Question

Two identical loops P and Q each of radius $5$cm are lying in perpendicular planes such that they have a common centre as shown in the figure. Find the magnitude and direction of the net magnetic field at the common centre of the two coils, if they carry currents equal to $3$A and $4$A respectively.

Answer 1

Radius of each coil $R=5cm=0.05m$

Magnetic field produced by coil $P$ and $Q$ is $B_1$ and $B_2$ in $x$ and $y$ direction, respectively.

Magnetic field produced by coil P, $B_1=\frac{{\mu }_o{I}_1}{2R}$ Tesla $\hat{x}$

$\therefore$ $B_1=\frac{3{\mu }_o}{2\left(0.05\right)}=30{\mu }_o$ Tesla$\hat{x}$

Magnetic field produced by coil Q, $B_2=\frac{{\mu }_o{I}_2}{2R}$ Tesla $\hat{y}$

$\therefore$ $B_2=\frac{4{\mu }_o}{2\left(0.05\right)}=40{\mu }_o$ Tesla$\hat{y}$

Total magnetic field produced at the center $B=\sqrt{B_1^2+B_2^2}$

$\therefore$ $B=\sqrt{(30{\mu }_o{)}^2+(40{\mu }_o{)}^2}$

$⟹$ $B=50{\mu }_o$ Tesla

Direction of magnetic field $\tan \theta =\frac{B_2}{B_1}$

$\therefore$ $\tan \theta =\frac{40{\mu }_o}{30{\mu }_o}=\frac{4}{3}$

$⟹$ $\theta =ta{n}^{-1}\left(4/3\right)={53}^o$