question_answer

Two wires AO and OC carry equal currents i as shown in figure. One end of both the wires extends to infinity. Angle AOC is\[\alpha \]. The magnitude of magnetic field at point P on the bisector of these two wires at a distance r from point O is

A) \[\frac{{{\mu }_{0}}}{2\pi }\frac{i}{r}\cot \left( \frac{\alpha }{2} \right)\]

B) \[\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}\cot \left( \frac{\alpha }{2} \right)\]

C) \[\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}\frac{\left( 1+\cos \frac{\alpha }{2} \right)}{\sin \left( \frac{\alpha }{2} \right)}\]

D) \[\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}\left( \frac{\alpha }{2} \right)\]

Correct Answer: C

Solution :

[c] \[x=r\sin \frac{\alpha }{2}\] \[\therefore {{B}_{p}}=2\left( \frac{{{\mu }_{0}}}{4\mu } \right)\left( \frac{i}{x} \right)\left[ \sin \left( 90{}^\circ -\frac{\alpha }{2} \right)+\sin 90{}^\circ  \right]\]\[=\frac{{{\mu }_{0}}}{2\pi }\frac{i}{r}\frac{\left( 1+\cos \frac{\alpha }{2} \right)}{\sin \frac{\alpha }{2}}\]