75-77] Infinite Plane 69 



An Infinite Plane. 



76. Suppose we have a plane extending to infinity in all directions, and 

 electrified with a charge a per unit area. From symmetry it is obvious that 

 the lines of force will be perpendicular to the plane at every point, so that 

 the tubes of force will be of uniform cross-section. Let us take as Gauss' 

 surface the tube of force which has as cross-section any element &> of area 

 of the charged plane, this tube being closed by two cross-sections each of 

 area w at distance r from the plane. If R is the intensity over either of 

 these cross-sections the contribution of each cross-section to Gauss' integral 

 is Ro), so that Gauss' Theorem gives at once 



whence R = ZTTCT. 



The intensity is therefore the same at all distances from the plane. 



The result that at the surface of the plane the intensity is 2-Trcr, may at 

 first seem to be in opposition to Coulomb's theorem ( 57) which states that 

 the intensity at the surface of a conductor is 4?ro-. It will, however, be seen 

 from the proof of this theorem, that it deals only with conductors in 

 which the conducting matter is of finite thickness ; if we wish to regard 

 the electrified plane as a conductor of this kind we must regard the 

 total electrification as being divided between the two faces, the surface 

 density being Jo- on each, and Coulomb's theorem then gives the correct 

 result. 



If the plane is not actually infinite, the result obtained for an infinite 

 plane will hold within a region which is sufficiently near to the plane for the 

 edges to have no influence. As in the former case of the cylinder, we can 

 obtain the potential within this region by integration. If r measures the 

 perpendicular distance from the plane 



so that F=(7-27r0T, 



and, as before, the constant of integration cannot be determined without 

 a knowledge of the conditions at the edges. 



77. It is instructive to compare the three expressions which have been 

 obtained for the electric intensity at points outside a charged sphere, cylinder 

 and plane respectively. Taking r to be the distance from the centre of the 



