A spherical shell has inner radius `R_1`, outer radius `R_2`, and mass M, distributed uniformly throughout the shell. The magnitude of the gravitational force exerted on the shell by a point mass particle of mass m, located at a distance d from the center, inside the inner radius, is:

Options

A. a.0
B. b.`GMm//R_1^2`
C. c.`GMm//d^2`
D. d.`GMm//(R_2^2-d^2)`

Explanation

To find the magnitude of the gravitational force exerted on a spherical shell by a point mass located inside the inner radius, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a spherical shell with an inner radius \( R_1 \) and an outer radius \( R_2 \).
- The shell has a uniform mass \( M \) distributed throughout it.
- A point mass \( m \) is located at a distance \( d \) from the center, where \( d < R_1 \).

2. **Applying Gauss's Law for Gravitation**:
- According to Gauss's Law for gravitation, the gravitational field \( E_g \) inside a spherical shell (at a distance less than the inner radius) is zero. This is because the mass enclosed within a Gaussian surface inside the shell is zero.
- Mathematically, we can express this as:
\[ \oint E_g \cdot dS = -4\pi G \cdot M_{\text{enclosed}} \]
- Since \( M_{\text{enclosed}} = 0 \) (there is no mass inside the Gaussian surface), we have:
\[ \oint E_g \cdot dS = 0 \]
- This implies that the gravitational field \( E_g \) inside the shell is:
\[ E_g = 0 \]

3. **Calculating the Gravitational Force**:
- The gravitational force \( F \) experienced by the point mass \( m \) due to the gravitational field \( E_g \) is given by:
\[ F = m \cdot E_g \]
- Since we have established that \( E_g = 0 \):
\[ F = m \cdot 0 = 0 \]

4. **Conclusion**:
- Therefore, the magnitude of the gravitational force exerted on the shell by the point mass located inside the inner radius is:
\[ F = 0 \, \text{N} \]

### Final Answer:
The gravitational force exerted on the shell by the point mass is \( 0 \, \text{N} \).

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