Question

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

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

Answer 1

The correct option is A $\frac{4{\mu }_{0}i}{\sqrt{5}\pi L}$

$|\overrightarrow{B}|$ at point $P$ is:

$B=\frac{{\mu }_{0}i}{4\pi R}(\sin {\theta }_1+\sin {\theta }_2)$

here $R=\frac{L}{4}$

$\sin {\theta }_1=\sin {\theta }_2=\frac{\frac{L}{2}}{\sqrt{{\left(\frac{L}{2}\right)}^2+{\left(\frac{L}{4}\right)}^2}}=\frac{4}{2\sqrt{5}}$

so $B=\frac{{\mu }_{0}i}{4\pi \frac{L}{4}}(\frac{4}{2\sqrt{5}}+\frac{4}{2\sqrt{5}})$

$\Rightarrow B=\frac{4{\mu }_{0}i}{\sqrt{5}\pi L}$ so $\left(A\right)$ is correct option.