464 BELL SYSTEM TECHNICAL JOURNAL 



In vacuo, therefore, there is a current-density (l/47r) times the 

 rate-of-change of Eo sin nt, and it has an ampHtude of (nEo/4:T) and 

 is 90° ahead of the field in phase. To this current-density corresponds 

 the speed of light in vacuo, the well-known constant c. The speed 

 of light in the non-ionized lower regions of the atmosphere differs so 

 little from c that we need never bother with the difference, which 

 henceforth will be ignored. 



When the waves pass out of ordinary air into the ionosphere, there 

 is still the displacement-current but now in addition there is the 

 current borne by moving electrons. Here is a second pitfall. It 

 may seem obvious that the electron-current must add on to the 

 displacement-current, creating a total current-density greater than 

 that in vacuo and therefore lowering the wave-speed. Not so at all ! 

 The point is, that when the electrons are truly free, the field sets 

 them into oscillation in such a curious way that when they become 

 adjusted, they are oscillating with their velocities 90° behind the field 

 in phase. Their contribution to the current, being proportional to 

 their velocity, is also 90° behind the field, and hence in perfect opposi- 

 tion of phase to the displacement-current. 



Therefore the electron-current density — call it le — is to be subtracted 

 from the displacement current-density! Accordingly I write, 



W2 _ w(l/47r)£o /jx 



c2 w(l/47r)£o - /. 



The reader may suppose that the factor cos nt, common to both 

 currents, has been divided out.- The quantity le is clearly propor- 

 tional to Eo and also to our familiar N the density of electrons, and 

 in fact the reader can undoubtedly work out with ease that it is 

 equal to NEoe^jmn. Here e and m stand for the charge and mass 

 of the ion, as is customary. Therefore we find: 



u'~ 1 1 (2) 



c^ 1 - ^TrNe-jmn- 1 - Ne-jirmf- 



The wave-speed is greater in the ionosphere than it is in vacuo or or- 

 dinary air. I now recall from the most elementary optics the principle 

 that when two media adjoin in which light has different wave-speeds, 

 and light passes through their common boundary into the medium 

 where its speed is greater, it is refracted away from the normal to the 

 boundary. Accepting for the moment the over-simplified model of the 



2 Actually Maxwell's theorem does refer to the amplitudes of the currents — but 

 if the currents are not exactly 0° or 180° apart in phase, the amplitude of one must 

 be taken as a complex quantity. 



