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, centred on the axis.
D. In the region `r gt 2R`, the magnetic field in 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 different regions around them. Let's break it down step by step.

### Step 1: Understanding the System
We have:
- An infinitely long hollow cylindrical conductor of radius \( R \) carrying a steady current \( I \).
- An infinite solenoid of radius \( 2R \) with \( n \) turns per unit length, also carrying a steady current \( I \).
- We need to determine the magnetic field at a point \( P \) located at a distance \( r \) from the common axis of the cylinder and the solenoid.

### Step 2: Magnetic Field Inside the Hollow Cylinder
For a hollow cylindrical conductor:
- Inside the conductor (i.e., for \( r < R \)), the magnetic field \( B \) is zero.
- Outside the conductor (i.e., for \( r > R \)), the magnetic field is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Magnetic Field Inside the Solenoid
For an infinite solenoid:
- Inside the solenoid (i.e., for \( r < 2R \)), the magnetic field \( B \) is constant and given by:
\[ B = \mu_0 n I \]
- Outside the solenoid (i.e., for \( r > 2R \)), the magnetic field is zero.

### Step 4: Analyzing Different Regions
Now, we will analyze the magnetic field in different regions based on the distance \( r \):

1. **Region 1: \( 0 < r < R \)**
- Inside the hollow cylinder, \( B = 0 \).
- Inside the solenoid, \( B = \mu_0 n I \).
- **Conclusion**: The magnetic field is non-zero and equal to \( \mu_0 n I \).

2. **Region 2: \( R < r < 2R \)**
- Outside the hollow cylinder, \( B = \frac{\mu_0 I}{2\pi r} \).
- Inside the solenoid, \( B = \mu_0 n I \).
- The magnetic fields from the solenoid and the hollow cylinder will have different directions. The net magnetic field will not be along the common axis.
- **Conclusion**: The magnetic field is not along the common axis.

3. **Region 3: \( 2R < r \)**
- Outside both the hollow cylinder and the solenoid, the magnetic field from the solenoid is zero.
- The magnetic field from the hollow cylinder is \( \frac{\mu_0 I}{2\pi r} \).
- **Conclusion**: The magnetic field is non-zero.

### Final Conclusion
Based on the analysis:
- **Option A**: Correct (non-zero magnetic field in \( 0 < r < R \)).
- **Option B**: Incorrect (the magnetic field is not along the common axis in \( R < r < 2R \)).
- **Option C**: Incorrect (the magnetic field is not tangential in \( 2R < r \)).
- **Option D**: Correct (non-zero magnetic field in \( r > 2R \)).

### Correct Statements:
The correct statements are **A** and **D**.

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