300 Mr. J. Frenkel on the Surface Electric 



Take a very thin layer MM X at a distance x > r from SSi- 

 It contains n dx nuclei per sq. cm., where dx is the thickness 

 of the layer and n the number of atoms per c.c, with a 

 positive charge endx, which obviously must be neutralized 

 by the electrons belonging to nuclei enclosed in two lay era 

 of thickness r on both sides of MMj (the part contributed 

 by MMj being negligibly small). 



Take another layer PPj of the same thickness dx at a 

 distance x<r from SS X . Half of the positive charge, due 

 to the nuclei enclosed in it, is neutralized by the electrons 

 belonging to the nuclei which are contained in a layer of 

 thickness r on the inward side of PP X ; the layer on the 

 other side of PP 2 is narrower {x<r) and the electrons 

 belonging to its nuclei neutralize but, a part of the other 

 half, which can be easily calculated. In fact, take a layer 

 of thickness d% at distance %<x from SSi ; it contains 

 ,„ 2tt sin <j>d<b 7 „ sin <f> c76 



nuclei whose electrons are enclosed within PP^ where <k> is 

 the angle between the line joining the nucleus with the 

 corresponding electron and the positive direction of ,r, 

 determined by the inequality x < rcoscf) < x + dx. Putting 



cos cf) — ?/, we find that — du — sin (f>d(j) = — - and 



,.. sinci j, 1 dx 7f , 

 n at • — cT~ dd> = ^ n — at. 



z z r 



Integrating this from f=0 to f =x, we find for the negative 



1 x 

 charge contributed to PP X by the layer PS — ^ne-dx, 



whence we conclude that PP X contains an excess of positive 



charge equal to - ne ( 1 — - f dx. 



Noticing that the negative charge in a layer QQ X on the 

 outer side of the surface at a distance x<0 from it, is 

 contributed by the electrons belonging to a layer of thickness 

 r— \x\ —r-\-x, we find, in the same way, that it is equal to 



— ^ne(l— -i^iy Thus, for the mean volume density in 



different planes parallel to the surface, we get 



= + \en(l- X \ 0<x<r, 



P x =—o en 



>r. 



