Classical Electrodynamics · Problem 2.5

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · work-done-removing-charge-grounded-sphere

Problem

(a) Show that the work done to remove the charge

q

from a distance

r > a

to infinity against the force, Eq. (2.6), of a grounded conducting sphere is

W = \frac{q^2 a}{8\pi\epsilon_0(r^2 - a^2)}

Relate this result to the electrostatic potential, Eq. (2.3), and the energy discussion of Section 1.11. (b) Repeat the calculation of the work done to remove the charge

q

against the force, Eq. (2.9), of an isolated charged conducting sphere. Show that the work done is

W = \frac{1}{4\pi\epsilon_0}\left[\frac{q^2 a}{2(r^2 - a^2)} - \frac{q^2 a}{2r^2} - \frac{qQ}{r}\right]

Relate the work to the electrostatic potential, Eq. (2.8), and the energy discussion of Section 1.11.

Intuitive explanation

PART (A)

When you move a charge away from a grounded sphere, you're working against the attractive force from the induced surface charges—the sphere stays at zero potential. The key is that work equals the charge times the change in potential:

W = q\Delta V

. Since the sphere is grounded, its potential is fixed at zero, so you only need to find how the potential at the charge's location changes as it moves from

r

to infinity.

PART (B)

Now the sphere is isolated and holds a fixed total charge

Q

, so its potential changes as you move the external charge away. The complication is that the sphere's potential depends on both its own charge and the presence of the external charge. Again use

W = q\Delta V_{\text{at charge}}

, but now

V

includes contributions from both the sphere's charge and the induced redistribution—this is why you'll see multiple terms reflecting the interaction energy and self-energy changes.