Question

Find the magnetic field at the centre $O$ of the loop shown in the figure.

Answer 1

As point - O lies in - line with all three wires magnetic field due to them

$\overrightarrow{B_2}=\overrightarrow{B_4}=\overrightarrow{B_6}=0$

We know,

${\overrightarrow{B}}_{arc}$= $\frac{{\mu }_{o}I\theta }{4\pi R}$

So, $\overrightarrow{B_1}=\frac{{\mu }_{o}i}{4\times 4\pi R}\left(\frac{\pi }{2}\right)=\frac{{\mu }_{o}i}{4\times 8R}$

$\overrightarrow{B_3}=\frac{{\mu }_{o}i\pi }{4\pi \left(2R\right)}=\frac{{\mu }_{o}i}{8R}$

$\overrightarrow{B_5}=\frac{{\mu }_{o}i}{4\pi \left(R\right)}\left(\frac{\pi }{2}\right)=\frac{{\mu }_{o}i}{8R}$

So, ${\overrightarrow{B}}_{net}={\sum }_0^{n=1}{\overrightarrow{B}}_n=\overrightarrow{B_1}+\overrightarrow{B_2}+\overrightarrow{B_3}+\overrightarrow{B_4}+\overrightarrow{B_5}+\overrightarrow{B_6}$

${\overrightarrow{B}}_{net}$= $0+0+0+\frac{{\mu }_{o}i}{8R}[\frac{1}{4}+1+1]$

${\overrightarrow{B}}_{net}=\frac{9{\mu }_{o}i}{32R}$