Discussion
For uniformly charged solid sphere of radius R, which of the following statements is correct about the magnitude of electric field at a distance r from the center of the sphere ?
Answer: C
Inside the sphere the field is proportional to r.
Apply Gauss's law to a sphere of radius r < R. It encloses the fraction \((r/R)^{3}\) of the total charge Q:
\[ \begin{aligned} E\cdot 4\pi r^{2} &= \frac{Q}{\varepsilon_{0}}\frac{r^{3}}{R^{3}} \\ E &= \frac{Qr}{4\pi\varepsilon_{0}R^{3}} \end{aligned} \]
so E grows linearly from zero at the centre to its maximum at the surface, then falls as \(1/r^{2}\) outside. E = 0 inside describes a conductor or a charged spherical shell, not a uniformly charged solid sphere. A constant non-zero field or a 1/r field inside does not satisfy Gauss's law for a uniform volume charge.
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