112 Systems of Conductors [OH. iv 



9. Four equal uncharged insulated conductors are placed symmetrically at the corners 

 of a regular tetrahedron, and are touched in turn by a moving spherical conductor at the 

 points nearest to the centre of the tetrahedron, receiving charges e 1} e 2 , e 3 , e. Shew that 

 the charges are in geometrical progression. 



10. In question 9 replace " tetrahedron " by " square," and prove that 



(e 1 -e 2 )(e 1 e 3 - e 2 2 ) = e (e 2 e 3 - e l e 4 ). 



11. Shew that if the distance x between two conductors is so great as compared with 

 the linear dimensions of either, that the square of the ratio of these linear dimensions to 

 x may be neglected, then the coefficient of induction between them is CC'/%, where (7, C' 

 are the capacities of the conductors when isolated. 



12. Two insulated fixed condensers are at given potentials when alone in the electric 

 field and charged with quantities EI , E% of electricity. Their coefficients of potential are 

 Piii Ji2j PM- But if they are surrounded by a spherical conductor of very large radius R 

 at potential zero with its centre near them, the two conductors require charges EI, E 2 to 



produce the given potentials. Prove, neglecting -^ , that 



13. Shew that the locus of the positions, in which a unit charge will induce a given 

 charge on a given uninsulated conductor, is an equipotential surface of that conductor 

 supposed freely electrified. 



14. Prove (i) that if a conductor, insulated in free space and raised to unit potential, 

 produce at any external point P a potential denoted by (P), then a unit charge placed at 

 P in the presence of this conductor uninsulated will induce on it a charge - (P) ; 



(ii) that if the potential at a point Q due to the induced charge be denoted by (PQ\ 

 then (PQ) is a symmetrical function of the positions of P and Q. 



15. Two small uninsulated spheres are placed near together between two large 

 parallel planes, one of which is charged, and the other connected to earth. Shew by 

 figures the nature of the disturbance so produced in the uniform field, when the line of 

 centres is (i) perpendicular, (ii) parallel to the planes. 



16. A hollow conductor A is at zero potential, and contains in its cavity two other 

 insulated conductors, B and C, which are mutually external : B has a positive charge, and 

 C is uncharged. Analyse the different types of lines of force within the cavity which are 

 possible, classifying with respect to the conductor from which the line starts, and the 

 conductor at which it ends, and proving the impossibility of the geometrically possible 

 types which are rejected. 



Hence prove that B and C are at positive potentials, the potential of C being less than 

 that of B. 



17. A portion P of a, conductor, the capacity of which is (7, can be separated from the 

 conductor. The capacity of this portion, when at a long distance from other bodies, is c. 

 The conductor is insulated, and the part P when at a considerable distance from the 

 remainder is charged with a quantity e and allowed to move under the mutual attraction 

 up to it ; describe and explain the changes which take place in the electrical energy of the 

 system. 



