Electrostatic energy of a sphere

Advanced

Energy of a uniformly charged sphere.

Key relationU = 3Q²/20πε₀R

A uniformly charged sphere has U = 3Q²/(20πε₀R) = (3/5)kQ²/R. A classical electron radius arises by setting this equal to mc².

Worked example

r_e = 2.8×10⁻¹⁵ m.

Common mistake

Includes self-energy, not just pair energy.

In more detail

The electrostatic energy of a uniformly charged sphere is U = 3kQ²/(5R). It represents the work needed to assemble the charge on the sphere.

Estimating the energy required to build charged droplets.
Nuclear physics models the Coulomb energy of nuclei this way.
Physicists estimate the energy of a charged nucleus by treating it as a uniformly charged sphere.

Why is the energy positive?

Because the like charges repel, so energy must be added to bring them together.

Does it depend on the sphere material?

No, only on charge and radius for a conductor or uniform insulator.

Why does a sphere of charge have energy?

Assembling the charge requires work against the repulsion of the parts, so the energy depends on the square of the charge.