Two identical conducting wires $AOB$ and $COD$ are placed at right angles to each other. The wire $AOB$ carries an electric current $I_1$ and $COD$ carries a current $I_2$ . The magnetic field on a point lying at a distance $d$ from $O$, in a direction perpendicular to the plane of the wires $AOB$ and $COD$, will be given by
AIEEE 2007,AIPMT 2014, Diffcult
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Clearly, the magnetic fields at a point $P,$ equidistant from ${A O B}$ and $C O D$ will have directions perpendicular to each other, as they are placed normal to each other.
${\text { Resultant field, } B=\sqrt{B_{1}^{2}+B_{2}^{2}}}$
${\text { But } B_{1}=\frac{\mu_{0} I_{1}}{2 \pi d} \text { and } B_{2}=\frac{\mu_{0} I_{2}}{2 \pi d}}$
${\therefore B=\sqrt{\left(\frac{\mu_{0}}{2 \pi d}\right)^{2}\left(I_{1}^{2}+I_{2}^{2}\right)}}$
$or, B=\frac{\mu_{0}}{2 \pi d}\left(I_{1}^{2}+I_{2}^{2}\right)^{1 / 2}$
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