A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be :

Options

A. `(2GM)/(9a^(2))`
B. `(GM)/(9a^(2))`
C. `(GM)/(3a^(2))`
D. `(2GM)/(3a^(2))`

Answer

To find the gravitational field at a distance \(3a\) from the center of a solid sphere of mass \(M\) and radius \(a\), surrounded by a uniform concentric spherical shell of thickness \(2a\) and mass \(2M\), we can follow these steps:

### Step 1: Understand the Configuration
We have a solid sphere of mass \(M\) and radius \(a\). Surrounding this sphere is a spherical shell of thickness \(2a\) and mass \(2M\). The inner radius of the shell is \(a\) and the outer radius is \(3a\).

### Step 2: Identify the Point of Interest
We need to calculate the gravitational field at a distance \(3a\) from the center of the solid sphere.

### Step 3: Apply the Gravitational Field Formula
For a point outside a solid sphere or a spherical shell, the gravitational field \(g\) can be calculated using the formula:
\[ g = \frac{GM}{r^2} \]
where \(G\) is the gravitational constant, \(M\) is the mass enclosed within the radius \(r\), and \(r\) is the distance from the center.

### Step 4: Calculate the Gravitational Field Due to the Solid Sphere
At a distance \(3a\), the gravitational field due to the solid sphere (mass \(M\)) is:
\[ g_{\text{sphere}} = \frac{GM}{(3a)^2} = \frac{GM}{9a^2} \]

### Step 5: Calculate the Gravitational Field Due to the Spherical Shell
The entire mass of the spherical shell (mass \(2M\)) can be treated as if it is concentrated at its center when calculating the gravitational field outside it. Therefore, at a distance \(3a\), the gravitational field due to the shell is:
\[ g_{\text{shell}} = \frac{G(2M)}{(3a)^2} = \frac{2GM}{9a^2} \]

### Step 6: Combine the Gravitational Fields
The total gravitational field at a distance \(3a\) from the center is the sum of the fields due to the solid sphere and the spherical shell:
\[ g_{\text{total}} = g_{\text{sphere}} + g_{\text{shell}} = \frac{GM}{9a^2} + \frac{2GM}{9a^2} = \frac{3GM}{9a^2} = \frac{GM}{3a^2} \]

### Final Answer
Thus, the gravitational field at a distance \(3a\) from the center is:
\[ \boxed{\frac{GM}{3a^2}} \]