Examples 283 



10. A hollow conductor has the form of a quarter of a sphere bounded by two 

 perpendicular diametral planes. Find the image of a charge placed at any point 

 inside. 



11. A conducting surface consists of two infinite planes which meet at right angles, 

 and a quarter of a sphere of radius a fitted into the right angle. If the conductor is at zero 

 potential, and a point-charge e is symmetrically placed with regard to the planes and the 

 spherical surface at a great distance / from the centre, shew that the charge induced on 

 the spherical portion is approximately 5ea 3 /rrf 3 . 



12. A point-charge is placed in front of an infinite slab of dielectric, bounded by a 

 plane face. The angle between a line of force in the dielectric and the normal to the face 

 of the slab is a ; the angle between the same two lines in the immediate neighbourhood of 

 the charge is /3. Prove that a, /3 are connected by the relation 



27 



13. An electrified particle is placed in front of an infinitely thick plate of dielectric. 

 Shew that the particle is urged towards the plate by a force 



K-l 



where d is the distance of the point from the plate. 



14. Two dielectrics of inductive capacities KI and < 2 are separated by an infinite plane 

 face. Charges e l5 e 2 are placed at points on a line at right angles to the plane, each at a 

 distance a from the plane. Find the forces on the two charges, and explain why they are 

 unequal. 



15. Two conductors of capacities c l5 c 2 in air are on the same normal to the plane 

 boundary between two dielectrics KJ, * 2 , a ^ great distances a, b from the boundary. They 

 are connected by a thin wire and charged. Prove that the charge is distributed between 

 them approximately in the ratio 



16. A thin plane conducting lamina of any shape and size is under the influence of a 

 fixed electrical distribution on one side of it. If cr\ be the density of the induced charge 

 at a point P on the side of the lamina facing the fixed distribution, and o- 2 that at the 

 corresponding point on the other side, prove that a-\ 0-2=0-0, where O-Q is the density at P 

 of the distribution induced on an infinite plane conductor coinciding with the lamina. 



17. An infinite plate with a hemispherical boss of radius a is at zero potential under 

 the influence of a point-charge e on the axis of the boss distant /from the plate. Find the 

 surface density at any point of the plate, and shew that the charge is attracted towards 

 the plate with a- force 



18. A conductor is formed by the outer surfaces of two equal spheres, the angle 

 between their radii at a point of intersection being 2?r/3. Shew that the capacity of the 

 conductor so formed is 



5\/3-4 



where a is the radius of either sphere. 



