Question

If the potential at the surface of a planet of mass M and radius R is assumed to be zero. The potential at the center of planet is .

• A. $\frac{GM}{2R}$
• B. $\frac{-GM}{2R}$
• C. zero
• D. not defined.

Answer 1

The correct option is B $\frac{-GM}{2R}$

Let gravitational potential at center be X
Gravitational potential due to Planet/Solid sphere at an internal point is $=\frac{-GM}{2R^3}(3R^2-r^2)$
(Assuming potential at infinity to be Zero.)

But given gravitational potential at the surface is zero.

Energy required to move 1kg mass from Planets surface to center of planet is constant irrespective of our assumptions or references.
Which is $=(\frac{-GM}{R}-\frac{-3GM}{2R})=\left(\frac{GM}{2R}\right)$
So change in potential in this case is ( zero(on surface) - X )
$\Rightarrow \frac{GM}{2R}=(0-X)\Rightarrow X=-\frac{GM}{2R}$