Two similar coils of radius r are placed concentrically with their planes at right angles to each other. The currents flowing in them are I and 3I, respectively. The resultant magnetic induction at the centre is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{\sqrt{10}\,\mu_0 I}{2r} \)
The field at the centre of a circular coil is \( \mu_0 I/2r \), directed along the axis of the coil. The planes of the two coils are at right angles, so their axes, and hence their fields at the common centre, are perpendicular; the fields add as vectors.
\( \displaystyle B_1 = \frac{\mu_0 I}{2r} \qquad B_2 = \frac{\mu_0 (3I)}{2r} = 3B_1 \)
\( \displaystyle B = \sqrt{B_1^{2} + B_2^{2}} = B_1\sqrt{1 + 9} = \frac{\sqrt{10}\,\mu_0 I}{2r} \)