Examples 285 



28. Shew that the force between two insulated spherical conductors of radius a placed 

 in an electric field of uniform intensity F perpendicular to their line of centres is 



c being the distance between their centres. 



29. Two uncharged insulated spheres, radii a, 6, are placed in a uniform field of force 

 so that their line of centres is parallel to the lines of force, the distance c between their 

 centres being great compared with a and b. Prove that the surface density at the point 

 at which the line of centres cuts the first sphere (a) is approximately 



F( 6b 3 15a6 3 28 2 6 3 57a 3 6 3 "1 



30. A conducting sphere of radius a is embedded in a dielectric (K] whose outer 

 boundary is a concentric sphere of radius 2a. Shew that if the system be placed in 

 a uniform field of force F t equal quantities of positive and negative electricity are 

 separated of amount 



5A+7" 



31. A sphere of glass of radius a is held in air with its centre at a distance c from a. 

 point at which there is a positive charge e. Prove that the resultant attraction is 



where /3 = (A'- 



32. A conducting spherical shell of radius a is placed, insulated and without charge,. 

 in a uniform field of force of intensity F. Shew that if the sphere be cut into two- 

 hemispheres by a plane perpendicular to the field, a force fya?F 2 is required to prevent 

 the hemispheres from separating. 



33. A spherical shell, of radii a, b and inductive capacity K, is placed in a uniform 

 field of force F. Shew that the force inside the shell is uniform and equal to 



_ 9KF _ 



9 A' -2 (K- 1) 2 (6 3 /a 3 -l) ' 



34. The surface of a conductor being one of revolution whose equation is 



4 + !_l 



rr' 12' 



where r, r' are the distances of any point from two fixed points at distance 8 apart, find 

 the electric density at either vertex when the conductor has a given charge. 



35. The curve 



1 9a ( a-\-x ax 1 _ 1 



16 x+a*+* x-a*+* a 



when rotated round the axis of x generates a single closed surface, which is made the 

 bounding surface of a conductor. Shew that its capacity will be a, and that the surface 

 density at the end of the axis will be e/3?ra 2 , where e is the total charge. 



36. Two equal spheres each of radius a are in contact. Shew that the capacity of the 

 conductor so formed is 2a log e 2. 



