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(A) The magnetic field at the centre of a full circular loop of radius $R$ carrying current $i$ is $B = \frac{\mu_0 i}{2R}.$For a semicircular loop,th

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$\frac{\mu_0 i}{2\sqrt{2} R}$

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(A) The magnetic field at the centre of a full circular loop of radius $R$ carrying current $i$ is $B = \frac{\mu_0 i}{2R}.$For a semicircular loop,the magnetic field at the centre is half of that,i.e.,$B_{semi} = \frac{\mu_0 i}{4R}.$Let the semicircular loop in the $x-y$ plane produce a magnetic field $B_{xy} = \frac{\mu_0 i}{4R}$ along the negative $z$-axis (using the right-hand rule).Similarly,the semicircular loop in the $x-z$ plane produces a magnetic field $B_{xz} = \frac{\mu_0 i}{4R}$ along the negative $y$-axis.Since these two fields are mutually perpendicular,the resultant magnetic field $B$ is given by:$B = \sqrt{B_{xy}^2 + B_{xz}^2} = \sqrt{\left(\frac{\mu_0 i}{4R}\right)^2 + \left(\frac{\mu_0 i}{4R}\right)^2}$$B = \sqrt{2 \left(\frac{\mu_0 i}{4R}\right)^2} = \frac{\mu_0 i}{4R} \sqrt{2} = \frac{\mu_0 i}{2\sqrt{2} R}.$

$A$ current loop consists of two identical semicircular parts each of radius $R,$ one lying in the $x-y$ plane and the other in the $x-z$ 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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