Why Does Expanding a Charged Spherical Shell Require Work?

[harini_5]

Homework Statement

A spherical shell of radius R with uniform charge Q is expanded to a radius H.find the work done by electric forces during the shell expansion
given answer is 1/(8pi epsilon) *Q^2(1/R-1/H)
my attempt
work=-change in P E
=Q(V1-V2)
=1/(4pi epsilon) *Q^2(1/R-1/H)
please explain the additional 1/2 factor

 

[rl.bhat]

Here you have to consider the energy stored in a spherical capacitor.
What is the capacity of a spherical conductor?
What is the energy stored in a capacitor?

 

[harini_5]

what is the mistake in my approach?
why not take change in potential energy as work done?

 

[rl.bhat]

what is the mistake in my approach?
why not take change in potential energy as work done?

Because the electric field in continuously changing during the expansion of the spherical conductor. To find the total work done you have to take the integration.
Energy stored in the capacitor= 1/2*Q^2/C
Capacity of the spherical conductor = 4πεοR