86 Conductors and Condensers [OH. in 



12. Three insulated concentric spherical conductors, whose radii in ascending order 

 of magnitude are a, b, c, have charges e l5 e 2 , e 3 respectively, find their potentials and shew 

 that if the innermost sphere be connected to earth the potential of the outermost is 

 diminished by 



13. A conducting sphere of radius a is surrounded by two thin concentric spherical 

 conducting shells of radii b and c, the intervening spaces being filled with dielectrics of 

 inductive capacities K and L respectively. If the shell b receives a charge E, the other 

 two being uncharged, determine the loss of energy and the potential at any point when 

 the spheres A and C are connected by a wire. 



14. Three thin conducting sheets are in the form of concentric spheres of radii 

 a + o?, a, a c respectively. The dielectric between the outer and middle sheet is of 

 inductive capacity K, that between the middle and inner sheet is air. At first the outer 

 sheet is uninsulated, the inner sheet is uncharged and insulated, the middle sheet is 

 charged to potential V and insulated. The inner sheet is now uninsulated without 

 connection with the middle sheet. Prove that the potential of the middle sheet falls to 



KVc (a + d] 



15. Two insulated conductors A and B are geometrically similar, the ratio of their 

 linear dimensions being as L to L'. The conductors are placed so as to be out of each 

 other's field of induction. The potential of A is V and its charge is E, the potential 

 of B is V and its charge is E'. The conductors are then connected by a thin wire. 

 Prove that, after electrostatic equilibrium has been restored, the loss of electrostatic 

 energy is 



(EL'-E'L}(V-V'} 

 L+L' 



16. If two surfaces be taken in any family of equipotentials in free space, and two 

 metal conductors formed so as to occupy their positions, then the capacity of the 



C 1 C 1 



condenser thus formed is * * , where C\j C 2 are the capacities of the external and 

 Cx-Og 



internal conductors when existing alone in an infinite field. 



17. A conductor (B) with one internal cavity of radius b is kept at potential U. A 

 conducting sphere (A\ of radius a, at great height above B contains in a cavity water 

 which leaks down a very thin wire passing without contact into the cavity of B through 

 a hole in the top of B. At the end of the wire spherical drops are formed, concentric 

 with the cavity ; and, when of radius d, they fall passing without contact through a small 

 hole in the bottom of B, and are received in a cavity of a third conductor (C) of capacity c 

 at a great distance below B. Initially, before leaking commences, the conductors A and C 

 are uncharged. Prove that after the rih drop has fallen the potential of C is 



f ar(b-dy -.I^IT. 



-ady }c 



where the disturbing effect of the wire and hole on the capacities is neglected. 



18. An insulated spherical conductor, formed of two hemispherical shells in contact, 

 whose inner and outer radii are b and b', has within it a concentric spherical conductor of 

 radius a, and without it another spherical conductor of which the internal radius is c. 

 These two conductors are earth-connected and the middle one receives a charge. Shew 

 that the two shells will not separate if 



