Theory and Measurement of Residual Charges. 4-43 



The force P is positive at A, vanishes for a certain value 

 of a , and is negative at B. If 



X=0, P=-2Trd(2x-a). 



Let p be the pressure due to the free electrons ; then, 

 since p = ~Kp, we have 



-.^ = -X-27rd(2tf-a)-P 

 p ax 



and d~P . 



*£6S=^-* 



Lftt n= number of free electrons per unit volume at any 

 point, and N the number of free electrons per unit volume 

 in the natural state. Then d — Ne, and p = ne, e being the 

 charge of an electron. We have then 



or 





where % . ,,, „ 47re 



X is written tor -r^-- 



This differential equation being independent of X will be 

 true for any uniform field applied to the dielectric, and in 

 the particular case we must have 



5 £^ = -X-27TN? (2ff-a)-P, 



n dx K ' 



where dP . 



-^ — = — 4:7rne. 

 ax 



Writing 2v = (dn\ 2 



\dx) ' 



the above differential equation becomes 



dn\n 2 /~* \ n ./' 



