Question

Figure shows infinitely long bent wires in the plane of paper and each carrying a current $i$ with $O$ as origin and $X$ and $Y$ axis as shown, co-ordinates of points $P$ & $Q$ are $(2m,0)$ and $(6m,0)$ respectively. Magnetic induction at $P$ is given by $\frac{\alpha {\mu }_{0}i}{32\pi }$. The value of $\alpha$ is

Answer 1

The currents passing through $P$ do not contribute to the magnetic field at P.
Only the wires perpendicular to the x-axis contribute to the magnetic field.
Hence, magnetic field at point $P$ is due to wires $AO$ and $BQ$

For wire AO, $\overrightarrow{B}=\frac{{\mu }_{o}i}{4\pi r}=\frac{{\mu }_{o}i}{4\pi 2}⨂$
For wire $BQ:{\overrightarrow{B}}_2=\frac{{\mu }_0}{4\pi }\frac{i}{4}⨂$
$Net\overrightarrow{B}={\overrightarrow{B}}_1+{\overrightarrow{B}}_2=\frac{{\mu }_{o}i}{4\pi }.\frac{3}{4}=\frac{6{\mu }_{o}i}{32\pi }$
$\therefore \alpha =6$