Question

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. $\frac{{\sqrt{\mu }}_{0}I}{2R}$
• C. $\frac{{\mu }_{0}I}{2R}$
• D. $\frac{{\mu }_{0}I}{R}$

Answer 1

Answer: (A)

Magnetic field induction due to vertical loop at the centre $O$ is
$B_1=\frac{{\mu }_{0}I}{2R}$
It acts in horizontal direction.
Magnetic field induction due to horizontal loop at the centre $O$ is
$B_2=\frac{{\mu }_{0}2I}{2R}$
It acts in vertically upward direction.
As $B_1$ and $B_2$ are perpendicular to each other,
therefore the resultant magnetic field induction at the centre $O$ is
$B_{\text{net}}=\sqrt{B_1^2+B_2^2}=\sqrt{{\left(\frac{{\mu }_{0}I}{2R}\right)}^2+{\left(\frac{{\mu }_{0}2I}{2R}\right)}^2}$
$B_{\text{net }}=\frac{{\mu }_{0}I}{2R}\sqrt{(1{)}^2+(2{)}^2}=\frac{\sqrt{5}{\mu }_{0}I}{2R}$