A cylinderical conductor of radius R is carrying constant current. The plot of the magnitude of the magnetic field, B with the distance, d from the centre of the conductor , is correctly represented by the figure:

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct plot of the magnetic field \( B \) with respect to the distance \( d \) from the center of a cylindrical conductor carrying a constant current, we will follow these steps:

### Step 1: Understand the Setup
We have a cylindrical conductor of radius \( R \) carrying a constant current \( I \). We need to analyze the magnetic field inside and outside the conductor.

### Step 2: Magnetic Field Inside the Conductor
For a point inside the cylindrical conductor at a distance \( d \) (where \( d < R \)), we can use Ampere's Circuital Law:

\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

The current density \( J \) can be defined as:

\[ J = \frac{I}{A} = \frac{I}{\pi R^2} \]

The enclosed current \( I_{\text{enc}} \) for a circular Amperian loop of radius \( d \) is:

\[ I_{\text{enc}} = J \cdot A_{\text{enc}} = J \cdot \pi d^2 = \frac{I}{\pi R^2} \cdot \pi d^2 = \frac{I d^2}{R^2} \]

Using Ampere's law, we have:

\[ B \cdot 2\pi d = \mu_0 \cdot \frac{I d^2}{R^2} \]

Solving for \( B \):

\[ B = \frac{\mu_0 I d}{2 \pi R^2} \]

This shows that the magnetic field \( B \) inside the conductor is directly proportional to the distance \( d \) from the center.

### Step 3: Magnetic Field Outside the Conductor
For a point outside the conductor at a distance \( d \) (where \( d > R \)), the entire current \( I \) is enclosed by the Amperian loop:

\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \]

Thus, we have:

\[ B \cdot 2\pi d = \mu_0 I \]

Solving for \( B \):

\[ B = \frac{\mu_0 I}{2 \pi d} \]

This shows that the magnetic field \( B \) outside the conductor is inversely proportional to the distance \( d \) from the center.

### Step 4: Plotting the Magnetic Field
1. **Inside the Conductor**: The magnetic field increases linearly with \( d \) until \( d = R \).
2. **Outside the Conductor**: The magnetic field decreases inversely with \( d \) after \( d = R \).

### Step 5: Conclusion
The correct representation of the magnetic field \( B \) as a function of distance \( d \) from the center of the cylindrical conductor is a graph that shows:
- A linear increase from the origin to \( d = R \).
- A hyperbolic decrease for \( d > R \).

Thus, the correct option that represents this behavior is **Option 4**.

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