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

8

Explanation

Capacities of conducting spheres are in the ratio of their radii `C_(1)` and `C_(2)` be the capacitites of `S_(1)` and `S_(2)` then therefore charge on `S_(1)` will be `Q - q_(1) ` say it is `Q_(1)`
`therefore (q_(1))/(q_(1)) = (q_(1))/(Q- q_(1)) = (C_(2))/(C_(1)) = (R)/(r) therefore q_(1) = Q ((R)/(R+r))`
This charge is again distributed in the same ratio. Therefore, charge on in second contact,
`q_(2) = Q (1+ (R)/(R + R)) ((R)/(R+r)) = Q [(R)/(R+r) + ((R)/(R+r))]`
Similarly
`q_(3) = - [(R)/(R+r)((R)/(R+r))^(3) ] and q_(0) - Q[(R)/(R+r) + (R)/(R+R))^(2) + ..... + ((R)/(R+r))^(n)]`
or `q_(0) = Q (R)/(r)[ (1-((R)/(R+r))^(n)] .... (ii)`
[S_(n) - (a(1-r^(n))/((1-r))]`
Therefore, electrostatic energy of after n such contacts `= (q_(pi)^(2))/(2 (4 pi epsi_(0)R))` or `U_(n) = (Q^(2) R)/( 8 pi epis_(0) r^(2))`
Where `q_(n)1` can be written from Eq .(ii)
Therefore, minimum value of speed v should be