Gravitational field due to a solid sphere

Options

A. remains constant throughout the sphere
B. increase inside the sphere and decreases outside the sphere
C. increases throughout with distance from the center
D. decreases throughout with distance from the centre

Explanation

To find the gravitational field due to a solid sphere, we can analyze the behavior of the gravitational field both inside and outside the sphere.

### Step-by-Step Solution:

1. **Understanding Gravitational Field**:
The gravitational field (E) at a distance r from the center of a solid sphere is influenced by the mass of the sphere (M) and the distance from its center.

2. **Gravitational Field Inside the Sphere**:
For a point inside a solid sphere (where r < R, R being the radius of the sphere), the gravitational field is given by:
\[ E_r = \frac{G \cdot M_{r}}{r^2} \]
where \(M_{r}\) is the mass enclosed within radius r. According to Gauss's law for gravitation, the mass enclosed within a radius r is proportional to the volume of the sphere of radius r:
\[ M_{r} = \frac{M}{R^3} \cdot r^3 \]
Therefore, the gravitational field inside the sphere becomes:
\[ E_r = \frac{G \cdot \left(\frac{M}{R^3} \cdot r^3\right)}{r^2} = \frac{G \cdot M}{R^3} \cdot r \]
This shows that the gravitational field inside the sphere increases linearly with r.

3. **Gravitational Field Outside the Sphere**:
For a point outside the solid sphere (where r > R), the gravitational field is given by:
\[ E_r = \frac{G \cdot M}{r^2} \]
This indicates that the gravitational field outside the sphere decreases with the square of the distance from the center of the sphere.

4. **Conclusion**:
- Inside the sphere, the gravitational field increases linearly with distance from the center.
- Outside the sphere, the gravitational field decreases with the square of the distance from the center.

Thus, the correct option is that the gravitational field increases inside the sphere and decreases outside the sphere.