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Question

A small current carrying loop having current i is placed in the plane of the paper as shown in figure. Another semicircular loop having current i0 is placed concentrically in the same plane as that of the small loop, the radius of semicircular loop being R(R>>a). Find the force applied (in newton) by the smaller ring on the bigger ring.
(Given R=1m,i=i0=40μ0A,a=0.1m)
167904.PNG

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Solution

Magnetic field due to smaller ring at a distance R (equatorial point) from the centre:
B=μ04πiπa3R3=μ0iπa24R3 where M=iπa2
B=μ04πiπa3R3=μ0iπa24R3
dF=i0Rdθμ0ia24R3
dFx=dFsinθ=μ0ii0a2sinθdθ4R2
Fx=μ0ii0a24R2π0sinθdθ
Fx=μ0ii0a24R2×2=μ0ii0a22R2;Fy=0
Fnet=Fx=μ0ii0a22R2=8N

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