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What is the magnetic field at O due to current in the B infinite wire forming a loop as shown in fig. ?

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$\frac{μ_{0}}{4 π} \frac{i}{d}$

$4 π \frac{2 i}{d}$

$\frac{μ_{0}}{4 π} \frac{i}{d} (\sin ϕ_{1} + \sin ϕ_{2})$

$\frac{μ_{0}}{4 π} \frac{i}{d} (\cos ϕ_{1} + \cos ϕ_{2})$

answer is A.

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Detailed Solution

Here required angles $θ_{1} and θ_{2} are (90^{\circ} - ϕ_{1})$ and $(90^{\circ} - ϕ_{2})$ Hence
B $= \frac{μ_{0} i}{4 πd} [\sin (90^{\circ} - ϕ_{1}) + \sin (90^{\circ} - ϕ_{2})]$
$= \frac{μ_{0} \dot{I}}{4 πd} (\cos ϕ_{1} + \cos ϕ_{2})$

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