13-05-10 : Electrostatics

As manmay explained this question to me, for every time the two spheres are brought in contact, their potentials equalize. So we have to use that fact to get the answer. The spheres are not identical.

We know that V = kQRadius

Now on the first touch,
k(Q - q₁)R = kq₁r

On the second touch,
k(Q - q₂)R = k(q₁ + q₂)r

And so on till the n^th touch where

k(Q - q_n)R = k(q₁ + q₂ + q₃ + .......+ q_n)r

We need a relation between q_n and Q.
Note from the first equation that
q₁ = QRR + r

Let RR + r be "a".

q₂ = Qa² and so on.
Therefore q_n = Qaⁿ

So our last equation becomes :
Q - QaⁿR = (Sum of GP)r

The sum of GP is given by :
Sum = Q(1 - aⁿ)1 - a

Ok got it :
U_n = (Total charge)²2C
Total charge acquired after n touches = Q₀ = q₁ + q₂ + q₃...

U_n = (q₁ + q₂ + q₃ + ..... + q_n)²2C
= Q²(a + a² + .... + aⁿ)²2C
= Q²a²(1 - aⁿ)²2C(1 - a)²

For the second part of the question, if n → ∞, the GP becomes an infinite one.
So the sum of the GP becomes a1 - a

Hence U_n = Q²a²2C(1 - a)²