Examples 287 



45. A spherical conductor of radius a is surrounded by a uniform dielectric K, which 

 is bounded by a sphere of radius b having its centre at a small distance y from the centre 

 of the conductor. Prove that if the potential of the conductor is F, and there are no 

 other conductors in the field, the surface density at a point where the radius makes an 

 angle B with the line of centres is 



KVb ( 6(K-l)ya?cos0 1 



Ia + b + 2 (#- 



46. A shell of glass of inductive capacity K, which is bounded by concentric spherical 

 surfaces of radii a, b (a < 6), surrounds an electrified particle with charge E which is at a 

 point Q at a small distance c from 0, the centre of the spheres. Shew that the potential 

 at a point P outside the shell at a distance r from Q is approximately 



E 2Ec(b*-a s )(K-l) z cos 6 



where 6 is the angle which QP makes with OQ produced. 



47. If the centres of the two shells of a spherical condenser be separated by a small 

 distance d, prove that the capacity is approximately 



ab 



48. A condenser is formed of two spherical conducting sheets, one of radius b 

 surrounding the other of radius a. The distance between the centres is c, this being so 

 small that (c/a) 2 may be neglected. The surface densities on the inner conductor at the 

 extremities of the axis of symmetry of the instrument are <TI, o- 2 , and the mean surface 

 density over the inner conductor is v. Prove that 



49. The equation of the surface of a conductor is r = a (l+P n ) where e is very small, 

 and the conductor is placed in a uniform field of force F parallel to the axis of harmonics. 

 Shew that the surface density of the induced charge at any point is greater than it would 

 be if the surface were perfectly spherical, by the amount 



47r(2?i + l) l 



50. A conductor at potential V whose surface is of the form r=a(l+cP n ) is sur- 

 rounded by a dielectric (K] whose boundary is the surface r=b (I+r)P n ), and outside this 

 the dielectric is air. Shew that the potential in the air at a distance r from the origin is 



KabV PI (2n + I)a n b 2n + 1 +(K-I}r,b n W n + l +(n+I}a* n + l } P n 



(K-\}a+b_r (1+n+nK) b*" + l +(K-l) (n+l) d* n + l 



where squares and higher powers of e and 77 are neglected. 



51. The surface of a conductor is nearly spherical, its equation being 



where e is small. Shew that if the conductor is uninsulated, the charge induced on 

 it by a unit charge at a distance / from the origin and of angular coordinates 0, is 

 approximately 



