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The mass density inside a solid sphere of radius $r$ varies as $\rho(r)=\rho_0\left(\frac{r}{R}\right)^\beta$, where $\rho_0$ and $\beta$ are constants and $r$ is the distance from the centre. Let $E_1$ and $E_2$ be gravitational fields due to sphere at distance $\frac{R}{2}$ and $2 R$ from the centre of sphere. If $\frac{E_2}{E_1}=4$, the value of $\beta$ is
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3
$$
\text { Mass enclosed in a sphere of radius } r \text { is }
$$
$$
\begin{aligned}
M_p & =\int \rho d V=\int_0^\rho \rho_0\left(\frac{r}{R}\right)^\beta \cdot 4 \pi r^2 d r \\
& =\left[\frac{4 \pi \rho_0}{R^\beta} \cdot \frac{r^{\beta+3}}{\beta+3}\right]_0^r
\end{aligned}
$$
So, mass enclosed in a sphere of radius $\frac{R}{2}$ is
$$
M_1=\frac{4 \pi \rho_0}{R^\beta} \frac{\left(\frac{R}{2}\right)^{\beta+3}}{(\beta+3)}=\frac{4 \pi \rho_0 R^3}{\beta+3)} \times \frac{1}{2^{\beta+3}}
$$
and mass enclosed in radius $R$ is
$$
M_2=\frac{4 \pi \rho_0}{R^\beta} \cdot \frac{R^{\beta+3}}{\beta+3}=\frac{4 \pi \rho_0 R^3}{\beta+3}
$$
So, gravitational field intensities are
$$
E_1=\frac{G M_1}{\left(\frac{R}{2}\right)^2}=\frac{4 G}{R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3} \times \frac{1}{2^{\beta+3}}
$$
and $E_2=\frac{G M_2}{(2 R)^2}=\frac{G}{4 R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3}$
As $\frac{E_2}{E_1}=4$ we get
$$
\begin{array}{cc}
& \frac{\frac{G}{4 R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3}}{\frac{4 G}{R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3} \times \frac{1}{2^{\beta+3}}}=4 \\
\Rightarrow \quad & \frac{2^{\beta+3}}{16}=4 \Rightarrow 2^{\beta+3}=64 \\
2^\beta .2^3=2^6 \Rightarrow 2^\beta=8 \Rightarrow 2^\beta=2^3 \Rightarrow \beta=3
\end{array}
$$
\text { Mass enclosed in a sphere of radius } r \text { is }
$$
$$
\begin{aligned}
M_p & =\int \rho d V=\int_0^\rho \rho_0\left(\frac{r}{R}\right)^\beta \cdot 4 \pi r^2 d r \\
& =\left[\frac{4 \pi \rho_0}{R^\beta} \cdot \frac{r^{\beta+3}}{\beta+3}\right]_0^r
\end{aligned}
$$
So, mass enclosed in a sphere of radius $\frac{R}{2}$ is
$$
M_1=\frac{4 \pi \rho_0}{R^\beta} \frac{\left(\frac{R}{2}\right)^{\beta+3}}{(\beta+3)}=\frac{4 \pi \rho_0 R^3}{\beta+3)} \times \frac{1}{2^{\beta+3}}
$$
and mass enclosed in radius $R$ is
$$
M_2=\frac{4 \pi \rho_0}{R^\beta} \cdot \frac{R^{\beta+3}}{\beta+3}=\frac{4 \pi \rho_0 R^3}{\beta+3}
$$
So, gravitational field intensities are
$$
E_1=\frac{G M_1}{\left(\frac{R}{2}\right)^2}=\frac{4 G}{R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3} \times \frac{1}{2^{\beta+3}}
$$
and $E_2=\frac{G M_2}{(2 R)^2}=\frac{G}{4 R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3}$
As $\frac{E_2}{E_1}=4$ we get
$$
\begin{array}{cc}
& \frac{\frac{G}{4 R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3}}{\frac{4 G}{R^2} \times \frac{4 \pi \rho_0 R^3}{\beta+3} \times \frac{1}{2^{\beta+3}}}=4 \\
\Rightarrow \quad & \frac{2^{\beta+3}}{16}=4 \Rightarrow 2^{\beta+3}=64 \\
2^\beta .2^3=2^6 \Rightarrow 2^\beta=8 \Rightarrow 2^\beta=2^3 \Rightarrow \beta=3
\end{array}
$$
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