Question

Two very small identical circular loops, (1) and (2), carrying equal currents I are placed vertically (with respect to the plane of the paper) with their geometrical axes perpendicular to each other as shown in the figure. Find the magnitude and direction of the net magnetic field produced at the point O.

Answer 1

We know, Magnetic field due to circular loop $=\frac{\mu 0}{4\pi }\frac{2\pi R^{2}I}{(x^2+R^2{)}^{3/2}}$

$|\overrightarrow{B}|=\frac{r_0{R}^{2}R}{2(x^2+R^2{)}^{3/2}}$

$|{\overrightarrow{B}}_{net}|=\sqrt{2}|\overrightarrow{B}|=\frac{\sqrt{2}{r}_0{R}^{2}I}{2(x^2+R^2{)}^{3/2}}$

So net magnetic field magnitude $\left(|{\overrightarrow{B}}_{net}|\right)$ and its direction is $\frac{r_0{R}^{2}I}{\sqrt{2}(x^2+R^2{)}^{3/2}}$

is along vector $\frac{-\hat{i}-\hat{j}}{\sqrt{2}}$