Question

A current $i$ is flowing in a straight conductor of length $L$. The magnetic induction at a point on its axis at a distance $\frac{L}{4}$ from its centre will be:

• A. $Zero$
• B. $\frac{{\mu }_{0}i}{2\pi L}$
• C. $\frac{{\mu }_{0}i}{\sqrt{2}L}$
• D. $\frac{4{\mu }_{0}i}{\sqrt{5}\pi L}$

Answer 1

Answer: (D)

$|\overrightarrow{B}|$ at point $P$ is:
$<br/>B=\frac{{\mu }_{0}i}{4\pi R}\left(\sin {\theta }_1+\sin {\theta }_2\right)<br/>$
here $R=\frac{L}{4}$
$\sin {\theta }_1=\sin {\theta }_2=\frac{\frac{L}{2}}{\sqrt{{\left(\frac{L}{2}\right)}^2+{\left(\frac{L}{4}\right)}^2}}=\frac{4}{2\sqrt{5}}$
$<br/>\text{ so }B=\frac{{\mu }_{0}i}{4\pi \frac{L}{4}}\left(\frac{4}{2\sqrt{5}}+\frac{4}{2\sqrt{5}}\right)<br/>$
$\Rightarrow B=\frac{4{\mu }_{0}i}{\sqrt{5}\pi L}$