Question

two straight long conductors AOB and COD are perpendicular to each other and carry currents I1 and I2 respectively. find the magnitude of the magnetic field at a point P at a distance a from the point O in a direction perpendicular to the plane ABCD.

Answer 1

Magnetic field at P due to current through AOB and COD will be

$B_1=\frac{{\mu }_0}{4\pi }\frac{2I_1}{a}and{B}_2=\frac{{\mu }_0}{4\pi }\frac{2I_2}{a}$

Here, $\overrightarrow{B_1}and\overrightarrow{B_2}$ are perpendicular to each other. The magnitude of resultant magnetic field will be

$$\begin{gathered} B=\sqrt{B_1^2+B_2^2}={\left[{\left(\frac{{\mu }_0}{4\pi }\frac{2I_1}{a}\right)}^2+{\left(\frac{{\mu }_0}{4\pi }\frac{2I_2}{a}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \\ =\frac{{\mu }_0}{2\pi a}{\left(I_1^2+I_2^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \end{gathered}$$