Question

Answer in brief:

A spherical shell of radius b with charge Q is expanded to a radius a. Find the work done by the electrical forces in the process.

Answer 1

Consider a spherical conducting shell of radius r placed in a medium of permittivity ε. On the charged conductor, the mechanical force per unit area is

f = `"F"/"dS" = sigma^2/(2ε)`

Where σ is the conductor's surface charge density.
Given, the charge on the spherical shell is Q, σ = Q/4πr². The force acts outward, normal to the surface.

Consider the force displaces a charged area element σdS through a small distance dx, then the work done by the force is

dW = Fdx = `(sigma^2/(2ε) "dS")` dx

During the displacement, the area element sweeps out a volume dV = dS × dx.

Since V = `4/3 pi "r"^3`, dV = 4πr² dr

∴ dW = `sigma^2/(2ε)"dV" = 1/(2ε)("Q"/(4pi"r"^2))^2 (4pi"r"^2 "dr")`

`= "Q"^2/(8piε) 1/"r"^2 "dr"`

Therefore, the work done by the force in expanding the shell from radius r = b to r = a is

W = `int "dW" = "Q"^2/(8piε)` \[\int\limits_b^a \frac{1}{r^2}\] dr

`= "Q"^2/(8piε) [- 1/"r"]_"b"^"a" = "Q"^2/(8piε) (1/"b" - 1/"a")`

This gives the required expression for the work done.