Question

Calculate the magnetic field at a point located at a distance $\frac{\sqrt{3}}{2}a$ from midpoint of the straight wire of length $a$ carrying a current of $I$. The point lies on the perpendicular bisector of the wire.

• A. $\frac{{\mu }_0}{4\pi }\frac{I}{a}$
• B. $\frac{{\mu }_{0}I}{4\pi a}\frac{2}{\sqrt{3}}$
• C. $\frac{{\mu }_{0}I}{2\pi a}\frac{2}{\sqrt{3}}$
• D. None of these

Answer 1

The correct option is B $\frac{{\mu }_{0}I}{4\pi a}\frac{2}{\sqrt{3}}$

From $\Delta PSO$,

$\tan \theta =\frac{PS}{SO}=\frac{a/2}{\sqrt{3}a/2}=\frac{1}{\sqrt{3}}$

$\therefore \theta ={30}^{\circ }$

So magnetic field due to wire $PQ$ at $O$ is given by

$B=\frac{{\mu }_0}{4\pi }\frac{I}{OS}[\sin {\theta }_1+\sin {\theta }_2]$

$B=\frac{{\mu }_0}{4\pi }\frac{I}{\left(\sqrt{3}a/2\right)}[\sin {30}^{\circ }+\sin {30}^{\circ }]$

$B=\frac{{\mu }_0}{4\pi }\frac{I}{a}\frac{2}{\sqrt{3}}$

Hence, option $\left(b\right)$ is correct.