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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
$\frac{\mu_0}{2\pi d}\left(\frac{I_1}{I_2}\right)$
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
$\frac{\mu_0}{2\pi d}(I_1^2 + I_2^2)^{1/2}$
Detailed Solution
The fields of the two wires at P are perpendicular: $B = \sqrt{B_1^2 + B_2^2}$
$B = \sqrt{\left(\frac{\mu_0I_1}{2\pi d}\right)^2 + \left(\frac{\mu_0I_2}{2\pi d}\right)^2} = \frac{\mu_0}{2\pi d}\sqrt{I_1^2 + I_2^2}$
$B = \sqrt{\left(\frac{\mu_0I_1}{2\pi d}\right)^2 + \left(\frac{\mu_0I_2}{2\pi d}\right)^2} = \frac{\mu_0}{2\pi d}\sqrt{I_1^2 + I_2^2}$