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Field Outside Uniformly Charged Solid Sphere

Field outside a uniformly charged solid sphere of radius R and total charge Q. Identical in form to a point charge at the centre.
Class 11Class 12
Derivation

Setup

A solid sphere of radius RR carries total charge QQ uniformly distributed throughout its volume. Volume charge density:

ρ=Q43πR3=3Q4πR3\rho = \frac{Q}{\frac{4}{3}\pi R^3} = \frac{3Q}{4\pi R^3}

Gaussian surface for r>Rr > R

Choose a concentric spherical Gaussian surface of radius r>Rr > R.

Entire charge QQ is enclosed. By spherical symmetry:

E⋅4πr2=Qε0E \cdot 4\pi r^2 = \frac{Q}{\varepsilon_0} E=Q4πε0r2(r>R)\boxed{E = \frac{Q}{4\pi\varepsilon_0 r^2} \quad (r > R)}

Identical to point charge

Outside, the solid sphere behaves as a point charge at its centre — regardless of how the charge is distributed internally (shell or solid), as long as the distribution is spherically symmetric.

Note
The field is continuous at $r = R$: approaching from outside gives $E = Q/(4\pi\varepsilon_0 R^2)$, which matches the surface value from the inside derivation.