In the following figure a wire bent in the form of a regular polygon of $n$ sides is inscribed in a circle of radius $a$. Net magnetic field at centre will be
KVPY 2012, Diffcult
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(b) Magnetic field at the centre due to one side
${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2i\sin \theta }}{r}$where $r = a\cos \theta $
So ${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2i\sin \theta }}{{a\cos \theta }} = \frac{{{\mu _0}i}}{{2\pi a}}\tan \theta $
Hence net magnetic field
${B_{net}} = n \times \frac{{{\mu _0}i}}{{2\pi a}}\tan \frac{\pi }{n}$.
${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2i\sin \theta }}{r}$where $r = a\cos \theta $
So ${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2i\sin \theta }}{{a\cos \theta }} = \frac{{{\mu _0}i}}{{2\pi a}}\tan \theta $
Hence net magnetic field
${B_{net}} = n \times \frac{{{\mu _0}i}}{{2\pi a}}\tan \frac{\pi }{n}$.
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