In a certain region of space, the gravitational field is given by `-(k)/(r)` where `r` is the distance and k is a constant. If the gravitaional potential at `r=r_(0)` be `V_(0)`, then what is the expression for the gravitaional potential (V)-

Options

A. `klog((r)/(r_(0)))`
B. `k log((r_(0))/(r))`
C. `V_(0)+klog((r)/(r_(0)))`
D. `V_(0)+klog((r_(0))/(r))`

Explanation

To find the expression for the gravitational potential \( V \) given the gravitational field \( E = -\frac{k}{r} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between gravitational field and potential**:
The gravitational field \( E \) is related to the gravitational potential \( V \) by the equation:
\[ E = -\frac{dV}{dr} \]
This means that the gravitational field is the negative gradient (rate of change) of the gravitational potential.

2. **Substitute the given gravitational field**:
We know that the gravitational field in this case is given by:
\[ E = -\frac{k}{r} \]
Therefore, we can write:
\[ -\frac{dV}{dr} = -\frac{k}{r} \]
This simplifies to:
\[ \frac{dV}{dr} = \frac{k}{r} \]

3. **Integrate to find the potential**:
To find \( V \), we need to integrate \( \frac{dV}{dr} \):
\[ dV = \frac{k}{r} dr \]
Integrating both sides gives:
\[ V = k \ln r + C \]
where \( C \) is the constant of integration.

4. **Determine the constant of integration**:
We are given that the gravitational potential at \( r = r_0 \) is \( V_0 \). Therefore, we can use this condition to find \( C \):
\[ V_0 = k \ln r_0 + C \]
Rearranging gives:
\[ C = V_0 - k \ln r_0 \]

5. **Substitute back to find the expression for \( V \)**:
Now, substituting \( C \) back into the expression for \( V \):
\[ V = k \ln r + (V_0 - k \ln r_0) \]
This simplifies to:
\[ V = V_0 + k \ln \left(\frac{r}{r_0}\right) \]

### Final Expression:
Thus, the expression for the gravitational potential \( V \) is:
\[ V = V_0 + k \ln \left(\frac{r}{r_0}\right) \]