Two similar coils of radius R are lying concentrically with their planes perpendicular to each other and carrying current I and 2 I respectively. What is the resultant magnetic field at the center?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{\sqrt{5}\,\mu_0 I}{2R} \)
The field at the centre of a circular coil of radius R carrying current I is \( B = \frac{\mu_0 I}{2R} \), directed along the axis of the coil. The coils are concentric with their planes at right angles, so the two fields at the centre are perpendicular to each other and add as vectors.
\( \displaystyle B_1 = \frac{\mu_0 I}{2R}, \qquad B_2 = \frac{\mu_0 (2I)}{2R} = \frac{\mu_0 I}{R} \)
\( \displaystyle B = \sqrt{B_1^{2} + B_2^{2}} = \frac{\mu_0 I}{R}\sqrt{\frac{1}{4} + 1} = \frac{\sqrt{5}\,\mu_0 I}{2R} \)