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 regarding the gravitational field \( E_1 \) and \( E_2 \) at distances \( r_1 \) and \( r_2 \) from the axis of an infinitely long solid cylinder of radius \( R \), we can follow these steps:

### Step 1: Understand the Gravitational Field Inside and Outside the Cylinder
The gravitational field \( E \) due to an infinitely long solid cylinder can be determined using Gauss's law. The behavior of the gravitational field differs depending on whether the point of interest is inside or outside the cylinder.

- **Inside the Cylinder** (\( r < R \)):
The gravitational field \( E \) is given by:
\[ E = \frac{\rho r}{2 \epsilon_0} \]
where \( \rho \) is the mass density of the cylinder.

- **Outside the Cylinder** (\( r > R \)):
The gravitational field \( E \) is given by:
\[ E = \frac{\rho R^2}{2 \epsilon_0 r^2} \]

### Step 2: Analyze the Conditions for \( r_1 \) and \( r_2 \)
Let’s consider two cases based on the distances \( r_1 \) and \( r_2 \):

1. **Case 1**: Both \( r_1 \) and \( r_2 \) are inside the cylinder (\( r_1 < R \) and \( r_2 < R \)):
- Since \( E \) is directly proportional to \( r \), if \( r_1 < r_2 \), then \( E_1 < E_2 \).

2. **Case 2**: Both \( r_1 \) and \( r_2 \) are outside the cylinder (\( r_1 > R \) and \( r_2 > R \)):
- Since \( E \) is inversely proportional to \( r^2 \), if \( r_1 < r_2 \), then \( E_1 > E_2 \).

3. **Case 3**: One point is inside and the other is outside the cylinder (\( r_1 < R \) and \( r_2 > R \)):
- Here, \( E_1 \) (inside) will be greater than \( E_2 \) (outside) since the field inside increases with \( r \) while the field outside decreases with \( r^2 \).

### Step 3: Conclusion
From the analysis above, we can conclude the following relationships:
- If \( r_1 < r_2 \) and both are inside, then \( E_1 < E_2 \).
- If \( r_1 < r_2 \) and both are outside, then \( E_1 > E_2 \).
- If \( r_1 < R < r_2 \), then \( E_1 > E_2 \).

Thus, the relationships between \( E_1 \) and \( E_2 \) depend on the relative positions of \( r_1 \) and \( r_2 \) with respect to \( R \).