Figure shows a closed loop $A O C B A$ in which current $I$ is flowing as shown. Given $O A = O B = O C = a$ . Find the magnetic field at point $B$ due to this loop.

- \frac{μ_{0} I}{4 π \sqrt{2} a} (^i +^k)

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- \frac{μ_{0} I}{4 π \sqrt{2} a} (^j +^k)

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\frac{{- μ}_{0} I}{2 \sqrt{2} π a} (^j +^k)

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None of these

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Solution

The correct option is B $- \frac{μ_{0} I}{4 π \sqrt{2} a} (^j +^k)$
At $B$ , magnetic field will be zero due to $C B$ and $A B$ parts.
$_{A O} = \frac{μ_{0}}{4 π} \frac{I}{a} [sin 0^{o} + sin 45^{o}] (-^j) = \frac{{- μ}_{0} I}{4 π \sqrt{2} a}^j$
$_{O C} = \frac{μ_{0}}{4 π} \frac{I}{a} [sin 0^{o} + sin 45^{o}] (-^k) = \frac{{- μ}_{0} I}{4 π \sqrt{2} a}^k$
Net magnetic field at $B$
$_{A O} +_{O C} = \frac{{- μ}_{0} I}{4 π \sqrt{2} a} (^j +^k)$

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Figure shows a closed loop AOCBA in which current I is flowing as shown. Given OA=OB=OC=a. Find the magnetic field at point B due to this loop.

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