$(a)\;\large\frac{V}{4}\qquad(b)\;\large\frac{V}{2}\qquad(c)\;\large\frac{3 V}{4}\qquad(d)\;None\;of\;these$

Answer : (d) None of these

Explanation :

$v=\Bigg \{ \begin{matrix} \large \frac{\rho (3 R^3-r^2)}{6\;\in_{0}\;} & r \leq R \\ \large\frac{\rho \;R^3}{3 \in_{0}\;r} & r \geq R \end{matrix}$

$v|_{r=R} = V =\large\frac{\rho\;R^2}{3 \in_{0}}$

$V_{1}=v|_{r=\large\frac{R}{2}} = V =\large\frac{\rho}{6 \in_{0}}\;(3 R^2-\large\frac{R^2}{4}) = \large\frac{11\;\rho\;R^2}{24 \in_{0}}$

$\large\frac{V_{1}}{V} =\large\frac{11\times3}{24} = \large\frac{11}{8}\;.$

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