A charge $q$ is uniformly distributed in the hollow sphere of radii $r_{1}$ and $r_{2} (r_{2} > r_{1})$ . The electric field at a point $P$ situated at distance $x$ from the centre (for $r_{1} < x < r_{2}$ ) is :

\frac{Q (x)}{4 π ε_{0} (r_{1}^{3} - r_{1}^{3})}

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\frac{Q (x^{3} - r_{1}^{3})}{4 π ε_{0} (r_{2}^{3} - r_{1}^{3})}

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\frac{Q (x^{3} - r_{1}^{3})}{4 π ε_{0} x^{2} (r_{2}^{3} - r_{1}^{3})}

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\frac{Q (r_{1}^{3})}{4 π ε_{0} x^{2} (r_{3}^{3} - r_{1}^{3})}

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Solution

The correct option is C $\frac{Q (x^{3} - r_{1}^{3})}{4 π ε_{0} x^{2} (r_{2}^{3} - r_{1}^{3})}$
If $ρ$ be the volume charge density, $Q = ρ . \frac{4}{3} π (r_{2}^{3} - r_{1}^{3}) . . . (1)$
Let $E$ be the electric field at distance $x$ .
using Gauss's law, $E .4 π x^{2} = \frac{Q_{e n c l o s e d}}{ϵ_{0}} = \frac{ρ . \frac{4}{3} π (x^{3} - r_{1}^{3})}{ϵ_{0}}$
$E = \frac{Q (x^{3} - r_{1}^{3})}{4 π ϵ_{0} x^{2} (r_{2}^{3} - r_{1}^{3})}$ using (1)

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