Question

A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is:
[A.] maximum at either end of the conductor
[B.] maximum at the axis of the conductor and minimum at the midpoint
[C.] minimum at the surface of the conductor
[D.] minimum at the axis of the conductor
[E.] same at all points in the cross-section of the conductor
Choose the correct answer from the options given below:

• A. D Only
• B. B, C Only
• C. A, D Only
• D. E Only

Hint:

Inside a current-carrying conductor with uniform current density, magnetic field increases linearly with distance from the axis.

Answer 1

The Correct Option is A

To determine the magnetic field inside a long cylindrical conductor carrying current, we consider Ampère's Law, which relates the magnetic field (\(\mathbf{B}\)) around a closed loop to the current passing through the loop.

The formula for Ampère's Law is:

\(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\)

Where:

• \(\oint \mathbf{B} \cdot d\mathbf{l}\) represents the line integral of the magnetic field around a closed loop.
• \(\mu_0\) is the permeability of free space.
• \(I_{\text{enc}}\) is the current enclosed by the loop.

We consider a loop of radius \(r\) inside the conductor. The current density is uniform, so the current enclosed by our loop is proportional to the area of the loop:

\(I_{\text{enc}} = J \pi r^2\)

Where \(J\) is the current density. The total current \(I\) is given by:

\(I = J \pi R^2\)

Where \(R\) is the radius of the conductor.

Using Ampère's Law and recognizing that symmetry allows us to assume \(\mathbf{B}\) is constant in magnitude at a radial distance from the conductor's axis:

\(B \cdot 2 \pi r = \mu_0 J \pi r^2\)

Simplifying for \(B\), we have:

\(B = \frac{\mu_0 J r}{2}\)

Since \(J = \frac{I}{\pi R^2}\), substituting this into the equation for \(B\) gives:

\(B = \frac{\mu_0 I r}{2 \pi R^2}\)

This equation shows that the magnetic field \(B\) varies linearly with \(r\), meaning it is minimum (zero) at the axis (\(r = 0\)) and increases as one moves towards the surface. Therefore, the magnetic field is minimum at the axis.

Given the options, this analysis matches with option D: "minimum at the axis of the conductor."