A conducting sphere S 1 of radius r is attached to an insulating handle. Another conducting 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 then recharged such that the charge on it is again Q & it is again brought into contact with S 2 & removed. This procedure is repeated n times
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Sol. 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
= 
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 ’
∴ 
= 
= 
= 
.
It implies that Q charge is to be distributed in S 2 and S 1 in the ratio of R/r.
∴ q 1 = Q 
......(1)
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 
This charge is again distributed in the same ratio. Therefore, charge on S 2 in second contact,
∴ q 2 = Q 
= Q 
Similarly, q 3 = Q 
and q n = Q 
or q n = Q 
..........(i)
Therefore, electrostatic energy of S 2 after n such contacts
U n = 
= 
or U n = 
where q n can be written from equation (1).
q n = 
as n → ∞
∴ 
= 
= 
∴ 
= 
= 
or 
= 
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