Question

A solid sphere of mass $M$ and radius $R$ has a spherical cavity of radius $\frac{R}{2}$ such that the centre of cavity is at a distance $\frac{R}{2}$ from the centre of the sphere. The force of attraction that this sphere would exert on a particle of mass $m$ which lies at a distance $d(>R)$ from the centre of the solid sphere on the straight line joining the centre of the sphere and the center of cavity is

• A. $\frac{GMm}{d^2}$
• B. $\frac{GMm}{d^2}[1-\frac{1}{8{(1-\frac{R}{2d})}^2}]$
• C. $\frac{GMm}{d^2}[1+\frac{1}{8{(1-\frac{R}{2d})}^2}]$
• D. $\frac{GMm}{8d^2}$

Answer 1

The correct option is B $\frac{GMm}{d^2}[1-\frac{1}{8{(1-\frac{R}{2d})}^2}]$

Given that,
Radius of solid sphere $=R$
Mass of the solid sphere $=M$

The density of the solid sphere, $\rho =\frac{M}{\frac{4}{3}\pi R^3}$


Radius of the spherical cavity $=\frac{R}{2}$.

Let $M^{\prime }$ be the mass of the spherical cavity.

$M^{\prime }=\frac{M}{\frac{4}{3}\pi R^3}\times \frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3$

$\Rightarrow M^{\prime }=\frac{M}{8}$

Let the gravitational field at $\text{P}$ due to the complete sphere is $\overrightarrow{E_{\text{com}}}$.

Let us assume that the solid sphere is not there and only removed sphere of radius $\frac{R}{2}$ is there.

Now the gravitational field due to this spherical cavity at $\text{P}$ is ${\overrightarrow{E}}_{\text{rem}}$.

Then, the net gravitational field at $\text{P}$ is

$\overrightarrow{E_{net}}=\overrightarrow{E_{com}}-\overrightarrow{E_{rem}}$

$\Rightarrow \overrightarrow{E_{net}}=\frac{GM}{d^2}-\frac{G{M}^{\prime }}{{(d-R/2)}^2}$

$\Rightarrow \overrightarrow{E_{net}}=\frac{GM}{d^2}-\frac{GM}{8{(d-R/2)}^2}$

$\Rightarrow \overrightarrow{E_{net}}=\frac{GM}{d^2}-\frac{GM}{8d^2{(1-\frac{R}{2d})}^2}$

$\Rightarrow \overrightarrow{E_{net}}=\frac{GM}{d^2}[1-\frac{1}{8{(1-\frac{R}{2d})}^2}]$

Force on particle $m$,

$F=m{E}_{net}=\frac{GMm}{d^2}[1-\frac{1}{8{(1-\frac{R}{2d})}^2}]$

Hence, option (b) is the correct answer.

Why this question: To enhance the understanding of electric field at any point due to the combination of solid and hollow bodies.