The magnetic field due to a conductor fo unifrom cross section of radius `a` and carrying a steady current is represented by

Options

A.
B.
C.
D.

Answer

To solve the problem of finding the magnetic field due to a conductor of uniform cross-section of radius \( a \) carrying a steady current \( I \), we can follow these steps:

### Step 1: Understand the Geometry
We have a cylindrical conductor with radius \( a \) carrying a steady current \( I \). We need to analyze the magnetic field both inside and outside the conductor.

### Step 2: Apply Ampere's Law
We will use Ampere's Law, which states that:
\[ \oint \mathbf{B} \cdot d\mathbf{L} = \mu_0 I_{\text{enc}} \]
where \( I_{\text{enc}} \) is the current enclosed by the path of integration.

### Step 3: Consider Inside the Conductor (\( r < a \))
For points inside the conductor (where \( r \) is the distance from the center and \( r < a \)):
- The current density \( J \) is uniform, given by:
\[ J = \frac{I}{\pi a^2} \]
- The enclosed current \( I_{\text{enc}} \) for a radius \( r \) is:
\[ I_{\text{enc}} = J \cdot \text{Area} = J \cdot \pi r^2 = \frac{I}{\pi a^2} \cdot \pi r^2 = I \frac{r^2}{a^2} \]
- Applying Ampere's Law:
\[ B \cdot (2\pi r) = \mu_0 I_{\text{enc}} = \mu_0 \left(I \frac{r^2}{a^2}\right) \]
- Thus,
\[ B = \frac{\mu_0 I r}{2 \pi a^2} \]
This shows that the magnetic field \( B \) inside the conductor is directly proportional to \( r \).

### Step 4: Consider Outside the Conductor (\( r \geq a \))
For points outside the conductor:
- The total current enclosed is simply \( I \).
- Applying Ampere's Law:
\[ B \cdot (2\pi r) = \mu_0 I \]
- Thus,
\[ B = \frac{\mu_0 I}{2 \pi r} \]
This shows that the magnetic field \( B \) outside the conductor is inversely proportional to \( r \).

### Step 5: Conclusion
From our analysis:
- Inside the conductor (\( r < a \)), \( B \) increases linearly with \( r \).
- Outside the conductor (\( r \geq a \)), \( B \) decreases inversely with \( r \).

### Step 6: Identify the Correct Representation
Given the options, the correct representation of the magnetic field as a function of distance from the center of the conductor will show a linear increase up to \( r = a \) and then a hyperbolic decrease for \( r > a \). Thus, the correct option is **A**.

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