Question

If a wire of length \( L \) carrying a current \( i \) is bent in the shape of a semi-circular arc as shown in the figure, then the magnetic field at the centre of the arc is:

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

Hint:

Always express the radius in terms of arc length when the wire is bent into a circular or semi-circular shape. Then use standard magnetic field formulas accordingly.

Answer 1

The Correct Option is A

Step 1: Relation between arc length and radius.
A semi-circular wire has length: \[ L = \pi R \Rightarrow R = \frac{L}{\pi} \] Step 2: Magnetic field at the center of a semi-circular current-carrying wire: \[ B = \frac{\mu_0 i}{4R} \] Step 3: Substitute \( R = \frac{L}{\pi} \) into the equation: \[ B = \frac{\mu_0 i}{4 . \frac{L}{\pi}} = \frac{\pi \mu_0 i}{4L} \]