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

B

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` ten
`C_2/C_1=R/r`
a. Charges are disturbed 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_2=q_1/(Q-q_1)=C_2/C_1=R/r`
It implies that `Q` charge is to be distributed in `S_2` and `S_1` in the ratio of `R//r`.
`:. q_1=Q(R/(R+r))`......i
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+2)(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]` ........ii
`[S_n=(a(1-r^n))/((1-r))]`
Therefore, electrostatic energy of `S_2` after `n` such contacts
`U_n=a_n^2/(2C)=(Qn)^(2)/(R+r)[1+R/(R+r)+...+...+(R/(R+r)^(n-1)]`
as `nrarroo`
`q_oo=(QR)/(R+r)(1/(1-R/(R+r)))`
`=(QR)/(R+r)((R+r)/r)=QR/r [S_oo=a/(1-r)]`
`:. U_oo=q_oo^2/(2C)=(Q^2R^2//r^2)/(8piepsilon_0R)`
or `U_oo=(Q^2R)/(8piepsilon_0r^2)`