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There is a solid sphere of radius $R$ having uniformly distributed charge. What is the relation between electric field $E$ inside the sphere and radius of the sphere $R$?

1. $E \propto R^2$
2. $E \propto R$
3. $E \propto \frac{1}{R^3}$
4. $E \propto \frac{1}{R^2}$

Correct answer: $E \propto R$

Solution

For a uniformly charged solid sphere, the electric field at a distance $r<R$ is $E=\frac{1}{4\pi\varepsilon_0}\frac{Qr}{R^3}$. If the point is taken at the surface, $r=R$, so the field varies as $E\propto R$ for fixed charge density. Hence the correct relation is linear in $R$.

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