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. **Understanding the Gravitational Field Outside the Shell**:
- For a thin spherical shell, the gravitational field intensity \( E \) at a distance \( r \) from the center (where \( r > R \)) is given by:
\[ E = \frac{GM}{r^2} \]
- Here, \( M \) is the mass of the shell, and \( r \) is the distance from the center of the shell.

2. **Given Condition**:
- We know that at a distance \( \frac{R}{2} \) outside the shell (which is at a distance \( R + \frac{R}{2} = \frac{3R}{2} \) from the center), the gravitational field intensity is \( E \).
- Thus, we can write:
\[ E = \frac{GM}{\left(\frac{3R}{2}\right)^2} = \frac{GM}{\frac{9R^2}{4}} = \frac{4GM}{9R^2} \]

3. **Finding the Gravitational Field Inside the Shell**:
- According to the shell theorem, the gravitational field inside a thin spherical shell is zero. This applies to any point inside the shell, including at a distance \( \frac{R}{2} \) from the center.
- Therefore, at a distance \( \frac{R}{2} \) from the center of the shell, the gravitational field intensity \( E' \) is:
\[ E' = 0 \]

4. **Conclusion**:
- The gravitational field intensity at a distance \( \frac{R}{2} \) from the center of the thin shell is \( 0 \).

### Final Answer:
\[ E' = 0 \]