Question

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

• A. $\frac{{\mu }_{0}i}{2\sqrt{2}R}$
• B. $\frac{{\mu }_{0}i}{2R}$
• C. $\frac{{\mu }_{0}i}{4R}$
• D. $\frac{{\mu }_{0}i}{\sqrt{2}R}$

Answer 1

Answer: (A)

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,
$B_{xy}=\frac{1}{2}\left(\frac{{\mu }_{0}i}{2R}\right)$ negative $z$ direction.
Similarly field due to loop in $x$-z plane, $B_{xz}=\frac{1}{2}\left(\frac{{\mu }_{0}i}{2R}\right)$ in negative y direction.
$\therefore$ Magnitude of resultant magnetic field,
$B=\sqrt{B_{xy}^2+B_{xz}^2}=\sqrt{{\left(\frac{{\mu }_{0}i}{4R}\right)}^2+{\left(\frac{{\mu }_{0}i}{4R}\right)}^2}$
$=\frac{{\mu }_{0}i}{4R}\sqrt{2}=\frac{{\mu }_{0}i}{2\sqrt{2}R}$