Question
Prove that gravitational potential difference is the work done in carrying a unit mass from one point to another.
✓
Answer
Gravitational potential at A $=-\frac{\text{GM}}{\text{r}_{\text{a}}}$
Gravitational potential at B $=-\frac{\text{GM}}{\text{r}_{\text{b}}}$
Difference in potential $=-\text{GM}\Big(\frac{1}{\text{r}_{\text{b}}}-\frac{1}{\text{r}_{\text{a}}}\Big)$
Work done in carrying a units mass from A to B is,
$\text{W}=\int^\limits{\text{r}_{\text{b}}}_\limits{\text{r}_{\text{a}}}\frac{\text{GM}}{\text{x}^2}$
$\text{W}=-\text{GM}\Big|\frac{1}{\text{x}}\Big|^{\text{r}_{\text{b}}}_{\text{r}_{\text{a}}}=-\text{GM}\Big(\frac{1}{\text{r}_{\text{b}}}-\frac{1}{\text{r}_{\text{a}}}\Big)$
So, work done and gravitational potential difference are same.
Gravitational potential at B $=-\frac{\text{GM}}{\text{r}_{\text{b}}}$
Difference in potential $=-\text{GM}\Big(\frac{1}{\text{r}_{\text{b}}}-\frac{1}{\text{r}_{\text{a}}}\Big)$
Work done in carrying a units mass from A to B is,
$\text{W}=\int^\limits{\text{r}_{\text{b}}}_\limits{\text{r}_{\text{a}}}\frac{\text{GM}}{\text{x}^2}$
$\text{W}=-\text{GM}\Big|\frac{1}{\text{x}}\Big|^{\text{r}_{\text{b}}}_{\text{r}_{\text{a}}}=-\text{GM}\Big(\frac{1}{\text{r}_{\text{b}}}-\frac{1}{\text{r}_{\text{a}}}\Big)$
So, work done and gravitational potential difference are same.
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