Question

Two straight long conductors $AOB$ and $COD$ are perpendicular to each other and carry currents $i_1$ and $i_2$. The magnitude of the magnetic induction at a point $P$ at a distance $d$ from the point $O$ in a direction perpendicular to the plane $ABCD$ is :

• A. $\frac{{\mu }_0}{2\pi d}(i_1+i_2)$
• B. $\frac{{\mu }_0}{4\pi d}(i_1-i_2)$
• C. $\frac{{\mu }_0}{2\pi d}\sqrt{(i_1^2+i_2^2)}$
• D. $\frac{{\mu }_0}{2\pi d}\left(\frac{i_1{i}_2}{i_1+i_2}\right)$

Answer 1

The correct option is C $\frac{{\mu }_0}{2\pi d}\sqrt{(i_1^2+i_2^2)}$

Magnetic field at point $P$ due to the wire $AOB$ is $B_1=\frac{{\mu }_0{I}_1}{2\pi d}$
Magnetic field at point $P$ due to the wire $COD$ is $B_2=\frac{{\mu }_0{I}_2}{2\pi d}$
But, $B_1\perp B_2$
$\therefore B=\sqrt{B_1^2+B_2^2}=\frac{{\mu }_0}{2\pi d}\sqrt{I_1^2+I_2^2}$