A long straight wire of a circular cross-section (radius a) carries a steady current I and the current I is uniformly distributed across this cross-section. Which of the following plots represents the variation of magnitude of magnetic field B with distance r from the centre of the wire?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the variation of the magnetic field \( B \) with distance \( r \) from the center of a long straight wire carrying a steady current \( I \) uniformly distributed across its circular cross-section, we can follow these steps:

### Step 1: Identify the regions based on distance \( r \)
We need to consider two regions:
1. Inside the wire (where \( r < a \))
2. Outside the wire (where \( r \geq a \))

### Step 2: Apply Ampère's Law for the inside region (\( r < a \))
For a point inside the wire, we can use Ampère's Law, which states:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]
Where \( I_{enc} \) is the current enclosed by the Amperian loop of radius \( r \).

The current density \( J \) is given by:
\[ J = \frac{I}{\pi a^2} \]
The enclosed current \( I_{enc} \) for a radius \( r \) is:
\[ I_{enc} = J \cdot A = J \cdot \pi r^2 = \frac{I}{\pi a^2} \cdot \pi r^2 = I \frac{r^2}{a^2} \]

Now, applying Ampère's Law:
\[ B(2\pi r) = \mu_0 I_{enc} = \mu_0 I \frac{r^2}{a^2} \]
Solving for \( B \):
\[ B = \frac{\mu_0 I r}{2\pi a^2} \]

### Step 3: Apply Ampère's Law for the outside region (\( r \geq a \))
For points outside the wire, the entire current \( I \) is enclosed:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I \]
Thus:
\[ B(2\pi r) = \mu_0 I \]
Solving for \( B \):
\[ B = \frac{\mu_0 I}{2\pi r} \]

### Step 4: Summarize the results
- For \( r < a \):
\[ B = \frac{\mu_0 I r}{2\pi a^2} \]
This indicates that \( B \) increases linearly with \( r \) inside the wire.

- For \( r \geq a \):
\[ B = \frac{\mu_0 I}{2\pi r} \]
This indicates that \( B \) decreases with \( r \) outside the wire.

### Step 5: Plot the results
- The plot for \( B \) vs \( r \) will show a linear increase for \( r < a \) and a hyperbolic decrease for \( r \geq a \).

### Conclusion
The correct plot representing the variation of the magnetic field \( B \) with distance \( r \) from the center of the wire will show a linear increase up to the radius \( a \) and then a decrease as \( r \) increases beyond \( a \).

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