Two spheres each of mass `M` and radius `R` are separated by a distance of `r`. The gravitational potential at the midpoint of the line joining the centres of the spheres is

Options

A. `-(GM)/(r )`
B. `-(2GM)/( r)`
C. `-(GM)/(2 r)`
D. `-(4GM)/( r)`

Explanation

To find the gravitational potential at the midpoint of the line joining the centers of two spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Setup**:
- We have two spheres, each with mass \( M \) and radius \( R \).
- The distance between the centers of the spheres is \( r \).
- The midpoint between the two spheres is point \( C \).

2. **Determine Distances**:
- Since point \( C \) is the midpoint, the distance from each sphere to point \( C \) is \( \frac{r}{2} \).

3. **Calculate Gravitational Potential Due to Each Sphere**:
- The gravitational potential \( V \) due to a mass \( M \) at a distance \( d \) is given by the formula:
\[ V = -\frac{GM}{d} \]
- For the first sphere (let's call it sphere A), the gravitational potential at point \( C \) is:
\[ V_1 = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r} \]
- For the second sphere (sphere B), the gravitational potential at point \( C \) is:
\[ V_2 = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r} \]

4. **Calculate the Net Gravitational Potential at Point C**:
- The net gravitational potential \( V_{\text{net}} \) at point \( C \) is the sum of the potentials due to both spheres:
\[ V_{\text{net}} = V_1 + V_2 = -\frac{2GM}{r} + -\frac{2GM}{r} \]
- Simplifying this gives:
\[ V_{\text{net}} = -\frac{4GM}{r} \]

5. **Conclusion**:
- Therefore, the gravitational potential at the midpoint of the line joining the centers of the spheres is:
\[ V_{\text{net}} = -\frac{4GM}{r} \]

### Final Answer:
The gravitational potential at the midpoint of the line joining the centers of the spheres is \( -\frac{4GM}{r} \).

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