136 Dielectrics and Inductive Capacity [CH. v 



5. The outer coating of a long cylindrical condenser is a thin shell of radius a, and 

 the dielectric between the cylinders has inductive capacity K on one side of a plane, 

 through the axis, and K' on the other side. Shew that when the inner cylinder is 

 connected to earth, and the outer has a charge q per unit length, the resultant force on 

 the outer cylinder is 



ira(K+K') 

 per unit length. 



6. A heterogeneous dielectric is formed of n concentric spherical layers of specific 

 inductive capacities K\, K^ ... K n , starting from the innermost dielectric, which forms a 

 solid sphere ; also the outermost dielectric extends to infinity. The radii of the spherical 

 boundary surfaces are 1} a 2 ,...a n _ l respectively. Prove that the potential due to a 

 quantity Q of electricity at the centre of the spheres at a point distant r from the centre 

 in the dielectric K s is 



5Yi_l 



K\r a, 



7. A condenser is formed by two rectangular parallel conducting plates of breadth 

 b and area A at distance d from each other. Also a parallel slab of a dielectric of thickness 

 t and of the same area is between the plates. This slab is pulled along its length from 

 between the plates, so that only a length x is between the plates. Prove that the electric 

 force sucking the slab back to its original position is 



where t' = t(K- l)/K, K is the specific inductive capacity of the slab, E is the charge, and 

 the disturbances produced by the edges are neglected. 



8. Three closed surfaces 1, 2, 3 are equipotentials in an electric field. If the space 

 between 1 and 2 is filled with a dielectric K, and that between 2 and 3 is filled with a 

 dielectric K'> shew that the capacity of a condenser having 1 and 3 for faces is (7, given by 



where A, B are the capacities of air-condensers having as faces the surfaces 1, 2 and 2, 3 

 respectively. 



9. The surface separating two dielectrics (K^ K 2 ) has an actual charge a- per unit 

 area. The electric forces on the two sides of the boundary are FI, F 2 at angles c 1? c 2 with 

 the common normal. Shew how to determine F%, and prove that 



10. The space between two concentric spheres radii a, b which are kept at potentials 

 A, B, is filled with a heterogeneous dielectric of which the inductive capacity varies as 

 the nth power of the distance from their common centre. Shew that the potential at any 

 point between the surfaces is 



A-B 



