Physics - Gravitation Question with Solution | TestHub

The gravitational field $E$ and potential $V$ are related by $E=-\frac{dV}{dr}$.

Given $E=-\frac{k}{r}$, we have $dV=-Edr=-(-\frac{k}{r})dr=\frac{k}{r}dr$.

Integrating from $d_i$ to $r$:

${\int }_{V_i}^{V}dV={\int }_{d_i}^r\frac{k}{r^{\prime }}d{r}^{\prime }$

$V-V_i=k[\ln r^{\prime }{]}_{d_i}^r$

$V-V_i=k(\ln r-\ln d_i)$

$V-V_i=k\ln \frac{r}{d_i}$

$V=k\ln \frac{r}{d_i}+V_i$