A solid sphere of radius `a` and mass `m` is surrounded by cocentric spherical shell of thickness `2a` and mass `2m` the gravitational field at a distance 3a from their centres is

Options

A. `(Gm)/(a^(2))`
B. `(Gm)/(3a^(2))`
C. `(Gm)/(5a^(2))`
D. `(Gm)/(4a^(2))`

Answer

To solve the problem, we need to find the gravitational field at a distance of \(3a\) from the center of a solid sphere of radius \(a\) and mass \(m\), which is surrounded by a concentric spherical shell of thickness \(2a\) and mass \(2m\).

### Step-by-Step Solution:

1. **Identify the Components**:
- We have a solid sphere with radius \(a\) and mass \(m\).
- We have a spherical shell with an outer radius of \(3a\) (since the shell has a thickness of \(2a\)) and mass \(2m\).

2. **Gravitational Field due to the Solid Sphere**:
- For a solid sphere, the gravitational field \(E\) at a distance \(r\) from the center (where \(r > a\)) is given by the formula:
\[ E_{\text{sphere}} = \frac{Gm}{r^2} \]
- Here, we are interested in the gravitational field at \(r = 3a\):
\[ E_{\text{sphere}} = \frac{Gm}{(3a)^2} = \frac{Gm}{9a^2} \]

3. **Gravitational Field due to the Spherical Shell**:
- For a spherical shell, the gravitational field outside the shell (where \(r > R\), \(R\) being the outer radius of the shell) is given by:
\[ E_{\text{shell}} = \frac{G M_{\text{shell}}}{r^2} \]
- The mass of the shell is \(2m\) and at \(r = 3a\):
\[ E_{\text{shell}} = \frac{G(2m)}{(3a)^2} = \frac{2Gm}{9a^2} \]

4. **Total Gravitational Field**:
- The total gravitational field at a distance \(3a\) from the center is the sum of the gravitational fields due to the solid sphere and the shell:
\[ E_{\text{total}} = E_{\text{sphere}} + E_{\text{shell}} = \frac{Gm}{9a^2} + \frac{2Gm}{9a^2} \]
- Combining these gives:
\[ E_{\text{total}} = \frac{Gm + 2Gm}{9a^2} = \frac{3Gm}{9a^2} = \frac{Gm}{3a^2} \]

5. **Final Answer**:
- Therefore, the gravitational field at a distance \(3a\) from the center is:
\[ E_{\text{total}} = \frac{Gm}{3a^2} \]

### Conclusion:
The gravitational field at a distance \(3a\) from the center is \(\frac{Gm}{3a^2}\).