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## Homework Statement

- How much work is required to squeeze a uniformly charged spherical shell from a radius of ##r## to a radius of ##r−dr##, if

(a) the total charge q is a constant,

(b) the sphere is kept at a constant potential, e.g. grounded.

(c) Are the answers the same or different? Explain.

## Homework Equations

## The Attempt at a Solution

I'd like some help in checking that I have applied the right concept to the problem in my solution, for part a) first at least.

There is electrostatic pressure acting on the charges on the sphere's surface. To squeeze these charges in , one has to overcome such this force.

The electrostatic pressure is

$$F/A = \frac{\epsilon _0}{2} E^2$$

The electric-field magnitude on the sphere is

$$E = k\frac{q}{r^2}$$

The force causing such a pressure is

$$F = 2\pi r^2 \epsilon _0 E^2$$

$$F = \frac{1}{8\pi \epsilon _0}\frac{q^2}{r^2}$$

And the infinitesimal amount of work needed to squeeze these charges by ##dr## is

$$W_{on} = -W_{by} = -\frac{1}{8\pi \epsilon _0}\frac{q^2}{r^2} dr$$

Does any of this make sense? Thanks in advance!