Examples 113 



18. A conductor having a charge $1 is surrounded by a second conductor with charge 

 Q 2 . The inner is connected by a wire to a very distant uncharged conductor. It is then 

 disconnected, and the outer conductor connected. Shew that the charges $/, $ 2 ', are now 



Qi'> 



- - 



m+n + mn' m+n 



where C, C(\+m) are the coefficients of capacity of the near conductors, arid Cn is the 

 capacity of the distant one. 



19. If one conductor contains all the others, and there are n+l in all, shew that 

 there are n+l relations between either the coefficients of potential or the coefficients of 

 induction, and if the potential of the largest be F , and that of the others V 1 , F 2> ... F n , 

 then the most general expression for the energy is |<7F 2 increased by a quadratic function 

 of FI- F , F 2 - F , ... V H - F ; where C is a definite constant for all positions of the 

 inner conductors. 



20. The inner sphere of a spherical condenser (radii a, b) has a constant charge E, 

 and the outer conductor is at potential zero. Under the internal forces the outer 

 conductor contracts from radius b to radius bi. Prove that the work done by the 

 electric forces is 



i F2 b ~ b i 



"~- 



21. If, in the last question, the inner conductor has a constant potential F, its charge 

 being variable, shew that the work done is 



1 

 27 



and investigate the quantity of energy supplied by the battery. 

 22. With the usual notation, prove that 



Pll+P23>Pl2+Pl3 



23. Shew that if p rr , p rs , p 88 be three coefficients before the introduction of a new 

 conductor, and p rr ', p r8 , p s8 ' the same coefficients afterwards, then 



( Prr PSB -prrPss'} 4 ( Prs -Prs}*> 



24. A system consists of p + q + 2 conductors, A^A^...A^ lt B^...B^ C, D. Prove 

 that when the charges on the A's and on (7, and the potentials of the 's and of C are 

 known, there cannot be more than one possible distribution in equilibrium, unless C is 

 electrically screened from D. 



25. A, B, C, D are four conductors, of which B surrounds A and D surrounds C. 

 Given the coefficients of capacity and induction 



(i) of A and B when C and D are removed, 

 (ii) of C and D when A and B are removed, 

 (iii) of B and Z> when A and C are removed, 

 determine those for the complete system of four conductors. 



26. Two equal and similar conductors A and B are charged and placed symmetrically 

 with regard to each other ; a third moveable conductor C is carried so as to occupy 



J. 8 



