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?

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?

Infinity Learn · Accepted Answer

Magnetic field at O due to wire $$1$$ isB→$$1=$$μ$$0i4$$$$π$$R⊙ and that due to wire $$2$$ is B→$$2=$$μ$$0i4$$$$π$$R⊙⇒B→$$=B$$→$$1+B$$→$$2=$$μ$$0i2$$$$π$$R⊙ Magnetic field at the centre due to circular arc subtending angle θ at the centre is given byB→$$arc=$$μ$$0i2R$$θ$$2$$π⊗ If B→net $$=0$$⇒B→$$+B$$→arc $$=0$$ ⇒  B→$$=-B$$→arc  ⇒  μ$$0i2$$$$π$$R⊙$$=-$$μ$$0i2R$$θ$$2$$π⊗ ⇒  μ$$0i2$$$$π$$$$R=$$μ$$0i2R$$θ$$2$$π⇒$$1$$π$$=$$θ$$2$$π⇒θ$$=2rad$$

Magnetic field due to current element

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