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Let $ρ$ (r)= $\frac{Q}{{πR}^{4}} r$ be the charge density distribution for a solid sphere of radius R and total charge Q for a point ’p’ inside the sphere at distance $r_{1}$ from the centre of the sphere, the magnitude of electric field is

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a

$\frac{Q {r_{1}}^{2}}{3 π ε_{0} R^{4}}$

b

$\frac{Q {r_{1}}^{2}}{4 π ε_{0} R^{4}}$

c

0

d

$\frac{Q}{4 π ε_{0} {r_{1}}^{2}}$

answer is C.

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Detailed Solution

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$\begin{array}{l} q = \frac{4}{3} π r^{3} ρ, d q = 4 π r^{2} d r . ρ, q = \int_{0}^{r_{1}} 4 π r^{2} d r . ρ \\ q = \int_{0}^{r_{1}} 4 π r^{2} d r \times \frac{Q r}{π R^{4}}, q = \frac{4 Q}{R^{4}} \times \frac{{r_{1}}^{4}}{4} = \frac{Q {r_{1}}^{4}}{R^{4}} \end{array}$

$\begin{array}{l} E = \frac{1}{4 π \in_{0}} \frac{q}{{r_{1}}^{2}} = \frac{1}{4 π \in_{0}} \times \frac{Q {r_{1}}^{4}}{R^{4} {r_{1}}^{2}} \end{array} = \frac{1}{4 π \in_{0}} \frac{Q {r_{1}}^{4}}{R^{4}}$

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Let ρ (r)= QπR4r be the charge density distribution for a solid sphere of radius R and total charge Q for a point ’p’ inside the sphere at distance r1 from the centre of the sphere, the magnitude of electric field is