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

(i) `U_(n)=q_(n)^(2)/(8pi in_(0) R)` (ii) `U_(oo)=(Q^(2)R)/(8pi in_(0) r^(2))` Here `q_(n)=(QR)/r[1-(R/(R+r))^(n)]`

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`
(i) Charges are distributed in the ratio of their capacities. Let in the first constant, charge acquired by `S_(2)`, is `q_(1)`,. Therefore, charge on `S_(1)` will be `Q-q`.
Say it is `q'_(1) :. q_(1)/(q'_(1))=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
`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_(1)=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)]`
`rArr q_(n)=QR/r[1-(R/(R+r))^(n)]`...(i) `[S_(n)=(a(1-r^(n)))/((1-r))]`
Therefore, electrostatic energy of `S_(2)` after n such contacts
`U_(n)=(q_(n)^(2))/(2C)=q_(n)^(2)/(2(4piepsilon_(0)R))rArr U_(n)=q_(n)^(2)/(8 piepsilon_(0)R)`
where `q_(n)` can be written from Eq. (2)
(ii) `q_(n)=(QR)/(R+r)[1+R/(R+r)+...+...+(R/(R+r))^(n-1)]`
as `n rarr oo`
`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^(2)R^(2)//r^(2))/(8piepsilon_(0)R)rArr U_(oo)=(Q^(2)R)/(8piepsilon_(0)r^(2))`