Field of a uniformly charged sphere

Advanced

Grows linearly inside.

Key relationE = kQr/R³

Outside E = kQ/r²; inside a uniformly charged insulating sphere E = kQr/R³, growing linearly from the centre.

Worked example

At r = R/2 the field is half the surface value.

Common mistake

A conducting sphere has zero field inside.

In more detail

Inside a uniformly charged solid sphere, the field grows linearly with distance from the centre. Outside, it matches a point charge.

Modelling charged spheres such as nuclei and charged droplets.
Designing electrostatic experiments with spherical electrodes.
Physicists model the field inside a uniformly charged sphere, which helps describe charged particles and nuclei.

Why does the field grow inside?

Because more of the charge is enclosed as the radius increases.

Where is the field largest?

At the surface of the sphere.

Why does the field rise inside the sphere?

Inside, only the charge closer to the centre contributes, and that amount increases with radius.