Question

The value of magnetic field at point \( O \) in the given figure is:

• A. \( \frac{\mu_0 I}{2\pi R} \)
• B. \( \frac{\mu_0 I}{\pi R} \)
• C. \( \frac{\mu_0 I}{4R} \)
• D. \( \frac{\mu_0 I}{R} \)

Hint:

Use the formula \( B = \frac{\mu_0 I \theta}{4\pi R} \) for a current-carrying arc. For a semicircle, use \( \theta = \pi \) radians.

Answer 1

The Correct Option is C

The wire forms a semicircular arc of radius \( R \), with current \( I \) flowing through it. The magnetic field at the center \( O \) of a circular arc carrying current is given by the formula: \[ B = \frac{\mu_0 I \theta}{4\pi R} \] where \( \theta \) is the angle subtended by the arc at the center in radians. Here, since the arc is a semicircle: \[ \theta = \pi \text{ radians} \] So, \[ B = \frac{\mu_0 I \cdot \pi}{4\pi R} = \frac{\mu_0 I}{4R} \] Thus, the magnetic field at point \( O \) is: \[ B = \frac{\mu_0 I}{4R} \]