Question

Let ${{V}_{G}}$​ and ${{E}_{G}}$​ denote gravitational potential and field respectively, the it is possible to have (This question has multiple correct answers)
A: ${{V}_{G}}=0,{{E}_{G}}=0$
B: ${{V}_{G}}\ne 0,{{E}_{G}}=0$
C: ${{V}_{G}}=0,{{E}_{G}}\ne 0$
D: ${{V}_{G}}\ne 0,{{E}_{G}}\ne 0$

Answer 1

The gravitational field is the space around a body where another body feels gravitational force due to this body and gravitational potential at a point in a gravitational field is the work done per unit mass needed to move a body to desired location. Proceed to answer by keeping this in mind.

Formulas used:
Magnitude of gravitational field strength:
$g=\dfrac{GM}{{{r}^{2}}}$
Value of gravitational potential:
$V=\dfrac{-GM}{r}$
where G is the gravitational constant, M is the mass of the body and r is the distance from the body.

Complete step by step answer:
When we take the value of the distance as infinity, that is $r=\infty$, and substitute in the formulas of gravitational field and gravitational potential, we get the value as zero. Thus,
${{V}_{G}}=0,{{E}_{G}}=0$ is possible. Hence, option A is correct.
Let us consider ${{V}_{\infty }}=\dfrac{GM}{R}$
This implies that
$\begin{aligned} & {{V}_{R}}=0 \\\ & \Rightarrow {{E}_{R}}=\dfrac{GM}{{{R}^{2}}} \\\ \end{aligned}$
Hence when ${{V}_{G}}\ne 0,{{E}_{G}}=0$. Option B is also correct.
Now when we consider a spherical shell , we can say that the electric field inside it is zero but the potential is $V=-\dfrac{GM}{R}$
Hence when ${{V}_{G}}=0,{{E}_{G}}\ne 0$. Hence option C is correct.
Now, when we consider a point at a distance r from mass m .
Then, $V=-\dfrac{GM}{r},E=\dfrac{GM}{{{r}^{2}}}$
Hence, ${{V}_{G}}\ne 0,{{E}_{G}}\ne 0$. Option D is also correct.

Hence we can conclude that all the given options are correct.

Note: In questions like this, we should analyse each option since the question has more than one option as a correct answer. Students shouldn’t stop answering after getting a single option as correct. They must verify the other options too.