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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
A
$\frac{\mu_0i}{2\sqrt{2}R}$
B
$\frac{\mu_0i}{2R}$
C
$\frac{\mu_0i}{4R}$
D
$\frac{\mu_0i}{\sqrt{2}R}$
Detailed Solution
Magnetic field at the centre of a full circular loop is $\frac{\mu_0i}{2R}$; a semicircle gives half of this: $B_1 = B_2 = \frac{\mu_0i}{4R}$
The field of each semicircle is perpendicular to its own plane. The semicircle in the x-y plane gives a field along the z-axis, and the one in the x-z plane gives a field along the y-axis.
The two fields are therefore perpendicular to each other and equal in magnitude.
Resultant: $B = \sqrt{B_1^2 + B_2^2} = \sqrt{2}\times\frac{\mu_0i}{4R}$
$B = \frac{\mu_0i}{2\sqrt{2}R}$
The field of each semicircle is perpendicular to its own plane. The semicircle in the x-y plane gives a field along the z-axis, and the one in the x-z plane gives a field along the y-axis.
The two fields are therefore perpendicular to each other and equal in magnitude.
Resultant: $B = \sqrt{B_1^2 + B_2^2} = \sqrt{2}\times\frac{\mu_0i}{4R}$
$B = \frac{\mu_0i}{2\sqrt{2}R}$