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
AIPMT 2010, Medium
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The loop mentioned in the question must look like one as shown in the figure.
Magnetic field at the centre due to semicircular loop lying in $x-y$ plane, $B_{x y}=\frac{1}{2}\left(\frac{\mu_{0} i}{2 R}\right) \quad$ negative $z$ direction.
Similarly field due to loop in $x-z$ plane,
$B_{x z}=\frac{1}{2}\left(\frac{\mu_{0} i}{2 R}\right)$ in negative $y$ direction.
$\therefore$ Magnitude of resultant magnetic field,
$B =\sqrt{B_{x y}^{2}+B_{x z}^{2}}=\sqrt{\left(\frac{\mu_{0} i}{4 R}\right)^{2}+\left(\frac{\mu_{0} i}{4 R}\right)^{2}}$
$=\frac{\mu_{0} i}{4 R} \sqrt{2}=\frac{\mu_{0} i}{2 \sqrt{2} R}$
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