question_answer

Find the magnetic field at point O due to    semi-infinite wire shown in figure.

A)  \[\frac{\sqrt{2}{{\mu }_{0}}I}{(4\pi R)}\]out of plane of paper

B)  \[\frac{\sqrt{2}{{\mu }_{0}}I}{4\pi R}\]into the plane of paper

C)  \[\frac{(\sqrt{2}-1){{\mu }_{0}}I}{4\pi R}\]out of plane of paper

D)  \[\frac{(\sqrt{2}-1){{\mu }_{0}}I}{4\pi R}\]into the plane of paper

Correct Answer: C

Solution :

\[B=\frac{{{\mu }_{0}}I}{4\pi d}[\sin \alpha +\sin \beta ]\]  [Magnetic field due to current carrying wire] \[=\frac{{{\mu }_{0}}I}{4\pi d(R/\sqrt{2})}\left[ \sin \left( 2\pi -\frac{\pi }{4} \right)+\sin {{90}^{o}} \right]\] \[=\frac{{{\mu }_{0}}I}{4\pi R}\times \sqrt{2}\left[ 1-\frac{1}{\sqrt{2}} \right]\] \[=\frac{{{\mu }_{0}}I}{4\pi R}\times (\sqrt{2}-1)\] (Out of plane of paper)