Magnetic field due to a moving charge

Question

Two identical long conducting wires AOB and COD are placed at right angle to each other, with one above other such that 0 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 0 along a direction perpendicular to the plane containing the wires. The magnetic field at the point P will be

Easy

Solution

The magnetic field at the point P, at a perpendicular distance d from 0 in a direction perpendicular to the plane ABCD due to currents through AOB and COD are perpendicular to each other. Hence,

$B={\left({B}_{1}^{2}+{B}_{2}^{2}\right)}^{1/2}={\left[{\left(\frac{{\mu}_{0}}{4\pi}\frac{2{I}_{1}}{d}\right)}^{2}+\frac{{\mu}_{0}}{4\pi}{\left(\frac{2{I}_{2}}{d}\right)}^{2}\right]}^{1/2}=\frac{{\mu}_{0}}{2\pi d}{\left({I}_{1}^{2}+{I}_{2}^{2}\right)}^{1/2}$

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