities Ep, Ez and H^ are functions 

 of p only. If the suggested substitution, is made in equations (2), 

 the factor e~^^ cancels out and we have 



r dE 



!^ = (g + .-cOpB.. 



The quantity T is called the longitudinal propagation constant or 

 simply the propagation constant when no confusion is possible.* 



Recalling the implied exponential time factor e^"', we see that the 

 complete exponential factor in the expressions for the field intensities 

 is e~^^+^"^ The propagation constant T is often a complex number 

 and can be represented in the form a + i^ where the real part is 

 called the attenuation constant and the imaginary part, the phase 

 constant. Thus, e~"^ measures the decrease in the amplitudes of the 

 intensities and g-»(^2-w'), the change of their phases in time as well as 

 in the z-direction. The latter factor suggests that we are dealing 

 with a wave moving in the positive direction of the 2-axis with a velocity 

 (co/jS). A wave moving in the opposite direction is obtained by re- 

 versing the sign of F. 



Perfectly Conducting Coaxial Cylinders ^ 

 Let us now consider one of the simplest problems which, though 

 purely academic in itself, will throw some light on what is likely to 

 happen under less ideal conditions. We suppose that a perfect 

 dielectric is enclosed between two perfectly conducting coaxial cylinders 

 (Fig. 1) whose radii ^ are b and a {b < a). Our problem is to. find the 

 symmetric electromagnetic fields which can exist in such a medium. 



* Another set of exponential solutions is obtained from this by changing r into — T. 



* For a thorough discussion of "complementary" waves in coaxial pairs the reader 

 is referred to John R. Carson [4]. 



^ Only the outer radius of the inner conductor and the inner radius of the outer 

 conductor need be considered because in perfectly conducting media electric states 

 are entirely surface phenomena. 



