The unit vectors $\hat i,\;\hat j\;{\rm{and }}\,\hat k$ are as shown below. What will be the magnetic field at $O$ in the following figure
Diffcult
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(d) The field at $O$ due to $AB$ is $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\hat k$ and that due to $DE$ is also $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\hat k$.
However the field due to $BCD$ is $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\left( {\frac{\pi }{2}} \right)\,\hat k$.
Thus the total field at $O$ is $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\left( {2 + \frac{\pi }{2}} \right)\,\hat k$
However the field due to $BCD$ is $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\left( {\frac{\pi }{2}} \right)\,\hat k$.
Thus the total field at $O$ is $\frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}\left( {2 + \frac{\pi }{2}} \right)\,\hat k$
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