"INVERSE SrARE" SYSTEMS 41 



?, is the angle between (IS and the cross-section perpendicular 

 to PF. The solid angle of the cone is therefore dw = 



Hence dS contributes m/w. 



The whole disc then gives intensity rr X solid angle subtended 

 by the disc. It' we draw a sphere radius PA the area of this 

 sphere cut off'bv AH is 8r.PA.CG, where PG is the radius through 

 C, and the solid angle which it subtends at P is 2irPA.CG/PA 8 

 = 2-PA(PA-PO PA- = 27r(l-cos a ), where a = APC. 



The intensity at P is therefore 27rer(l cos a). If the radius 

 of the disc is very large compared with PC 1 , a is very nearly 90, 

 and the intensity IN very nearly STTO-. 



Whatever the form of the disc may be, so long as it is plane 

 the intensity will still be ^TTT if the radius to the nearest point of 

 the edge makes an angle with PC indistinguishable from !)() . 

 lor the whole of the area outside that distance subtends a 

 vanishing solid angle at P. 



Intensity due to a very long uniformly charged 

 cylinder near the middle. In the plane perpendicular to the 

 axis of the cylinder and bisecting it the inten.Mtv is evidently 

 radial. It will l>e radial, too. at a distance from this plane small 

 compared with the distance from either end, if the distance from 

 the lao small compared \\ith the distance from the ends. 



f 11. I 'ids. and ( ' is the central point, take a 



F CD E' 



point I*, not (jiiite in the plane through C. Draw PD perpen- 

 dicular to the axis and make DF---DE'. The intensity due to 

 11. bei identlv radial. That due to EF is in comparison negligible, 

 since EF is )>\ supposition not large, and it is very distant from 

 1' -.mpared with the part of the cylinder immediately under P. 

 II-nce tli<- intensity at P is radial. Further, if the position of P 

 ( -liaiiiM -s bv a small amount parallel to the axis the intensity is 

 oulv changed by a removal of a small length from one end of the 

 cvlindcr to the other with negligible effect. Draw a cylinder 

 radius ; and length PQ = 1. I Jfi, co-axial with the charged 

 c\linder, and applv (iatis^s theorem to the surface thus formed. 

 The intensity I, as we have just seen, is normal over the circular 

 ice and e\ ci \ u liei < the > ime, and it has no normal component 

 the flat ends of the cylinder I\). The (juantity of charge 

 within is 2-7ro<r, where a is the radius of the charg< d surface. 



Then /V/S 



and l = 



