A particle of mass M is placed at the centre of a spherical shell of same mass and radius a. What will be the magnitude of the gravitational potential at a point situated at a/2 distance from the centre ?

Options

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

Explanation

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

### Step 1: Understand the setup
We have a spherical shell of mass \( M \) and radius \( a \). A particle of mass \( M \) is placed at the center of this shell. We need to find the gravitational potential at a point \( P \) which is at a distance \( \frac{a}{2} \) from the center of the shell.

### Step 2: Recall the formula for gravitational potential
The gravitational potential \( V \) at a distance \( r \) from a mass \( M \) is given by the formula:
\[ V = -\frac{GM}{r} \]
where \( G \) is the gravitational constant.

### Step 3: Calculate the gravitational potential due to the spherical shell
According to the shell theorem, the gravitational potential inside a uniform spherical shell is constant and equal to the potential at the surface. Therefore, at a distance \( \frac{a}{2} \) from the center, the potential due to the shell (mass \( M \)) is:
\[ V_{\text{shell}} = -\frac{GM}{a} \]

### Step 4: Calculate the gravitational potential due to the mass at the center
The gravitational potential at point \( P \) due to the mass \( M \) located at the center (which is at distance \( \frac{a}{2} \) from point \( P \)) is:
\[ V_{\text{center}} = -\frac{GM}{\frac{a}{2}} = -\frac{2GM}{a} \]

### Step 5: Combine the potentials
The total gravitational potential \( V_P \) at point \( P \) is the sum of the potentials due to the shell and the mass at the center:
\[ V_P = V_{\text{shell}} + V_{\text{center}} = -\frac{GM}{a} - \frac{2GM}{a} \]
\[ V_P = -\frac{3GM}{a} \]

### Step 6: Conclusion
Thus, the magnitude of the gravitational potential at the point situated at a distance \( \frac{a}{2} \) from the center is:
\[ \boxed{-\frac{3GM}{a}} \]

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