If gravitational field Intensity is E at distance R/2 outside from then surface of a thin shell of radius R , the gravitational field intensity at distance R/2 from its centre is

Options

A. Zero
B. 2E
C. `(2E)/3`
D. `(3E)/2`

Answer

To solve the problem, we need to determine the gravitational field intensity at a distance \( \frac{R}{2} \) from the center of a thin spherical shell of radius \( R \), given that the gravitational field intensity is \( E \) at a distance \( \frac{R}{2} \) outside the surface of the shell.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a thin spherical shell of radius \( R \).
- The gravitational field intensity \( E \) is given at a distance \( \frac{R}{2} \) outside the shell. This means the distance from the center of the shell to this point is \( R + \frac{R}{2} = \frac{3R}{2} \).

2. **Apply Gauss's Law for Gravitation**:
- According to Gauss's law for gravitation, the gravitational field intensity \( g \) at a distance \( r \) from the center of a spherical shell is given by:
\[ g = -\frac{G M}{r^2} \]
- However, this applies only outside the shell. Inside the shell, the gravitational field intensity behaves differently.

3. **Evaluate the Gravitational Field Inside the Shell**:
- For a hollow thin spherical shell, the gravitational field inside the shell (at any point within the shell) is zero. This is a result of the symmetry of the shell and can be derived from Gauss's law:
- If we consider a Gaussian surface inside the shell, the mass enclosed by this surface is zero since all the mass is on the shell's surface.
- Thus, by Gauss's law:
\[ \oint \vec{E} \cdot d\vec{A} = -4\pi G M_{\text{enclosed}} = 0 \]
- This implies that the gravitational field intensity \( E \) inside the shell is zero.

4. **Conclusion**:
- Therefore, at a distance \( \frac{R}{2} \) from the center of the shell, the gravitational field intensity is:
\[ E_{\text{inside}} = 0 \]

### Final Answer:
The gravitational field intensity at a distance \( \frac{R}{2} \) from the center of the shell is **0**.