Question

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

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

Answer 1

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

At a point inside the solid sphere: $V_{inside}=\frac{-GM}{2R^3}(3R^2-x^2)$

$V_{inside}=-GM(\frac{3}{2R}-\frac{r^2}{2R^3})$; (if x$=$R)

At a point on the surface of the solid sphere:$V_{surface}=\frac{-GM}{R}$; (If x$=$R)

At a point outside the solid sphere:$V_{outside}=\frac{-GM}{x}$; (If x$=$R)

At the centre: $x=0$

$\Rightarrow V_c=-\frac{3}{2}\frac{GM}{R}=\frac{3}{2}{V}_{surface}$ in case of solid sphere potential is maximum at centre.

We have,

Gravitational potential on Earth $=V$

Now,

Gravitaional potential at the centre of the Earth $=V_C=\frac{3}{2}{V}_{surface}=\frac{3}{2}V$