Question

A current $ I $ flows in an infinitely long wire with cross section in the form of semi-circular ring of radius 1 m. The magnitude of the magnetic induction along its axis is

• A. \( \frac{\mu_0 I}{2r} \, \text{T} \)
• B. \( \frac{\mu_0 I}{4r} \, \text{T} \)
• C. \( \frac{\mu_0 I}{2\pi r} \, \text{T} \)
• D. \( \frac{\mu_0 I}{2\pi r^2} \, \text{T} \)

Hint:

When dealing with magnetic induction in current-carrying loops, use the Biot-Savart law to determine the magnetic field produced by specific configurations, such as semi-circular loops or solenoids.

Answer 1

The Correct Option is B

For an infinitely long wire bent into a semi-circular shape, we can use the Biot-Savart law to find the magnetic field at the center of the semi-circular loop. 
The formula for the magnetic induction along the axis of a semi-circular loop of current is given by: \[ B = \frac{\mu_0 I}{4r} \] Where: - \( \mu_0 \) is the permeability of free space, - \( I \) is the current, - \( r \) is the radius of the semi-circular loop. Given that the radius is 1 m and the magnetic field along the axis is directed along the line passing through the center of the semi-circle, the magnitude of the magnetic induction is \( \frac{\mu_0 I}{4r} \). 
Thus, the magnetic induction is \( \frac{\mu_0 I}{4r} \, \text{T} \).