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Q)

Find the magnitude of the magnetic field at the origin O due to a very long conductor carrying current I of the shape .

$\begin{array}{1 1} \frac{\mu_0 I}{2r} \bigg[\frac{\hat i}{\pi}+\frac{k}{2} \bigg] \\\frac{\mu_0 I}{r} \bigg[\frac{\hat i}{\pi}+\frac{k}{2} \bigg] \\ \frac{\mu_0 I}{4r} \bigg[\frac{\hat i}{\pi}+\frac{k}{2} \bigg] \\ \frac{\mu_0 I}{4r} \bigg[\frac{\hat i}{2}+\frac{k}{\pi2} \bigg] \end{array} $

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A)
Solution :
$\overrightarrow{B_0} =\overrightarrow{B_1}+ \overrightarrow{B_2}+\overrightarrow{B_3}$
$B_3= 0$
$\overrightarrow{B_0} =\overrightarrow{B_1}+ \overrightarrow{B_2}$
$\qquad=\large\frac{\mu_0}{4 \pi}.\frac{I}{r} $$(-\hat i) +(- \hat {k})Z +\frac{\mu_0}{4 \pi }. \frac{I}{R}$$(-\hat {k}).(\frac{\pi}{2})$
$\qquad= \large\frac{\mu_0}{4 \pi}.\frac{I}{r} $$\bigg[-\hat i +\large\frac{\pi \hat k}{2}\bigg]$
$\qquad= \frac{\mu_0 I}{4r} \bigg[\frac{\hat i}{\pi}+\frac{k}{2} \bigg] $
Answer : $ \frac{\mu_0 I}{4r} \bigg[\frac{\hat i}{\pi}+\frac{k}{2} \bigg] $
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