A particle of mass M is situated at the centre of a spherical shell of same mass and radius 'a'. The gravitational potential at a point situated at `(a)/(2)` distance from the centre, will be

Options

A. `-(3GM)/(a)`
B. `-(2GM)/(a)`
C. `-(GM)/(a)`
D. `-(4GM)/(a)`

Explanation

To find the gravitational potential at a point located at a distance of \( \frac{a}{2} \) from the center of a spherical shell of mass \( M \) and radius \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a spherical shell of mass \( M \) and radius \( a \).
- There is a particle of mass \( M \) located at the center of the shell.
- We need to find the gravitational potential at a point that is \( \frac{a}{2} \) away from the center.

2. **Gravitational Potential Due to a Point Mass**:
- The gravitational potential \( V \) due to a point mass \( m \) at a distance \( r \) is given by the formula:
\[ V = -\frac{Gm}{r} \]
- Here, \( G \) is the gravitational constant.

3. **Calculate the Potential Due to the Central Mass**:
- For the mass \( M \) located at the center, the distance from the point at \( \frac{a}{2} \) is \( \frac{a}{2} \).
- Therefore, the gravitational potential \( V_1 \) at that point due to the central mass \( M \) is:
\[ V_1 = -\frac{GM}{\frac{a}{2}} = -\frac{2GM}{a} \]

4. **Gravitational Potential Inside the Shell**:
- According to the shell theorem, the gravitational potential inside a uniform spherical shell is constant and equal to the potential at the surface of the shell.
- The potential at the surface of the shell (at distance \( a \)) is:
\[ V_{shell} = -\frac{GM}{a} \]
- Since the point \( \frac{a}{2} \) is inside the shell, the potential due to the shell at this point is also:
\[ V_{shell} = -\frac{GM}{a} \]

5. **Total Gravitational Potential at the Point**:
- The total gravitational potential \( V \) at the point \( \frac{a}{2} \) is the sum of the potentials due to the central mass and the shell:
\[ V = V_1 + V_{shell} = -\frac{2GM}{a} - \frac{GM}{a} \]
- Simplifying this expression gives:
\[ V = -\frac{3GM}{a} \]

### Final Answer:
The gravitational potential at a point situated at a distance \( \frac{a}{2} \) from the center of the spherical shell is:
\[ V = -\frac{3GM}{a} \]