Question

Two similar coils of radius R and number of turns N are lying concentrically with their planes at right angles to each other. The currents flowing in them are $I$ and $I\sqrt{3}$ respectively. Then the resultant magnetic induction at the centre will be :
(in $Wb/m^2)$

• A. $\frac{{\mu }_{0}NI}{2R}$
• B. $\frac{{\mu }_{0}NI}{R}$
• C. $\sqrt{3}{\mu }_0\frac{NI}{2R}$
• D. $\sqrt{5}\frac{{\mu }_{0}NI}{2R}$

Answer 1

The correct option is B $\frac{{\mu }_{0}NI}{R}$

Magnetic induction at the center of coil, $B=\frac{{\mu }_{o}NI}{2R}$ (along the axis)
For coil 1, $B_1=\frac{{\mu }_{o}NI}{2R}\hat{x}$
For coil 2, $B_2=\frac{{\mu }_{o}N\sqrt{3}I}{2R}\hat{y}$
Resultant magnetic induction $B_T=\frac{{\mu }_{o}NI}{2R}\sqrt{1+{\sqrt{3}}^2}$
$B_T=\frac{{\mu }_{o}NI}{R}$