Consider a uniform spherical charge distribution of radius $R_{1}$ centred at the origin O. In this distribution, a spherical cavity of radius $R_{2}$ , centred at P with distance $O P = a = R_{1} - R_{2}$ (see figure) is made. If the electric field inside the cavity at position $\vec{r}$ is $\vec{E} (\vec{r})$ , then the correct statement(s) is (are)

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$\vec{E}$ is uniform, its magnitude is independent of $R_{2}$ but its direction depends on $\vec{r}$

$\vec{E}$ is uniform, its magnitude depends on $R_{2}$ and its direction depends on $\vec{r}$

$\vec{E}$ is uniform, its magnitude is independent of a but its direction depends on $\vec{a}$

$\vec{E}$ is uniform and both its magnitude and direction depend on $\vec{a}$

answer is D.

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Detailed Solution

The sphere with cavity can be assumed as a complete sphere with +ve charge, radius $R_{1}$ + another complete sphere with –ve charge, radius $R_{2}$ .

Inside a spherical charge distribution for an internal point $\vec{E} = \frac{ρ}{3 ε_{0}} \vec{r}$

$E_{+} \to E$ due to total +ve charges

$E_{-} \to E$ due to total $- v e$ charges

Electric filed at any point inside cavity is given by,

$\overset{}{\vec{E}} = {\overset{}{\vec{E}}}_{+} + {\overset{}{\vec{E}}}_{-} \Rightarrow \vec{E} = \frac{ρ}{3 ε_{0}} \vec{r_{1}} + \frac{(- ρ)}{3 ε_{0}} \vec{r_{2}} = \frac{ρ}{3 ε_{0}} (\vec{r_{1}} - \vec{r_{2}}) = \frac{ρ}{3 ε_{0}} \vec{a}$