Question

Two long wires $AO$ and $OC$ carry equal currents $i$ a shown in the 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. $\frac{{\mu }_0}{2\pi }\frac{i}{r}\frac{(1+\cos \frac{\alpha }{2})}{\sin \left(\frac{\alpha }{2}\right)}$
• D. $\frac{{\mu }_0}{4\pi }\frac{i}{r}\left(\frac{\alpha }{2}\right)$

Answer 1

Answer: (C)

$\frac{{\mu }_0}{2\pi }\frac{i}{r}\frac{(1+\cos \frac{\alpha }{2})}{\sin \left(\frac{\alpha }{2}\right)}$

$x=r\sin \frac{\alpha }{2}$

$\therefore B_P=2\left(\frac{{\mu }_0}{4\pi }\right)\left(\frac{i}{x}\right)[\sin ({90}^o-\frac{\alpha }{2})+\sin {90}^o]$

$=\frac{{\mu }_0}{2\pi }\frac{i}{r}\frac{(1+\cos \frac{\alpha }{2})}{\sin \frac{\alpha }{2}}$