= Electric field of a uniformly charged ball
{title2=$E_r(r)=Qr/(4\pi\varepsilon_0R^3)\quad(0\leq r\leq R)$}
= Electric field of a uniformly charged solid sphere
{synonym}
Distribute a positive <electric charge> $Q$ uniformly through a <closed ball> of radius $R$ in vacuum, with no external field. Its volume <charge density> is $3Q/(4\pi R^3)$. For $0<r<R$, the enclosed fraction is $(r/R)^3$. Applying <Gauss's law> on a concentric <Gaussian surface> gives
$$
E_r(r)=\begin{cases}\dfrac{Qr}{4\pi\varepsilon_0R^3},&0\leq r\leq R,\\[4pt]\dfrac{Q}{4\pi\varepsilon_0r^2},&r\geq R.\end{cases}
$$
The <electric field> grows linearly inside and decays as an inverse square outside. With $E_R=Q/(4\pi\varepsilon_0R^2)$, its values at $r/R=1/2,1,2$ are $E_R/2,E_R,E_R/4$. The field vanishes at the centre by <spherical symmetry> and is continuous at the surface because this ideal volume distribution has no singular surface layer. Its interior behavior differs from a <conductor in electrostatic equilibrium>, where mobile charge redistributes.
\Image[/gauss-charged-ball-and-shell.png]
{title=A uniform volume charge and a uniform thin-shell charge have the same exterior field but different interior fields}
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