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Question

Two small identical circular loops, marked (1) and (2), carrying equal currents, are placed with the geometrical axes perpendicular each other as shown on figure. Find the magnitude and direction of the net magnetic field produced at the point O.
1781128_aa4dea0cb294498f9b08f4b29f046deb.png

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Solution

Magnetic field due to coil 1 at point O
B1=μ0iR22(R2+x2)3/2 along OC1
Magnetic field due to coil 2 at point O
B2=μ0iR22(R2+x2)3/2 along C2O
Both B1 and B2 are mutually perpendicular, so magnetic field at O is
B=B21+B22=2B1 (as B1=B2)
=2μ0iR22(R2+x2)3/2
As R<<x
B=2μ0iR22.x3=μ04π.22μ0i(πR2)x3
=μ04π.22μ0i Ax3
where A=πR2 is area of loop.
tanθ=B2B1 tanθ=1 (B2=B1)
θ=π4
B directed at an angle π4 with the direction of magnetic fields B1

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