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



68. An accumulator is formed of two confocal prolate spheroids, and the specific 

 inductive capacity of the dielectric is A7/csr, where tzr is the distance of any point from the 

 axis. Prove that the capacity of the accumulator is 



irKl/\Og 



where a, b and a\ , 6 : are the semi-axes of the generating ellipses. 



69. A thin spherical bowl is formed by the portion of the sphere x i +y i +z L = cz 

 bounded by and lying within the cone 2 + p = -^ > and is put in connection with the earth 



by a fine wire. is the origin, and (7, diametrically opposite to 0, is the vertex of the 

 bowl ; Q is any point on the rirn, and P is any point on the great circle arc CQ. Shew 

 that the surface density induced at P by a charge E placed at is 



EC CQ 



r . . dd 



where 



70. Three long thin wires, equally electrified, are placed parallel to each other so that 

 they are cut by a plane perpendicular to them in the angular points of an equilateral 

 triangle of side *j3c ; shew that the polar equation of an equipotential curve drawn on the 



plane is 



r 6 + c 6 - 2r 3 c 3 cos 30 = constant, 



the pole being at the centre of the triangle and the initial line passing through one of the 

 wires. 



71. A flat piece of corrugated metal (y=asinm#) is charged with electricity. Find 

 the surface density at any point, and shew that it exceeds the average density approxi- 

 mately in the ratio my : 1. 



72. A long hollow cylindrical conductor is divided into two parts by a plane through 

 the axis, and the parts are separated by a small interval. If the two parts are kept at 

 potentials F x and F 2 , the potential at any point within the cylinder is 



Fj + F 2 Fj F 2 _ l 2ar cos 

 2 TT a*-r 2 ' 



where r is the distance from the axis, and is the angle between the plane joining the 

 point to the axis and the plane through the axis normal to the plane of separation. 



73. Shew that the capacity per unit length of a telegraph wire of radius a at height h 

 above the surface of the earth, is 



74. An electrified line with charge e per unit length is parallel to a circular cylinder, 

 of radius a and inductive capacity K, the distance of the wire from the centre of the 

 cylinder being c. Shew that the force on the wire per unit length is 



K-\ 



75. A cylindrical conductor of infinite length, whose cross-section is the outer 

 boundary of three equal orthogonal circles of radius a has a charge e per unit length. 

 Prove that the electric density at distance r from the axis is 



e (3r 2 + a 2 ) (3r 2 - a 2 - \/6ar) (3r 2 - a 2 + >/6ar) 



