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Two similar coils of radius R are lying concentrically with their planes at right angles to each other. The currents flowing in them are I and 2I, respectively. The resultant magnetic field induction at the centre will be
A
$\frac{\mu_0I}{R}$
B
$\frac{\sqrt5\mu_0I}{2R}$
C
$\frac{3\mu_0I}{2R}$
D
$\frac{\mu_0I}{2R}$
Detailed Solution
Field at the centre of each coil: $B_1 = \frac{\mu_0I}{2R}$ and $B_2 = \frac{\mu_0(2I)}{2R}$
The planes are perpendicular, so the two fields are perpendicular to each other.
$B = \sqrt{B_1^2 + B_2^2} = \sqrt{\left(\frac{\mu_0I}{2R}\right)^2 + \left(\frac{2\mu_0I}{2R}\right)^2} = \frac{\sqrt5\mu_0I}{2R}$
The planes are perpendicular, so the two fields are perpendicular to each other.
$B = \sqrt{B_1^2 + B_2^2} = \sqrt{\left(\frac{\mu_0I}{2R}\right)^2 + \left(\frac{2\mu_0I}{2R}\right)^2} = \frac{\sqrt5\mu_0I}{2R}$