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Gravitational Potential

In a certain ...

Question

In a certain region of space, the gravitational field is given by $- \frac{k}{r}$ , where r is the distance and k is a constant. If the gravitational potential at $r = r_{0}$ be $V_{0}$ , then what is the expression for the gravitational potential (V).

A

k log

(\frac{r}{r_{0}})

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B

k log

(\frac{r_{0}}{r})

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C

V_{0} + k l o g (\frac{r}{r_{0}})

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D

V_{0} + k l o g (\frac{r_{0}}{r})

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Solution

The correct option is B $V_{0} + k l o g (\frac{r}{r_{0}})$
Gravitational potential $V = - \int \to E . \to d r$
$V_{r} - V_{o} = - \int_{r_{o}}^{r} \to E . \to d r$
$V_{r} - V_{o} = - \int_{r_{o}}^{r} - | E | d r$
$V_{r} - V_{o} = \int_{r_{o}}^{r} | E | d r$
$V_{r} - V_{o} = \int_{r_{o}}^{r} \frac{k}{r} d r$
$V_{r} - V_{o} = k l o g (r) {∣ ∣ ∣}_{r_{o}}^{r}$
$V_{r} - V_{o} = k l o g (r) - k l o g (r_{o})$ (using $l o g a - l o g b = l o g [\frac{a}{b}]$ )
$V_{r} = V_{o} + k l o g (\frac{r}{r_{o}})$

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