Question

Current \( I \) flows through a conducting wire of radius \( a \). The magnetic field \( B \) at a distance \( r \) from the centre of the wire (where \( r>a \) and \( \mu \) is the permeability of free space) is:

• A. \( \frac{\mu I}{2\pi a^2} \)
• B. \( \frac{\mu I r}{2\pi a^2} \)
• C. \( \frac{\mu I}{2\pi r} \)
• D. \( \frac{\mu I}{\pi r^2} \)

Hint:

The magnetic field around a long straight conductor is inversely proportional to the distance from the wire.

Answer 1

The Correct Option is C

To determine the magnetic field \(B\) at a distance \(r\) from the center of a conducting wire with current \(I\) flowing through it, we use Ampère's Law. According to Ampère's Law, the line integral of the magnetic field \(B\) around a closed path is equal to the permeability of free space \(\mu\) times the current enclosed by the path:

\[\oint B \cdot dl = \mu I_{enc}\] 

For a long straight wire, we consider a circular path of radius \(r\) centered on the wire. The magnetic field \(B\) is tangent to this circle and has uniform magnitude along the path, so:

\[B \cdot 2\pi r = \mu I\]

Solving for \(B\), we find:

\[B = \frac{\mu I}{2\pi r}\]

This yields the magnetic field at a distance \(r\) from the center of the wire, where \(r > a\). Hence, the correct expression for the magnetic field is:

\[\frac{\mu I}{2\pi r}\]

Answer 2

For a long straight conducting wire, the magnetic field at a distance \( r \) from the centre of the wire (for \( r>a \)) is given by Ampere’s law: \[ B = \frac{\mu I}{2\pi r} \] Where:
- \( B \) is the magnetic field at distance \( r \),
- \( I \) is the current in the wire,
- \( r \) is the distance from the wire,
- \( \mu \) is the permeability of free space.