Question
In a certain region of space, gravitational field is given by $$I = - \left( {\frac{K}{r}} \right).$$ Taking the reference point to be at $$r = {r_0}$$ with $$V = {V_0},$$ find the potential.
A.
$$K\log \frac{r}{{{r_0}}} + {V_0}$$
B.
$$K\log \frac{{{r_0}}}{r} + {V_0}$$
C.
$$K\log \frac{r}{{{r_0}}} - {V_0}$$
D.
$$\log \frac{{{r_0}}}{r} - {V_0}r$$
Answer :
$$K\log \frac{r}{{{r_0}}} + {V_0}$$
Solution :
We know that intensity is negative gradient of potential,
i.e., $$I = - \left( {\frac{{dV}}{{dr}}} \right)$$ and as here $$I = - \left( {\frac{K}{r}} \right),$$ so,
$$\eqalign{
& \frac{{dV}}{{dr}} = \frac{K}{r},\,\,{\text{i}}{\text{.e}}{\text{.,}}\,\,\int d V = K\int {\frac{{dr}}{r}} \cr
& {\text{or}}\,\,V - {V_0} = K\log \frac{r}{{{r_0}}} \cr
& {\text{so}}\,\,V = K\log \frac{r}{{{r_0}}} + {V_0} \cr} $$