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Q)

A current carrying loop consists of 3 identical quarter circles of radius R, lying in the positive quadrants of $x-y ,y-z$ and $z-x$ planes , with their centers at the origin joined together . Find the direction and magnitude to $\overrightarrow{B}$ at the origin .

$\begin{array}{1 1} \frac{\mu_0}{8R} (\hat i +\hat j+ \hat k) \\ \frac{\mu_0}{4R} (\hat i +\hat j+ \hat k) \\ \frac{\mu_0}{2R} (\hat i +\hat j+ \hat k) \\ \frac{\mu_0}{4 \pi R } (\hat i +\hat j+ \hat k) \end{array} $

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A)
Solution :
$\overrightarrow{B}=\overrightarrow{B_{xy}} +\overrightarrow{B_{yz}}+\overrightarrow{B_{zx}}$
$\qquad=\bigg(\large\frac{\mu_0}{4 \pi}.\frac{I}{R} \frac{\pi}{2} $$\hat k \bigg)+\bigg(\large\frac{\mu_0}{4 \pi}.\frac{I}{R} \frac{\pi}{2} $$\hat i \bigg)+\bigg(\large\frac{\mu_0}{4 \pi}.\frac{I}{R} \frac{\pi}{2} $$\hat j \bigg)$
$\qquad= \large\frac{\mu_0}{4 \pi}.\frac{I}{R} \frac{\pi}{2} $$(\hat i +\hat j+\hat k)$
$\qquad= \large\frac{\mu_0}{8R}.$$(\hat i+\hat j+\hat k)$
Answer : $\frac{\mu_0}{8R} (\hat i +\hat j+ \hat k)$
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