Q.
Two wires $PQ\ and\ QR,$ carry equal currents $i$ as shown in figure. One end of both the wires extends to infinity $\angle PQR = \ \theta.$ The magnitude of the magnetic field at O on the bisector angle of these two wires at a distance $r\ from\ point\ Q\ $is
A) $\frac{\mu_{0}}{4\pi}\frac{i}{r}\sin\left( \frac{\theta}{2} \right)$
B) $\frac{\mu_{0}}{4\pi}\frac{i}{r}\cot\left( \frac{\theta}{2} \right)$
C) $\frac{\mu_{0}}{4\pi}\frac{i}{r}\tan\left( \frac{\theta}{2} \right)$
D) $\frac{\mu_{0}}{4\pi}\frac{i}{r}\ \frac{(1 + \cos\theta/2)}{(\sin{\theta/2})}$
Explanation
(d) Perpendicular to $O\ \ from\ PQ.$
Or $QR,\ \ a = r\sin{\theta/2}$
Magnetic field induction at $O$ due to current through $PQ\ \ and\ QR$ is
$B = \frac{\mu_{0}}{4\pi}\frac{i}{a}\ \lbrack\sin(90{^\circ} - \theta/2$)+ sin$90{^\circ}\rbrack\ \times 2\ $
$= \frac{\mu_{0}}{2\pi}\ \frac{i}{\sin{\theta/2}}(\cos{\theta/2 + 1) = \frac{\mu_{0}}{2\pi}\frac{i}{r}\ \frac{(1 + \cos{\theta/2)}}{\sin{\theta/2}}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.