question_answer

An infinite current carrying conductor carries current i and lies parallel to Z-axis and situated at point P as shown in figure. Find  \[\int\limits_{A}^{B}{\vec{B}\,.d\vec{\ell }}.\]

A) \[\frac{{{\mu }_{0}}i}{24}\]                 

B) \[\frac{{{\mu }_{0}}i}{16}\]

C) \[\frac{{{\mu }_{0}}i}{12}\]                 

D) \[\frac{{{\mu }_{0}}i}{8}\]

Correct Answer: A

Solution :

[a] \[B=\frac{{{\mu }_{0}}i}{2\pi \sqrt{{{x}^{2}}+{{a}^{2}}}}\] \[\int{\overrightarrow{B}}.\overrightarrow{d\ell }=\int\limits_{a/\sqrt{5}}^{a}{\frac{{{\mu }_{0}}idx\cos \,(90-\theta )}{2\pi \,\sqrt{{{x}^{2}}+{{a}^{2}}}}}\] \[=\frac{{{\mu }_{0}}ia}{2\pi }\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)_{\frac{a}{\sqrt{3}}}^{a}=\frac{{{\mu }_{0}}i}{2\pi }\left[ \frac{\pi }{4}-\frac{\pi }{6} \right]=\frac{{{\mu }_{0}}i}{24}\]