Solving Irodov Electrodynamics Homework: Work Required to Transfer Charge q

[zorro]

Homework Statement

A point charge q is located at the centre O of a spherical uncharged conducting layer provided witha small orifice as shown in figure. The inside and outside radii of the layer are equal to a and b resp. What amount of work has to be performed to slowly transfer a charge q from the point O to infinity through the orifice?

The Attempt at a Solution

The point charge induces a negative charge (-q) on inner surface and an equal positive charge (q) on outer surface.
We can calculate the potential at point O.
V1 = -kq/a + kq/b
taking potential at infinity as 0,
V2=0
Now work done = ( V2 - V1 )q
W = kq²(1/a - 1/b)

The answer given is slightly different. There is an additional 2 in the denominator.
Any help appreciated.

 

[Dickfore]

As you move the charge, the charge distribution on the sphere changes. This gives an additional factor of 2.

What you should do is calculate the force acting on the charge from the sphere if the charge is at distance $x$ from its center. Then, you should integrate this force with respect to $x$ from $0$ to $\infty$. Use the method of images.

 

[zorro]

That would be very tedious.
I found a better method anyways.
Thanks!