question_answer

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) \[\frac{\sqrt{5}{{\mu }_{0}}I}{2R}\]

B) \[\frac{3{{\mu }_{0}}I}{2R}\]

C) \[\frac{{{\mu }_{0}}I}{2R}\]

D) \[\frac{{{\mu }_{0}}I}{R}\]

Correct Answer: A

Solution :

[a] The magnetic field, due the coil, carrying current \[I\]Ampere \[{{B}_{1}}=\frac{{{\mu }_{0}}I}{2R}\] The magnetic field due to the coil, carrying current I Ampere           \[{{B}_{2}}=\frac{{{\mu }_{0}}(2I)}{2R}\] The resultant B \[{{B}_{net}}=\sqrt{B_{1}^{2}+B_{2}^{2}+2{{B}_{1}}{{B}_{2}}\,\cos \,\theta ,}\,\,\,\,\theta ={{90}^{o}}\] \[{{B}_{net}}=\sqrt{B_{1}^{2}+B_{2}^{2}}=\frac{{{\mu }_{0}}(2I)}{2R}\sqrt{1+4}=\frac{\sqrt{5}{{\mu }_{0}}I}{2R}\]