Question

The given figure shows infinitely long bent wires in the $x-y$ plane of paper and each carrying a current $i$. Here $O$ is the origin. Co-ordinates of points $P$ and $Q$ are $(2m,0)$ and $(6m,0)$, respectively. Magnetic induction at $P$ is given by $\frac{\alpha {\mu }_{0}i}{32\pi }$. Find the value of $\alpha$.

Answer 1

Magnetic field at $P$ is only due to wire segments $AO$ and $BQ$ for rest of the segments of wires , the point $P$ lies on their axis. so magnetic field by them is zero at $P$.

$\overrightarrow{B}$ at $P$ due to $AO$

${\overrightarrow{B}}_{AO}=\frac{\left({\mu }_{0}i\right)}{4\pi \left(r\right)}=\frac{\left({\mu }_{0}i\right)}{4\pi \left(2\right)}$ inwards

$\overrightarrow{B}$ at $P$ due to $BQ$

${\overrightarrow{B}}_{B}Q=\frac{\left({\mu }_{0}i\right)}{4\pi \left(r\right)}=\frac{\left({\pi }_{0}i\right)}{4\pi \left(4\right)}$ inwards

Since both fields are in same direction net magnetic field at $P$ will be

${\overrightarrow{B}}_P={\overrightarrow{B}}_{A}O+{\overrightarrow{B}}_{B}Q$

${\overrightarrow{B}}_P=\frac{\left(3{\mu }_{0}i\right)}{16\pi }$

${\overrightarrow{B}}_P=\frac{\left(6{\mu }_{0}i\right)}{32\pi }$

Hence, $\alpha$ value is $6$.