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Magnetic field due to current element

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Question

A wire carrying current i has the configuration shown in figure. Two semi-infinite straight sections, each tangent to the same circle, are connected by a circular arc, of angle θ, along the circumference of the circle, with all sections lying in the same plane. What must θ (in rad) be in order for B to be zero at the center of circle?

Moderate
Solution

Magnetic field at O due to wire 1 is

B1=μ0i4πR and that due to wire 2 is B2=μ0i4πRB=B1+B2=μ0i2πR 

Magnetic field at the centre due to circular arc subtending angle θ at the centre is given by

Barc=μ0i2Rθ2π If Bnet =0B+Barc =0   B=-Barc    μ0i2πR=-μ0i2Rθ2π   μ0i2πR=μ0i2Rθ2π1π=θ2πθ=2rad


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