A wire of radius R having uniform cross section in which steady current l is flowing. Then

Options

A. a. Magnetic field increases linearly with the distance from the axis of the wire
B. b. Magnetic field is maximum at the surface of the wire
C. c. Magnetic field decreases hyperbolicaly, outside the wire, with the distance from the axis
D. d. Magnetic field remains constant as the wire is infinite

Answer

To solve the problem regarding the magnetic field around a wire with a steady current, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a wire of radius \( R \) with a steady current \( I \) flowing through it. We need to analyze the magnetic field \( B \) at different distances from the wire.

2. **Applying Ampere's Law**:
According to Ampere's Law, the line integral of the magnetic field \( B \) around a closed loop is equal to \( \mu_0 \) times the current enclosed by that loop:
\[ \oint B \cdot dl = \mu_0 I_{\text{enclosed}} \]
We will consider two cases: inside the wire (where \( r < R \)) and outside the wire (where \( r > R \)).

3. **Case 1: Inside the Wire (\( r < R \))**:
- The current density \( J \) is uniform, so the current enclosed by a circle of radius \( r \) is:
\[ I_{\text{enclosed}} = I \cdot \frac{\pi r^2}{\pi R^2} = I \cdot \frac{r^2}{R^2} \]
- Using Ampere's Law:
\[ B \cdot (2\pi r) = \mu_0 I_{\text{enclosed}} = \mu_0 I \cdot \frac{r^2}{R^2} \]
- Solving for \( B \):
\[ B = \frac{\mu_0 I r}{2 \pi R^2} \]
- This shows that \( B \) increases linearly with \( r \) (as \( r \) increases, \( B \) increases).

4. **Case 2: Outside the Wire (\( r > R \))**:
- The total current \( I \) is enclosed by any circle of radius \( r \):
\[ I_{\text{enclosed}} = I \]
- Again using Ampere's Law:
\[ B \cdot (2\pi r) = \mu_0 I \]
- Solving for \( B \):
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- This shows that \( B \) decreases inversely with \( r \) (as \( r \) increases, \( B \) decreases).

5. **Maximum Magnetic Field**:
- At the surface of the wire (\( r = R \)):
\[ B = \frac{\mu_0 I}{2 \pi R} \]
- This is the maximum value of the magnetic field.

6. **Conclusion**:
- From the analysis, we conclude:
- The magnetic field increases linearly with the distance from the axis of the wire inside the wire.
- The magnetic field is maximum at the surface of the wire.
- Outside the wire, the magnetic field decreases inversely with distance.
- The magnetic field does not remain constant as the wire is infinite.

### Final Answers:
- Correct options are:
- A: Magnetic field increases linearly with the distance from the axis of the wire.
- B: Magnetic field is maximum at the surface of the wire.