A steady current `I` flows along an infinitely long hollow cylindrical conductor of radius `R`. This cylinder is placed coaxially inside an infinite solenoid of radius `2 R`. The solenoid has a `n` turns per unit length and carries a steady current `I`. Consider a point `p` at a distance `r` from the common axis . The correct statement(s) is (are)

Options

A. In the region `0 lt r lt R`, the magnetic field is non-zero.
B. In the region `R lt r lt 2R`, the magnetic field is along the common axis.
C. In the region `R lt r lt 2R`, the magnetic field is tangential to the circle of radius `r` centered on the axis.
D. In the region `r lt 2R`, the magnetic field is non-zero.

Explanation

To solve the problem, we need to analyze the magnetic fields produced by both the hollow cylindrical conductor and the infinite solenoid at various distances from the common axis.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a hollow cylindrical conductor of radius \( R \) carrying a steady current \( I \).
- This conductor is placed coaxially inside an infinite solenoid of radius \( 2R \), which also carries a steady current \( I \) and has \( n \) turns per unit length.

2. **Magnetic Field Inside the Hollow Cylinder**:
- According to Ampere's Law, the magnetic field inside a hollow cylindrical conductor (for \( r < R \)) is zero because there is no current enclosed by the Amperian loop.
- Therefore, for \( 0 < r < R \), the magnetic field \( B = 0 \).

3. **Magnetic Field Outside the Hollow Cylinder but Inside the Solenoid**:
- For \( R < r < 2R \), we need to consider the contributions from both the hollow cylinder and the solenoid.
- The magnetic field due to the hollow cylinder (for \( r > R \)) can be calculated using Ampere's Law:
\[ B_{\text{cylinder}} = \frac{\mu_0 I}{2\pi r} \]
- The magnetic field inside the solenoid (for \( r < 2R \)) is given by:
\[ B_{\text{solenoid}} = \mu_0 n I \]
- Therefore, for \( R < r < 2R \), the total magnetic field is the vector sum of both contributions.

4. **Magnetic Field Outside the Solenoid**:
- For \( r > 2R \), the magnetic field due to the solenoid is zero (as the magnetic field outside an infinite solenoid is zero).
- However, the hollow cylindrical conductor still contributes a magnetic field:
\[ B_{\text{cylinder}} = \frac{\mu_0 I}{2\pi r} \]
- Thus, for \( r > 2R \), the magnetic field is non-zero due to the hollow cylinder.

5. **Evaluating the Statements**:
- **Statement A**: "In the region \( 0 < r < R \), the magnetic field is non-zero."
- **False** (since \( B = 0 \) in this region).
- **Statement B**: "In the region \( R < r < 2R \), the magnetic field is along the common axis."
- **False** (the magnetic field is not purely along the axis due to the contributions from both the solenoid and the hollow cylinder).
- **Statement C**: "In the region \( R < r < 2R \), the magnetic field is tangential to the circle of radius \( R \) centered on the axis."
- **False** (the magnetic field is not purely tangential).
- **Statement D**: "In the region \( r > 2R \), the magnetic field is non-zero."
- **True** (the magnetic field exists due to the hollow cylindrical conductor).

### Final Conclusion:
- The correct statements are **D** only.