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` ?

Explanation

Capacities of conducting spheres are in the ratio of their radii. Let `C_1` and `C_2` be the capacities of `S_1` and `S_2` then
`C_2/C_1=R/r`
a. Charges are distributed in the ratio of their capacities. Let in the first contact, charge, acquired by `S_2` is `q_1`. Therefore, charge on `S_1` will be `Q-q_1`. Say it is `q_1^'`
`:. q_1/q_1^'=q_1/(Q-q_1)=C_2/C_1=R/v`
`:. q_1=Q(R/(R+r))`
In the second contact `S_1` again acquires the same charge `Q`
Therefore, total charge in `S_1` and `S_2` will be
`Q+q_1=Q(1+R/(R+r))`
This charge is again distributed in the same ratio. Therefore, charge on `S_2` in second contact,
`q_2=Q(1+R/(R+r))(R/(R+r))`
`=Q[R/(R+r)(R/(R+r))^2]`
Similarly, `q_3=Q[R/(R+r)+(R/(R+r))^2+(R/(R+r))^3]`
and `q_n=Q[R/(R+r)(R/(R+r))^2+...+(R/(R+r))^n]`
or `q_n=QR/r[1-(R/(R+r))^n]`
`[S_n=(a(1-r^n))/((1-r))]`
Therefore electrostatic energy of `S_2` after `n` such contacts
`=q_n^2/(2(4piepsilon_0R))` or `U_n=q_n^2/(8piepsilon_0R)`
where `q_n`can be written from eqn ii
b. As `nrarroo`
`q_oo=QR/r`
`:. U_oo=q_oo^2/(2C)=(Q^2R^2//r^2)/(8piepsilon_0R)`
or `U_oo=(Q^2R)/(8piepsilon_0r^2)`