Q. A current loop, having two circular arcs joined by two radial lines is shown in the figure. It carries a current of 10 A. The magnetic field at point O will be close to :

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Solution:

$\vec{B} = \frac{\mu_{0} i}{4\pi} \theta \left[ \frac{1}{r_{1} } - \frac{1}{r_{2}}\right]\hat{k} $
$ r_{1} = 3cm = 3 \times10^{-2}m$
$ r_{2} = 5 cm = 5 \times10^{-2} m$
$ \theta= \frac{\pi}{4} , i = 10 A $
$\Rightarrow \vec{B} = \frac{4\pi \times10^{-7}}{16} \times10 \left[ \frac{1}{3\times10^{-2}} - \frac{1}{5 \times10^{-2}}\right] \hat{k} $
$ \Rightarrow \left|\vec{B} \right| = \frac{\pi}{3} \times10^{-5} T$
$ \approx 1 \times10^{-5} T $

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