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 $\left(\frac{r}{r_0}\right)$
• B. k log $\left(\frac{r_0}{r}\right)$
• C. $V_0+klog\left(\frac{r}{r_0}\right)$
• D. $V_0+klog\left(\frac{r_0}{r}\right)$

Answer 1

Answer: (C)

$V_0+klog\left(\frac{r}{r_0}\right)$

Gravitational potential $V=-\int \overrightarrow{E}.\overrightarrow{dr}$

$V_r-V_o=-{\int }_{r_o}^r\overrightarrow{E}.\overrightarrow{dr}$

$V_r-V_o=-{\int }_{r_o}^r-|E|dr$

$V_r-V_o={\int }_{r_o}^r|E|dr$

$V_r-V_o={\int }_{r_o}^r\frac{k}{r}dr$

$V_r-V_o=klog\left(r\right){|}_{r_o}^r$

$V_r-V_o=klog\left(r\right)-klog\left(r_o\right)$ (using $loga-logb=log\left[\frac{a}{b}\right]$)

$V_r=V_o+klog\left(\frac{r}{r_o}\right)$