Which one of the following plots represents the variation of the gravitational field on a particle with distance `r` due to a thin spherical shell of raduis `R`? (`r` is measured from the centre of the spherical shell).

Options

A.
B.
C.
D.

Explanation

To determine the variation of the gravitational field \( E \) on a particle with distance \( r \) due to a thin spherical shell of radius \( R \), we can analyze the gravitational field both inside and outside the shell.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a thin spherical shell with radius \( R \).
- We need to find the gravitational field at a distance \( r \) from the center of the shell.

2. **Inside the Shell** (\( r < R \)):
- According to the shell theorem, the gravitational field inside a thin spherical shell is zero.
- Therefore, for any point inside the shell (i.e., \( r < R \)), the gravitational field \( E \) is:
\[ E = 0 \]

3. **Outside the Shell** (\( r \geq R \)):
- For points outside the shell, the gravitational field behaves as if all the mass of the shell were concentrated at its center.
- The gravitational field \( E \) at a distance \( r \) from the center is given by:
\[ E = \frac{G M}{r^2} \]
- Here, \( M \) is the total mass of the shell, and \( G \) is the gravitational constant.

4. **Graphing the Gravitational Field**:
- For \( r < R \): The gravitational field \( E = 0 \).
- For \( r = R \): The gravitational field starts to decrease as \( \frac{1}{r^2} \).
- For \( r > R \): The gravitational field decreases inversely with the square of the distance.

5. **Conclusion**:
- The graph of the gravitational field \( E \) versus distance \( r \) will be zero for \( r < R \) and will decrease as \( \frac{1}{r^2} \) for \( r \geq R \).
- Thus, the correct plot representing this variation is option **B**.