Following curve shows the variation of intesity of gravitational field `(vec(I))` with distance from the centre of solid sphere(r) :

Options

A. (a)
B. (b)
C. (c)
D. (d)

Answer

To solve the problem of the variation of gravitational field intensity with distance from the center of a solid sphere, we need to analyze two cases: when the distance \( r \) is greater than the radius of the sphere \( R \) and when \( r \) is less than \( R \).

### Step-by-Step Solution:

**Step 1: Analyze the case when \( r > R \) (Outside the sphere)**

1. When we are outside the solid sphere, we can treat the entire mass \( M \) of the sphere as if it were concentrated at a point at its center.
2. The gravitational field intensity \( \vec{I} \) at a distance \( r \) from the center is given by the formula:
\[ \vec{I} = -\frac{GM}{r^2} \hat{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the sphere, and \( \hat{r} \) is the unit vector pointing away from the center.

**Step 2: Analyze the case when \( r < R \) (Inside the sphere)**

1. Inside the sphere, the gravitational field intensity depends on the mass enclosed within a radius \( r \).
2. The mass \( M' \) enclosed within radius \( r \) can be calculated using the ratio of volumes:
\[ \frac{M}{\frac{4}{3} \pi R^3} = \frac{M'}{\frac{4}{3} \pi r^3} \]
This leads to:
\[ M' = M \cdot \frac{r^3}{R^3} \]
3. The gravitational field intensity \( \vec{I} \) at a distance \( r \) from the center is then given by:
\[ \vec{I} = -\frac{GM'}{r^2} \hat{r} = -\frac{G \left(M \cdot \frac{r^3}{R^3}\right)}{r^2} \hat{r} \]
Simplifying this gives:
\[ \vec{I} = -\frac{GM}{R^3} r \hat{r} \]
This shows that the gravitational field intensity inside the sphere is directly proportional to \( r \).

**Step 3: Summary of the results**

- For \( r > R \):
\[ \vec{I} \propto \frac{1}{r^2} \]
- For \( r < R \):
\[ \vec{I} \propto r \]

### Conclusion

The variation of gravitational field intensity with distance from the center of the solid sphere shows that:
- Outside the sphere, the intensity decreases with the square of the distance.
- Inside the sphere, the intensity increases linearly with distance from the center.