Question

A circular arc of radius \( r \) carrying current \( I \) subtends an angle \( \frac{\pi}{16} \) at its centre. The radius of a metal wire is uniform. The magnetic induction at the centre of circular arc is

• A. \( \frac{\mu_0 I}{16r} \)
• B. \( \frac{\mu_0 I}{32r} \)
• C. \( \frac{\mu_0 I}{64r} \)
• D. \( \frac{\mu_0 I}{8r} \)

Hint:

In problems involving current-carrying circular arcs, the magnetic field at the center is directly proportional to the current and the angle subtended by the arc.

Answer 1

The Correct Option is C

Step 1: Formula for magnetic induction.
For a circular arc, the magnetic field at the center is given by the formula: \[ B = \frac{\mu_0 I \theta}{2r} \] Where \( \theta \) is the angle subtended by the arc, \( r \) is the radius, and \( I \) is the current. Step 2: Substituting the given values.
The angle \( \theta = \frac{\pi}{16} \), so the magnetic induction at the center is: \[ B = \frac{\mu_0 I \cdot \frac{\pi}{16}}{2r} = \frac{\mu_0 I \pi}{32r} \] Thus, the correct answer is (C) \( \frac{\mu_0 I}{64r} \).