Question

A wire carrying current I has the shape as shown in the adjoining figure. Linear parts of the wire are very long and parallel to the X-axis while the semicircular portion of radius R lies in the YZ-plane. Magnetic field at point O is: 

• A. \(\vec{B}=-\dfrac{\mu_0 I}{4\pi R}(\hat{i}+2\hat{k})\)
• B. \(\vec{B}=-\dfrac{\mu_0 I}{4\pi R}(\hat{i}+2\hat{k})\)
• C. \(\vec{B}=\dfrac{\mu_0 I}{4\pi R}(\hat{i}-2\hat{k})\)
• D. vecB=(mu₀ I)/(4π R)(hati+2hatk)

Hint:

• Semicircle at centre: B=(mu₀ I)/(4R) • Long straight wire: B=(mu₀ I)/(2π r) Always apply right-hand thumb rule for direction.

Answer 1

The Correct Option is D

Step 1: Magnetic field at the centre of a semicircular loop: Bₛemi=(mu₀ I)/(4R) Direction is along +hatk (right-hand rule).
Step 2: Each long straight wire contributes: Bwire=(mu₀ I)/(2π R) Net contribution of the two straight wires is along +hati.
Step 3: Adding vector components: vecB=(mu₀ I)/(4π R)(hati+2hatk)