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. $k\text{ln}\frac{r}{r_0}$
• B. $k\text{ln}\frac{r_0}{r}$
• C. $V_0+k\text{ln}\frac{r}{r_0}$
• D. $V_0+k\text{ln}\frac{r_0}{r}$

Answer 1

The correct option is C $V_0+k\text{ln}\frac{r}{r_0}$

Given that,
Gravitational field, $E=-\frac{k}{r}$
The relation between gravitational $\left(E\right)$ field and gravitational potential is given by
$dV=-\overrightarrow{E}.\overrightarrow{dr}$
The change in potential is given by,
$\Rightarrow \underset{V_0}{\overset{V}{\int }}dV=-\underset{r_0}{\overset{r}{\int }}\frac{-k}{r}dr$
$\Rightarrow V-V_0=k{\left\{\text{ln}r\right\}}_{r_0}^r$
$\Rightarrow V=V_0+k\text{ln}\frac{r}{r_0}$

Hence, option (c) is correct.