286 Methods for the Solution of Special Problems [CH. vm 



37. Two spheres of radii a, b are in contact, a being large compared with b. Shew 

 that if the conductor so formed is raised to potential F, the charges on the two spheres are 



/ 7T 2 fi 2 



Va(\--" 



38. A conducting sphere of radius a is in contact with an infinite conducting plane. 

 Shew that if a unit point-charge be placed beyond the sphere and on the diameter through 

 the point of contact at distance c from that point, the charges induced on the plane and 

 sphere are 



TTOL TTd , 7TO, . TTd 



-- cot and cot -- 1 . 



39. Prove that if the centres of two equal uninsulated spherical conductors of radius 

 a be at a distance 2c apart, the charge induced on each by a unit charge at a point 

 midway between them is 



where c=acosha. 



40. Shew that the capacity of a spherical conductor of radius a, with its centre at a 

 distance c from an infinite conducting plane, is 



00 



a sinh a 2J cosech wa, 

 where c=acosha. 



41. An insulated conducting sphere of radius a is placed midway between two 

 parallel infinite uninsulated planes at a great distance 2c apart. Neglecting ( - ) , shew 

 that the capacity of the sphere is approximately 



a jl+|log2J. 



42. Two spheres of radii r 1} r 2 , touch each other, and their capacities in this position 

 are c l5 c 2 . Shew that 



{oo "1 

 / 2 I^ 



where /= 



43. A conducting sphere of radius a is placed in air, with its centre at a distance c 

 from the plane face of an infinite dielectric. Shew that its capacity is 



a sinh a 5) ( jf y ) cosech na, 

 where a=c/a. 



44. A point-charge e is placed between two parallel uninsulated infinite conducting 

 planes, at distances a and b from them respectively. Shew that the potential at a point 

 between the planes which is at a distance z from the charge and is on the line through the 

 charge perpendicular to the planes is 



\ r'( 2a z 



) \2a + 2b 



