A current $i$ is flowing in a straight conductor of length $L$. The magnetic induction at a point distant $\frac{L}{4}$ from its centre will be
Diffcult
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(a) By using $B = \frac{{{\mu _0}}}{{4\pi }}.\frac{i}{a}(\sin {\varphi _1} + \sin {\varphi _2})$
$ \Rightarrow B = \frac{{{\mu _0}}}{{4\pi }}.\frac{i}{{(L/4)}}(2\sin \varphi )$
Also $\sin \varphi = \frac{{L/2}}{{\sqrt 5 L/4}} = \frac{2}{{\sqrt 5 }}$$ \Rightarrow B = \frac{{4{\mu _0}i}}{{\sqrt 5 \pi L}}$
$ \Rightarrow B = \frac{{{\mu _0}}}{{4\pi }}.\frac{i}{{(L/4)}}(2\sin \varphi )$
Also $\sin \varphi = \frac{{L/2}}{{\sqrt 5 L/4}} = \frac{2}{{\sqrt 5 }}$$ \Rightarrow B = \frac{{4{\mu _0}i}}{{\sqrt 5 \pi L}}$
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