Two conducting sphere of radii a and b are placed at separation d. It is given that `d gt gt a and d gt gt b` so that charge distribution on both the sphere remains spherically symmetric. Assume that a charge `+ q` is given to the sphere of radius a and `-q` is given to the sphere of radius b.
(i) Write the electrostatic energy (U) of the system and calculate the capacitance of the system using the expression of U.
(ii) Prove that the capacitance of the system is given by `(1)/(C)=((1)/(4 pi in_(0)a)+(1)/(4 pi in_(0)b))"if" d rarr infty`

Answer

(i) `U=(q^(2))/(8 pi in_(0))((1)/(a)+(1)/(b)-(2)/(d));C=(4 pi in_(0))/((1)/(a)+(1)/(b)-(2)/(d))`