Question

If $V$ is the gravitational potential on the surface of the earth, then what will be its value at the centre of the earth?

• A. $2V$
• B. $3V$
• C. $\frac{3V}{2}$
  
• D. $\frac{2V}{3}$

Answer 1

The correct option is C $\frac{3V}{2}$


The gravitational potential due to a solid sphere $($like earth$)$ of radius $R$ at a distance $r$ from its centre is given by $$V=-\frac{GM}{2R^3}(3R^2-r^2);(r\le R)$$ Let $V_S$ be the potential at the surface.

At the surface of earth, $r=R$,
$\therefore V_S=-\frac{GM}{R}=V\left(\text{say}\right)......\left(1\right)$

Let $V_C$ be the potential at center.

At the centre, $r=0$,

$\therefore V_C=-\frac{3}{2}\frac{GM}{R}$

$\Rightarrow V_C=\frac{3V}{2}$ $[$from $1]$

Hence, option (c) is the correct answer.