Which of the following most closely depicts the correct variation of the gravitation potential, $$V(r)$$ with distance $$r$$ due to a large planet of radius $$R$$ and uniform mass density? (figures are not drawn to scale)

For any point outside or on the surface, a spherical mass behaves as if all its mass $$M$$ is concentrated at its center.

$$V_{out} = -\frac{GM}{r}$$

This represents a rectangular hyperbola in the negative quadrant. As $$r \rightarrow \infty, V \rightarrow 0$$. At the surface ($$r = R$$), $$V_s = -\frac{GM}{R}$$.

For a uniform solid sphere, the potential at an internal point is given by:

$$V_{in} = -\frac{GM}{2R^3}(3R^2 - r^2)$$

This is a parabolic relation with respect to $$r$$. At the center ($$r = 0$$): $$V_c = -\frac{3GM}{2R} = 1.5 V_s$$. The potential at the center is $$1.5$$ times the potential at the surface (and more negative).

All these are correctly shown in option (c).