ELECTRIFIED SURFACE SEPARATING TWO DIELECTRICS. 27 



the bases. If F and F' are the forces in the two media near this 

 element, we shall have 



= F' n </S - F n </S, 

 whence 



Hence, the difference between the perpendicular components of the 

 force on the two sides of an electrified surface is equal to the product of 

 4?r by the density on the surface. 



This theorem includes that of Coulomb (36) as a particular case, 

 in which S is the surface of a conductor. 



40. Let us consider two points P and P' in the middle of the 

 bases of the preceding cylinder. The action F at the point P is 

 made up of the action of all the masses external to the element, and 

 of the action - </> of this element. In passing from P to P' the action 

 of the external masses does not appreciably change, but that of the 

 element changes its sign, and becomes + <. As this force, moreover, 

 is perpendicular by symmetry, the perpendicular component of the 

 force varies by 2<, which gives 



whence 



(f> = 27TCT. 



As the perpendicular component of the force is alone modified, 

 the tangential components of the forces F and F' are equal on both 

 sides the surface. If / and /' are the angles of the forces with the 

 perpendicular on the same side of the surface, we have then 



F' sin / ' = F sin i. 



The preceding equation gives, moreover, 



F' cos /' - F cos / = 47TO- ; 

 from which we have 



tan i' 47TO- 477-0- 



r ~ . 



tan i P cos i F 



The forces undergo therefore a sort of refraction on meeting an 

 electrified surface. It may be remarked in passing, that as the law of 

 refraction is determined by the ratio between the tangents of the 

 angles of the perpendicular and the forces, there could never be any 

 phenomenon analogous to total reflection. 



