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

(i) When `S_1` and `S_2` come in contact, there is transfer of charges till the potentials of the two spheres become equal. During first contact,
`V_1 = V_2 (q_1 " charge shifts from " S_1 " to " S_2)`
`rArr (K(Q - q_1))/(r) = (Kq_1)/rR " or " q_1 = Q (R/(R + r))`
During second contact, again
`V_1 = V_2`
`rArr (K[Q - (q_2 - q_1)]]/r = (K q_2)/R`
`[(q_2 - q_1) "charge shifts from" S_1 "to" S_2]`
`:. q_2 = Q[(R)/(R + r)+ ((R)/(R + r))^2]`
On third contact, again
`V_1 = V_2`
`rArr (K[Q-(q_3 - q_2)])/(r) = (K q_3)/(R)`
`[(q_3 - q_2) "charge shifts from" S_1 "to" S_2]`
`:. q_3 = Q[(R)/(R + r)+ ((R)/(R + r))^2 + ((R)/(R + r))^3]`
On nth contact, by symmertry
`V_1 = V_2`
`rArr (K[Q - (q_n - q_(n-1))])/(r) = (K q_n)/(R)`
`[(q_n - q_(n-1) "charge shift from" S_1 "to" S_2]`
`q_n = Q[(R)/(R + r) + ((R)/(R + r))^2 +...+ ((R)/(R + r))^n]`
=`(Q R)/(r) [1 -((R)/(R + r))^n]`
The eletrostatic energy of `S_2` after n contacts is
`U_n = (1)/(2) (q_n^2)/(C) = (1)/(2) xx (1)/(4 pi epsilon_0 R) xx {(Q R)/(r) [1 -((R)/(R + r))^n]}^2`
(ii) The limiting value is
` underset(n rarr oo)(Lt) U_n = underset(n rarr oo)(Lt) [(1)/(2) xx (1)/(4 pi epsilon_0 R){(Q R)/(r)[1 -((R)/(R + r))^n]}^2]`
=`(Q^2 R)/(2( 4 pi epsilon_0)r^2)`.