A conducting sphere `S_1` of radius `r` is attached to an insulating handle. Another conduction sphere `S_2` of radius `R` is mounted on an insulating stand. `S_2` is initially uncharged. `S_1` is given a charge `Q` brought into contact with `S_2` and removed. `S_1` is recharge such that the charge on it is again `Q` and it is again brought into contact with `S_2` and removed. This procedure is repeated `n` times.
a. Find the electrostatic energy of `S_2` after `n` such contacts with `S_1`.
b. What is the limiting value of this energy as `nrarroo` ?

Answer

A, B, D

Explanation

Limiting value of energy as `nrarroo`. Let us calculate `q_n` when n tends to `oo`.
For G.P., `S_(oo)=(a)/(1-eta)` where `eta=` common ratio
`:.` `q_(oo)=(QR)/(R+r)[(1)/(1-(R)/(R+r))]` or `q_(oo)=(QR)/(r)`
`:.` `U_(oo)=(q_(oo)^2)/(2C)=((QR)/(r))^2xx(1)/(2xx(4piepsilon_0)xx(R))`
or `U_(oo)=(Q^2R^2)/(r^2xx2xx4piepsilon_0R)` or `U_(oo)=(Q^2R)/(2(4piepsilon_0)r^2)`