284 Methods for the Solution of Special Problems [CH. vni 



19. Within a spherical hollow in a conductor connected to earth, equal point-charges 

 e are placed at equal distances / from the centre, on the same diameter. Shew that each 

 is acted on by a force equal to 



1 



20. A hollow sphere of sulphur (of inductive capacity 3) whose inner radius is half its 

 outer is introduced into a uniform field of electric force. Prove that the intensity of the 

 field in the hollow will be less than that of the original field in the ratio 27 : 34. 



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

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

 hemispheres by a plane perpendicular to the field, these hemispheres tend to separate and 

 require forces equal to T 9 g 2 ^ 2 to keep them together. 



22. An uncharged insulated conductor formed of two equal spheres of radius a 

 cutting one another at right angles, is placed in a uniform field of force of intensity F, 

 with the line joining the centres parallel to the lines of force. Prove that the charges 

 induced on the two spheres are *g-Fa z and - 



23. A conducting plane has a hemispherical boss of radius a, and at a distance / from 

 the centre of the boss and along its axis there is a point-charge e. If the plane and the 

 boss be kept at zero potential, prove that the charge induced on the boss is 



24. A conductor is bounded by the larger portions of two equal spheres of radius a 

 cutting at an angle JTT, and of a third sphere of radius c cutting the two former or- 

 thogonally. Shew that the capacity of the conductor is 



25. Two infinite plane uninsulated conductors meet at an angle 60, and there is a 

 charge e at a point equidistant from each, and distant r from the line of intersection : find 

 the electrification at any point of the planes. Shew that at a point in a principal plane 

 through the charged point at a distance r \^3 from the line of intersection, the surface 

 density is 



471-r 2 \4 7 



26. If a particle charged with a quantity e of electricity be placed at the middle point 

 of the line joining the centres of two equal spherical conductors kept at zero potential, 

 shew that the charge induced on each sphere is 



- 2em (1 - m + m? - 3w 3 +4m 4 ), 



neglecting higher powers of m, which is the ratio of the radius to the distance between the 

 centres of the spheres. 



27. Two insulating conducting spheres of radii a, b, the distance c of whose centres 

 is large compared with a and 6, have charges E , E 2 respectively. Shew that the potential 

 energy is approximately 



