34 Electrostatics Field of Force [OH. u 



Let us take any small element dS of the closed surface in the neighbour- 

 hood of a point Q on the surface and join each point of its boundary to the 

 point Pi. Let the small cone so formed cut off an element of area do- from 



Fm. 7. 



a sphere drawn through Q with P l as centre, and an element of area dco from 

 a sphere of unit radius drawn about P l as centre. Let the normal to the 

 closed surface at Q in the direction away from P T make an angle 6 with P X Q. 



The intensity at Q due to the charge e l at P x is e^P-^Q 2 in the direction 

 P X Q so that the component of the intensity along the normal to the surface 

 in the direction away from P l is 



. COS ft 



The contribution to I IN^S from the element of surface is accordingly 



-^- cos 6 dS, 



the -f or sign being taken according as the normal at Q in the direction 

 away from P 1 is the outward or inward normal to the surface. 



Now cos 6 dS is' equal to da, the projection of dS on the sphere through Q 

 having P l as centre, for the two normals to dS and da- are inclined at an 

 angle 0. Also da- = PiQ 2 da). For do-, dco are the areas cut off by the same 

 cone on spheres of radii PjQ and unity respectively. Hence 



^ cos# dS = -^-^ = e, dco. 



If Pj is inside the closed surface, a line from P l to any point on the unit 

 sphere surrounding Pj may either cut the closed surface only once as at Q 

 in which case the normal to the surface at Q in the direction away from Pj 

 is the outward normal to the surface or it may cut three times, as at 

 Q', Q", Q'" i n which case two of the normals away from P l (those at Q', Q"' in 

 fig. 8) are outward normals to the surface, while the third normal away 

 from P! (that at Q" in the figure) is an inward normal or it may cut five, 



