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The magnetic field at the origin due to the current flowing in the wire is

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$- \frac{μ_{0} I}{8 π a} (\hat{i} + \hat{k})$

$\frac{μ_{0} I}{4 π a \sqrt{2}} (\hat{i} - \hat{k})$

$\frac{μ_{0} I}{2 π a} (\hat{i} + \hat{k})$

$\frac{μ_{0} I}{8 π a} (- \hat{i} + \hat{k})$

answer is C.

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Detailed Solution

$B_{O D} = 0 B_{O B} = 0 B_{A B} = \frac{μ_{0} I}{4 π a \sqrt{2}} [\cos 45 ° (- \hat{i}) + \cos 45 ° \hat{k}] = \frac{μ_{0} I}{8 π a} (- \hat{i} + \hat{k})$

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