124] Examples 111 



EXAMPLES. 



1. If the algebraic sum of the charges on a system of conductors be positive, then on 

 one at least the surface density is everywhere positive. 



2. There are a number of insulated conductors in given fixed positions. The 

 capacities of any two of them in their given positions are Cj and C 2 , and their mutual 

 coefficient of induction is B. Prove that if these conductors be joined by a thin wire, the 

 capacity of the combined conductor is 



3. A system of insulated conductors having been charged in any manner, charges are 

 transferred from one conductor to another till they are all brought to the same potential F. 

 Shew that 



where s 1} s 2 are the algebraic sums of the coefficients of capacity and induction respectively, 

 and E is the sum of the charges. 



4. Prove that the effect of the operation described in the last question is a decrease 

 of the electrostatic energy equal to what would be the energy of the system if each of the 

 original potentials were diminished by F. 



5. Two equal similar condensers, each consisting of two spherical shells, radii a, 6, 

 are insulated and placed at a great distance r apart. Charges e, e' are given to the inner 

 shells. If the outer surfaces are now joined by a wire, shew that the loss of energy is 

 approximately 



6. A condenser is formed of two thin concentric spherical shells, radii a, b. A small 

 hole exists in the outer sheet through which an insulated wire passes connecting the 

 inner sheet with a third conductor of capacity c, at a great distance r from the condenser. 

 The outer sheet of the condenser is put to earth, and the charge on the two connected 

 conductors is E. Prove that approximately the force on the third conductor is 



7. Two closed equipotentials F 1? F are such that Fj contains F , and F P is the 

 potential at any point P between them. If now a charge E be put at P t and both 

 equipotentials be replaced by conducting shells and earth connected, then the charges 

 EIJ EQ induced on the two surfaces are given by 



E E E 



8. A conductor is charged from an electrophorus by repeated contacts with a plate, 

 which after each contact is recharged with a quantity E of electricity from the electro- 

 phorus. Prove that if e is the charge of the conductor after the first operation, the 

 ultimate charge is 



Ee 

 E-e' 



