Let `E_1 and E_2` denotes gravitational field at distance `r_1 and r_2` from the axis of an infinitely long solid cylinder of the radius R. Which of the following must hold true ?

Options

A. `E_1 lt E_2 ` if `r_1 lt r_2 lt R`
B. `E_1 gt E_2` if `R lt r_1 lt r_2`
C. `E_1 gt E_2`, if `r_1 = R - E , r_2 = R +E` (E is positive constant `ltR` )
D. All of the above

Answer

To solve the problem of determining the relationship between the gravitational fields \( E_1 \) and \( E_2 \) at distances \( r_1 \) and \( r_2 \) from the axis of an infinitely long solid cylinder of radius \( R \), we can analyze the gravitational field both inside and outside the cylinder using Gauss's law.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have an infinitely long solid cylinder with radius \( R \).
- We need to find the gravitational field at two different distances \( r_1 \) and \( r_2 \) from the axis of the cylinder.

2. **Applying Gauss's Law**:
- Gauss's law for gravity states that the gravitational flux through a closed surface is equal to the enclosed mass divided by \( G \) (gravitational constant).
- The gravitational field \( E \) is related to the mass enclosed by the Gaussian surface.

3. **Case 1: Inside the Cylinder (\( r < R \))**:
- For a point inside the cylinder, the gravitational field \( E \) is directly proportional to the distance \( r \) from the axis:
\[ E = \frac{G M_{\text{enc}}}{2 \pi r} \]
- Here, \( M_{\text{enc}} \) is the mass enclosed within radius \( r \). Since the mass density \( \rho \) is uniform, we can express \( M_{\text{enc}} \) as:
\[ M_{\text{enc}} = \rho \cdot \text{Volume} = \rho \cdot \pi r^2 L \]
- Thus, the gravitational field inside the cylinder is:
\[ E \propto r \]

4. **Case 2: Outside the Cylinder (\( r > R \))**:
- For a point outside the cylinder, the gravitational field \( E \) is inversely proportional to the distance \( r \):
\[ E = \frac{G M_{\text{total}}}{2 \pi r} \]
- Here, \( M_{\text{total}} \) is the total mass of the cylinder, which remains constant regardless of \( r \) as long as \( r > R \).
- Thus, the gravitational field outside the cylinder is:
\[ E \propto \frac{1}{r} \]

5. **Comparing \( E_1 \) and \( E_2 \)**:
- If \( r_1 < R \) and \( r_2 < R \) (both points inside the cylinder), then since \( E \propto r \):
\[ \text{If } r_1 < r_2 \Rightarrow E_1 < E_2 \]
- If \( r_1 > R \) and \( r_2 > R \) (both points outside the cylinder), then since \( E \propto \frac{1}{r} \):
\[ \text{If } r_1 < r_2 \Rightarrow E_1 > E_2 \]

6. **Conclusion**:
- The relationships derived show that:
- Inside the cylinder, the gravitational field increases with distance from the axis.
- Outside the cylinder, the gravitational field decreases with distance from the axis.