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. $\frac{\mu_0 i}{2\sqrt2 R}$
• B. $\frac{\mu_0 i}{2R}$
• C. $\frac{\mu_0 i}{4R}$
• D. $\frac{\mu_0 i}{\sqrt2 R}$

Answer 1

Answer: (A)

$\frac{\mu_0 i}{2\sqrt2 R}$

Answer 2

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, $B_{xy}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ negative z direction.
Similarly field due to loop in x-z plane,
$B_{xz}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ in negative y direction.
$\therefore$ Magnitude of resultant magnetic field,
$B=\sqrt{{B_{xy}^2}+{B_{xz}^2}}=\sqrt{\big(\frac{\mu_0 i}{4R}\big)^2+\big(\frac{\mu_0 i}{4R}\big)^2}$
$\, \, \, \, =\frac{\mu_0 i}{4R}\sqrt2=\frac{\mu_0 i}{2 \sqrt2 R}$