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

A semicircular wire of radius \( R \) carries a current \( I \). The magnetic field at the center \( O \) of a circular arc with current is \( B = \frac{\mu_0 I \theta}{4\pi R} \), where \( \theta \) is the angle in radians subtended by the arc. For a semicircle, \( \theta = \pi \) radians. Substituting this, we get \( B = \frac{\mu_0 I \cdot \pi}{4\pi R} = \frac{\mu_0 I}{4R} \). Therefore, the magnetic field at \( O \) is \( B = \frac{\mu_0 I}{4R} \).