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A current loop consists of two identical semicircular parts each of radius R, one lying in the x-y plane and the other in x-z plane. If the current in the loop is i, the resultant magnetic field due to the two semicircular parts at their common centre is

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$\frac{μ_{0} i}{4 R}$

$\frac{μ_{0} i}{2 \sqrt{2} R}$

$\frac{μ_{0} i}{\sqrt{2} R}$

$\frac{μ_{0} i}{2 R}$

answer is B.

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Detailed Solution

Magnetic field at the centre due to semicircular loop lying in x-y plane,

$B_{xy} = \frac{1}{2} (\frac{μ_{0} i}{2 R}) in z - direction$

Magnetic field due to loop in x-z plane,

$B_{xz} = \frac{1}{2} (\frac{μ_{0} i}{2 R}) in y - direction$

Magnitude of resultant magnetic field,

$B_{net} = \sqrt{B_{xy}^{2} + B_{xz}^{2}} = \sqrt{{(\frac{μ_{0} i}{4 R})}^{2} + {(\frac{μ_{0} i}{4 R})}^{2}} = \frac{μ_{0} i}{4 R} \sqrt{2} = \frac{μ_{0} i}{2 \sqrt{2} R}$

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