Question

Two identical long conducting wires $AOB$ and $COD$ are placed at right angle to each other, with one above other such that $O$ is their common point for the two. The wires carry $I_1$ and $I_2$ currents, respectively. Point $P$ is lying at distance $d$ from $O$ along a direction perpendicular to the plane containing the wires. The magnetic field at the point $P$ will be

• A. $\frac{\mu_0}{2 \pi d}\big(\frac{I_1}{I_2}\big)$
• B. $\frac{\mu_0}{2 \pi d}(I_1 + I_2)$
• C. $\frac{\mu_0}{2 \pi d}(I_1^2 - I_2^2)$
• D. $\frac{\mu_0}{2 \pi d}(I_1^2 + I_2^2)^{1/2}$

Answer 1

Answer: (D)

$\frac{\mu_0}{2 \pi d}(I_1^2 + I_2^2)^{1/2}$

Answer 2

The magnetic field at the point P, at a perpendicular distance d from O in a direction perpendicular to the plane ABCD due to currents through AOB and COD are perpendicular to each other. Hence, $B=(B_1^2+B_2^2)^{1/2}$ $=\Big[\big(\frac{\mu_0}{4 \pi}\frac{2I_1}{d}\big)^2+\big(\frac{\mu_0}{4 \pi}\frac{2I_2}{d}\big)^2\Big]^{1/2}$ $\, \, \, \, =\frac{\mu_0}{2 \pi d}(I_1^2 + I_2^2)^{1/2}$