Question

If V is the gravitational potential on the surface of the earth, then what is its value at the centre of the earth?
$
  {\text{A}}{\text{. 2}}V \\
  {\text{B}}{\text{. 3}}V \\
  {\text{C}}{\text{. }}\dfrac{3}{2}V \\
  {\text{D}}{\text{. }}\dfrac{2}{3}V \\
 $

Answer 1

Hint: We need the expression for the gravitational potential for the distances less than or equal to the radius of earth. From that we need to find out the gravitational potential at the surface of earth and at the centre of earth. Then by comparing the two expressions, we can get the required answer.
Formula used:
The expression for the gravitational potential for the distances less than or equal to the radius of earth is given as
${V_{inside}} = - \dfrac{{GM}}{{2{R^3}}}\left( {3{R^2} - {x^2}} \right)$

Complete step by step answer:
We know that the gravitational potential for the distances less than or equal to the radius of earth is given as
${V_{inside}} = - \dfrac{{GM}}{{2{R^3}}}\left( {3{R^2} - {x^2}} \right)$
Here G represents the gravitational constant, M is the mass of the earth, R is the radius of the earth while x denotes the distance from the centre of earth at which the gravitational potential is to be measured.
If V is the gravitational potential at the surface of the earth then let us find this value using the above expression by putting $x = R$. Doing so, we get
$V = - \dfrac{{GM}}{{2{R^3}}}\left( {3{R^2} - {R^2}} \right) = - \dfrac{{GM}}{{2{R^3}}} \times 2{R^2} = - \dfrac{{GM}}{R}$
Now let us find out this value at the centre of earth for which we have $x = 0$. This can be done in the following way.
${V_c} = - \dfrac{{GM}}{{2{R^3}}}\left( {3{R^2} - 0} \right) = - \dfrac{{GM}}{{2{R^3}}} \times 3{R^2} = - \dfrac{3}{2}\dfrac{{GM}}{R}$
Comparing this value with the value obtained on the surface of earth, we get
${V_c} = \dfrac{3}{2}V$
This is the required answer.

Hence, the correct answer is option C.

Note:
The gravitational potential represents the amount of work done per unit the mass of an object to bring that object from infinity to that particular point in the gravitational field. The obtained value of gravitational potential signifies that the gravitational potential at the centre of earth is greater than that at the surface of earth.