288 Methods for the Solution of Special Problems [OH. vm 



52. A uniform circular wire of radius a charged with electricity of line density e 

 surrounds an insulated concentric spherical condenser of radius c ; prove that the 

 electrical density at any point of the surface of the conductor is 



53. A dielectric sphere is surrounded by a thin circular wire of larger radius 6- 

 carrying a charge E. Prove that the potential within the sphere is 



rY 



54. If within a conductor formed by a cone of semi-vertical angle cos" 1 PQ and two- 

 spherical surfaces r=a, r=6 with centres at the vertex of the cone, a charge q on the axis 

 at distance / from the vertex gives potential F, and if we write 



A. 



the summation with respect to m extending to all positive integers, and that with respect 

 to n to all numbers integral or fractional for which P n (^ ) = 0, determine A mn . Effecting 

 the summation with respect to m, shew that when r < /, 



and that when r > r', 



55. A spherical shell of radius a with a little hole in it is freely electrified to potential 

 F. Prove that the charge on its inner surface is less than VSl&ira, where S is the area of 

 the hole. 



56. A thin spherical conducting shell from which any portions have been removed is 

 freely electrified. Prove that the difference of densities inside and outside at any point is 

 constant. 



57. Electricity is induced on an uninsulated spherical conductor of radius a, by a 

 uniform surface distribution, density o-, over an external concentric non-conducting 

 spherical segment of radius c. Prove that the surface density at the point A of the 

 conductor at the nearer end of the axis of the segment is 



c(o4-a)/ AB\ 

 a* \ ADJ ' 



where B is the point of the segment on its axis, and D is any point on its edge. 



58. Two conducting discs of radii a, a', are fixed at right angles to the line which 

 joins their centres, the length of this line being r, large compared with a. If the first 

 have potential F and the second is uninsulated, prove that the charge on the first is 



59. A spherical conductor of diameter a is kept at zero potential in the presence of a. 

 fine uniform wire, in the form of a circle of radius c in a tangent plane to the sphere with 



