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 \( g \) on a particle with distance \( r \) due to a thin spherical shell of radius \( R \), we can analyze the gravitational field in two distinct regions: inside the shell and outside the shell.

### Step-by-Step Solution:

1. **Understanding the Region Inside the Shell**:
- When the particle is located inside the thin spherical shell (i.e., \( r < R \)), the gravitational field \( g \) is zero. This is a result of the shell theorem, which states that a uniform spherical shell of mass exerts no net gravitational force on a particle located inside it.
- Therefore, for \( r < R \):
\[ g = 0 \]

2. **Understanding the Region Outside the Shell**:
- When the particle is outside the shell (i.e., \( r \geq R \)), the gravitational field can be calculated using the formula:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the shell. Since the mass \( M \) remains constant, we can see that \( g \) is inversely proportional to the square of the distance \( r \).
- Thus, for \( r \geq R \):
\[ g \propto \frac{1}{r^2} \]

3. **Plotting the Graph**:
- Now, we can plot the gravitational field \( g \) against the distance \( r \):
- For \( r < R \), \( g = 0 \).
- At \( r = R \), the gravitational field starts to increase as \( g \) becomes proportional to \( \frac{1}{r^2} \).
- As \( r \) increases beyond \( R \), \( g \) decreases following the \( \frac{1}{r^2} \) relationship.

4. **Identifying the Correct Plot**:
- The graph will start at zero for \( r < R \) and will then show a decreasing curve for \( r \geq R \). This behavior matches with option B in the provided choices.

### Final Answer:
The correct plot representing the variation of the gravitational field \( g \) on a particle with distance \( r \) due to a thin spherical shell of radius \( R \) is option B.

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