Examples 137 



11. A condenser is formed of two parallel plates, distant h apart, one of which is 

 at zero potential. The space between the plates is filled with a dielectric whose inductive 

 capacity increases uniformly from one plate to the other. Shew that the capacity per unit 

 area is 



where K^ and K 2 are the values of the inductive capacity at the surfaces of the plate. The 

 inequalities of distribution at the edges of the plates are neglected. 



12. A spherical conductor of radius a is surrounded by a concentric spherical 

 conducting shell whose internal radius is 6, and the intervening space is occupied by a 



dielectric whose specific inductive capacity at a distance r from the centre is ~^- . If the 

 inner sphere is insulated and has a charge J, the shell being connected with the earth, 

 prove that the potential in the dielectric at a distance r from the centre is log - ^ . 



13. A spherical conductor of radius a is surrounded by a concentric spherical shell of 

 radius b, and the space between them is filled with a dielectric of which the inductive 

 capacity at distance r from the centre is ne~ I)2 p~ s where p=r/a. Prove that the capacity 



of the condenser so formed is 



52 



_ 



14. If the specific inductive capacity varies as e~, where r is the distance from 

 a fixed point in the medium, verify that a solution of the differential equation satisfied by 

 the potential is 



/a\ 2 f - r 

 (r) I 6 - l -a- 



and hence determine the potential at any point of a sphere, whose inductive capacity is 

 the above function of the distance from the centre, when placed in a uniform field of 

 force. 



15. Shew that the capacity of a condenser consisting of the conducting spheres r=a, 

 r=b, and a heterogeneous dielectric of inductive capacity K=f (6, $), is 



16. In an imaginary crystalline medium the molecules are discs placed so as to be 

 all parallel to the plane of xy. Shew that the components of intensity and polarisation 

 are connected by equations of the form 



