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.
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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.2√2μ0i(πR2)x3 =μ04π.2√2μ0iAx3 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