Two long straight conductors AOB and COD are perpendicular to each other and carry currents ${\mathrm{I}}_1 \mathrm{and} {\mathrm{I}}_2$. the magnitude of the magnetic inductions at a point ${}^{\text{'}}P^{\text{'}}$ at a distance ${}^{\text{'}}\mathrm{a}^{\text{'}}$ from the point O in a direction perpendicular to the plane ACBD is

Given data:$B=\frac{{\mu }_{0}i}{2\pi r}$

There are two straight and long conductors, AOB and COD, which are perpendicular to each other. These conductors carry currents of $i_1$ and $i_2$, respectively.

Formula used:

Detailed Solution:

The conductors AOB and COD, which are perpendicular to each other, are depicted in the figure. A distance a above O.

$B_1=\frac{{\mu }_0{i}_1}{2\pi a}$ and $B_1=\frac{{\mu }_0{i}_2}{2\pi a}$

As $B_1$ and $B_2$ are perpendicular, we get

$$\begin{gathered} B=\sqrt{{B_1}^2+{B_2}^2} \\ =\frac{{\mu }_0}{2\pi a}\sqrt{{i_1}^2+{i_2}^2} \end{gathered}$$

Therefore, the magnitude is $\frac{{\mu }_0}{2\pi a}\sqrt{{i_1}^2+{i_2}^2}$.