Two wires AO and OC carry equal currents i as shown in figure. One end of both the wire extends to infinity. Angle AOC is α. The magnitude of magnetic field at a point P on the bisector of these two wires at a distance r from point O is

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Two wires AO and OC carry equal currents i as shown in figure. One end of both the wire extends to infinity. Angle AOC is $α$ . The magnitude of magnetic field at a point P on the bisector of these two wires at a distance r from point O is

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$\frac{μ_{0}}{2 π} \frac{i}{r} cot (\frac{α}{4})$

$\frac{μ_{0}}{4 π} \frac{i}{r} cot (\frac{α}{2})$

$\frac{μ_{0}}{2 π} \frac{i}{r} \frac{(1 + \cot \frac{α}{2})}{\sin (\frac{α}{2})}$

$\frac{μ_{0}}{4 π} \frac{i}{r} (\frac{α}{2})$

answer is A.

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Detailed Solution

$\begin{array}{l} x = r \sin \frac{α}{2} \\ ∴ B_{P} = 2 (\frac{μ_{0}}{4 π}) (\frac{i}{x}) [\sin (90^{0} - \frac{α}{2}) + \sin 90^{0}] \\ = \frac{μ_{0}}{2 π} \frac{i}{r} \frac{(1 + \cos \frac{α}{2})}{\sin (\frac{α}{2})} = \frac{μ_{0} i}{2 πr} cot \frac{α}{4} \end{array}$