Examples 289 



its centre at the point of contact, which has a charge E of electricity ; prove that the 

 electrical density induced on the sphere at a point whose direction from the centre of the 

 ring makes an angle -^ with the normal to the plane is 



r(a 2 + c 2 sec 2 \ls - 2ac tan <ds cos 0) ~ * dtf. 



60. Prove that the capacity of a hemispherical shell of radius a is 



/! , 1\ 



a U+~)- 



\* T/ 



61. Prove that the capacity of an elliptic plate of small eccentricity e and area A is 

 approximately 



V \n)n \ I *M + M)' 



62. A circular disc of radius a is under the influence of a charge q at a point in its 

 plane at distance b from the centre of the disc. Shew that the density of the induced 

 distribution at a point on the disc is 



q /W^tf 



I^B? V 2 -r 2 ' 

 where r, R are the distances of the point from the centre of the disc and the charge. 



63. An ellipsoidal conductor differs but little from a sphere. Its volume is equal to 

 that of a sphere of radius r, its axes are 2r(l+a), 2r (1 + /3), 2r(l+y). Shew that neg- 

 lecting cubes of a, /3, y, its capacity is 



64. A prolate conducting spheroid, semi-axes a, 6, has a charge E of electricity. Shew 

 that repulsion between the two halves into which it is divided by its diametral plane is 



Determine the value of the force in the case of a sphere. 



65. One face of a condenser is a circular plate of radius a : the other is a segment of 

 a sphere of radius R, R being so large that the plate is almost flat. Shew that the 

 capacity is \KR log tjto where ^, t are the thickness of dielectric at the middle and edge 

 of the condenser. Determine also the distribution of the charge. 



66. A thin circular disc of radius a is electrified with charge E and surrounded by a 

 conductor with charge E^ placed so that the edge of the disc is the locus of the focus S of 

 the generating ellipse. Shew that the energy of the system is 



B being an extremity of the polar axis of the spheroid, and C the centre. 



67. If the two surfaces of a condenser are concentric and coaxial oblate spheroids of 

 small ellipticities and t' and polar axes 2c and 2c', prove that the capacity is 



neglecting squares of the ellipticities ; and find the distribution of electricity on each 

 surface to the same order of approximation. 



19 



