36 STATIC ELECTRICITY 



If a tube of force passes through a charge and 7 is 

 the charge within the tube, the product intensity x 

 cross-section changes by 4^. Let S X S 2 , I'm- ::'.). t>< MU-II a 

 tube having charge q within it between S x and S 2 . Applying 



Fio. 29. 



Gauss's theorem to the tube and its ends SjS r it is evident that 



TO _ T Gl 



J The 2 intensity just outside a conducting surface is 

 normal to the surface. Let , Fiic- '50, represent the section of 

 a small circular area of a conducting surfa. (harmed with a p T 

 unit area, and let a be so small thai it ma\ In regard -d M pi 

 the surface density of itfl charge as uniform. Let it becircul.. 

 l^Pjj be two points Oil the a\is of ,, respectively at e.|iial ({Malices 

 just outside and just inside the surface and M> near that \\ 1', is 

 indefinitely small compared with the radius of . 



Now the intensity at ////// point may )< -1 as tin i> Miltant 



of the intensity due" to , and the intensity due to tin tin- 



surface and to oth> in the 



system which \\ r ma\ di ii- 

 tixely hv S. At 1', the inteiisitx is 



point within I 



ductor, so that the intensity thue 

 due to S is eijual and oppu- 



that due to <. 1-Vom 

 that due to K is normal. So also 

 must that due to S be normal. At 

 l\ the intensity due t> is theaame 

 as at P r since the distain < 

 the points is negliffiUe comjMired 

 FIG. 30. with the distance of ilie nearest part* 



of S. The intensitx at \\ dut 



is equal and opposite to its value at P 2 and is tli 1 to 



and in the same direction as the intensity due to S. and hoth are 

 normal to the surface. 



The intensity just outside a conductor is 47r<r, where & 

 is the charge per unit area or the surface density. I 



