Which of the following statements are correct?

Options

A. Field inside an infinitely long solenoid is constant.
B. Field Inside a long cylindrical conductor carrying current parallel to its axis is `prop` r
C. Field Inside a long pipe carrying current parallel to its length is zero
D. Field outside a long current carrying wire is `prop (1//r^2)`

Answer

To determine which statements are correct regarding magnetic fields in various configurations, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Statement 1: Field inside an infinite long solenoid is constant.**
- The magnetic field inside an ideal infinite long solenoid is given by the formula:
\[ B = \mu_0 n I \]
where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current flowing through the solenoid.
- Since this field does not depend on the position inside the solenoid, it is constant.
- **Conclusion:** This statement is **correct**.

2. **Statement 2: Field inside a long cylindrical conductor carrying a current parallel to its axis is proportional to R.**
- For a long cylindrical conductor with current \( I \) flowing parallel to its axis, the magnetic field at a distance \( r \) (where \( r < R \), the radius of the cylinder) can be derived using Ampere's Law:
\[ B = \frac{\mu_0 I_{\text{enclosed}}}{2 \pi r} \]
where \( I_{\text{enclosed}} \) is the current enclosed by the Amperian loop of radius \( r \).
- The enclosed current is proportional to the area, hence:
\[ I_{\text{enclosed}} = I \cdot \left(\frac{r^2}{R^2}\right) \]
- Substituting this into the equation for \( B \):
\[ B = \frac{\mu_0 \cdot I \cdot \left(\frac{r^2}{R^2}\right)}{2 \pi r} = \frac{\mu_0 I r}{2 \pi R^2} \]
- This shows that the magnetic field is indeed proportional to \( r \).
- **Conclusion:** This statement is **correct**.

3. **Statement 3: Field inside a long pipe carrying a current parallel to its length is zero.**
- For a hollow cylindrical conductor (pipe) with current flowing along its surface, the magnetic field inside the hollow part (where \( r < R \)) can be analyzed using Ampere's Law.
- Since there is no current enclosed within a loop inside the hollow region, we have:
\[ B = 0 \]
- **Conclusion:** This statement is **correct**.

4. **Statement 4: Field outside a long current-carrying wire is proportional to \( \frac{1}{R^2} \).**
- For a long straight wire carrying current \( I \), the magnetic field at a distance \( r \) from the wire is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- This shows that the magnetic field is inversely proportional to \( r \), not \( r^2 \).
- **Conclusion:** This statement is **incorrect**.

### Final Conclusion:
- **Correct Statements:** 1, 2, and 3.
- **Incorrect Statement:** 4.