The gravitational potential at surface of earth is -v0. The mass of earth is M and its radius is R. Find gravitational potential at centre of earth.

1. zero
2. -v₀
3. (-v₀ + GM/2R)
4. (-v₀ + GM/2R)

Dear Student,

Please find below the solution to the asked query:

Gravitational potential due to a solid sphere for the point lies inside the sphere

We calculate potential in two parts. For this we consider a concentric smaller sphere of radius “r” such that point “P” lies on the surface of sphere. Now, the potential due to whole sphere is split between two parts :



Expression for the gravitational field at the point inside the solid sphere is,

$V_r=-\frac{GM}{2a^3}\left(3a^2-r^2\right)$
Given that, the potential at the surface of the solid sphere is "-V₀". So,

$V_a=-V_0=-\frac{GM}{2a^3}\left(3a^2-a^2\right)=-\frac{GM}{a}$
At the centre of the solid sphere,

$$\begin{gathered} V_C=-\frac{GM}{2a^3}\left(3a^2-0^2\right)=-\frac{3GM}{2a} \\ \Rightarrow V_C=-\frac{GM}{a}-\frac{GM}{2a} \\ \Rightarrow V_C=-V_0-\frac{GM}{2a} \end{gathered}$$

 

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