Question

Two wires AO and OC carry currents $i$ as shown in figure. One end of both the wires extends to infinity $\angle AOC=\alpha$ , the magnitude of magnetic field at a point P on the bisector of the two wires at a distance $r$ from O is

• A. $\frac{{\mu }_{0}i}{2\pi r}\cot \left(\frac{\alpha }{2}\right)$
• B. $\frac{{\mu }_{0}i}{4\pi r}\cot \left(\frac{\alpha }{2}\right)$
• C. $\frac{{\mu }_{0}i}{2\pi r}\frac{(1+\cos \frac{\alpha }{2})}{\sin \left(\frac{\alpha }{2}\right)}$
• D. $\frac{{\mu }_{0}i}{4\pi r}\sin \left(\frac{\alpha }{2}\right)$

Answer 1

The correct option is C $\frac{{\mu }_{0}i}{2\pi r}\frac{(1+\cos \frac{\alpha }{2})}{\sin \left(\frac{\alpha }{2}\right)}$

$B=B_1+B_2$
$=2B_1$
$=2\frac{{\mu }_{0}i}{4\pi r\sin \frac{d}{2}}(\cos \frac{d}{2}+\cos 0^{\circ })$
$=\frac{{\mu }_{0}i}{2\pi r}\frac{(1+\cos \frac{d}{2})}{\sin \left(\frac{d}{2}\right)}$