N'.^, = iV ( 1 - ^. ) . (19) 



600 BELL SYSTEM TECHNICAL JOURNAL 



Considering the former first: suppose that initially we have a 

 uniform distribution of electrons, A^ per unit volume; and that at a 

 certain moment it is suddenly distorted, by shifting every particle a 

 distance ^ parallel to the .r-direction, this distance being a function 

 of the original position x of the particle. Fix attention on a column 

 of unit cross-section; initially, between two planes x and x + dx, 

 there were Ndx electrons; they suddenly move over and occupy the 

 space between the planes nc + ^ and x + dx -\- ^ + {d^/dx)dx, so that 

 the density between these two planes, or let me say the number of 

 electrons per unit volume at jc + ?. is given by the formula: 



d^ 

 dx 



We now introduce Poisson's relation between net density p of 

 electric charge and space-derivatives of field strength. As by assump- 

 tion all shifts of electrons are parallel to the ^---direction, so also is the 

 field-vector; we put X for its magnitude, and write Poisson's equation 



thus: 



dX/dx = 47rp. (20) 



If the electrons were the only charged particles, we should have to 

 put N'e for p; and there would be a field of strength different from 

 zero and varying from place to place, even w^hen the distribution of 

 the electrons was uniform. If however there is also a uniform distribu- 

 tion of positive ions, N per unit volume, and this is not affected when 

 that of the electrons becomes non-uniform, then for p we need set 

 only the second term on the right-hand side of (19), multiplied by e; 

 we get 



dX/dx = - 4TrNe{d^/dx), (21) 



and integrating, with the boundary-condition ^^ X = at ^ = 0, 



X = - AirNe^, (22) 



so that an electron shifted from its original location, by virtue of 

 such a mass-distortion of the formerly uniform distribution, is indeed 

 subjected to a restoring force proportionate to its shift. 

 Putting m{(P^ldf)le lor A' in equation (22), we get: 



d'^ldt'' = - (47rAV/w)^ = - «i'^. -(23) 



showing that there is a tendency to oscillations — "plasma-electron 

 oscillations," as Tonks and Langmuir call them — of frequency vi 

 thus given: 



J.J = njlir = ylNe^wm = 8980ViV. (24) 



" This seems to be demanded by symmetry if ^ Is a sinusoidal function of x: 

 otherwise the case is more obscure. 



