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A current $I$ flows in a long straight wire with cross-section having the form of a thin half-ring of radius $R$ (figure). Find the induction of the magnetic field at the point $O$ .

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$\frac{μ_{0} i}{π^{2} R}$

$\frac{μ_{0} i}{2 π^{2} R}$

$\frac{μ_{0} i}{2 πR}$

$\frac{μ_{0} i}{4 πR}$

answer is A.

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Detailed Solution

Take two elements, as shown in figure. The current in each element,

$d i = \frac{i}{π} d θ$

The resultant field

$B = \int_{0}^{π / 2} 2 \sin θ d B = 2 \int_{0}^{π / 2} (\frac{μ_{0}}{2 π} . \frac{i}{R}) \sin θ d θ = \frac{μ_{0} i}{π^{2} R}$

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