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(B) The relationship between gravitational potential $V$ and gravitational field $E$ is given by $dV = -\vec{E} \cdot d\vec{r}$.Given $E = -\frac{K}{r

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$V = V_0 + K \ln \left( \frac{r}{r_0} \right)$

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(B) The relationship between gravitational potential $V$ and gravitational field $E$ is given by $dV = -\vec{E} \cdot d\vec{r}$.Given $E = -\frac{K}{r}$,we have $dV = -(-\frac{K}{r}) dr = \frac{K}{r} dr$.Integrating both sides from the reference point $(r_0, V_0)$ to an arbitrary point $(r, V)$:$\int_{V_0}^{V} dV = \int_{r_0}^{r} \frac{K}{r} dr$.$V - V_0 = K [\ln(r)]_{r_0}^{r}$.$V - V_0 = K (\ln(r) - \ln(r_0))$.$V - V_0 = K \ln \left( \frac{r}{r_0} \right)$.Therefore,$V = V_0 + K \ln \left( \frac{r}{r_0} \right)$.

In a certain region of space,the gravitational field is given by $E = -\frac{K}{r}$,where $r$ is the distance from a fixed point and $K$ is a constant. Taking the reference point to be at $r = r_0$ with potential $V = V_0$,the potential at a distance $r$ is:

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