Question

A current $i$ ampere flows in a circular arc of wire which subtends an angle $\left(\frac{3\pi }{2}\right)$ radius at its centre, whose radius is $R$. The magnetic induction $B$ at the centre is :

• A. $\frac{{\mu }_{0}i}{R}$
• B. $\frac{{\mu }_{0}i}{2R}$
• C. $\frac{2{\mu }_{0}i}{R}$
• D. $\frac{3{\mu }_{0}i}{8R}$

Answer 1

The correct option is D $\frac{3{\mu }_{0}i}{8R}$

$\text{Step 1: Magnetic field due to the arc subtending an angle}$

At the centre :

Magnetic field due to complete circle i.e for angle $2\pi$ : $B=\frac{{\mu }_{o}i}{2R}$

$\Rightarrow$Magnetic field due to unit angle: $B=\frac{1}{2\pi }\times \frac{{\mu }_{o}i}{2R}$

$\Rightarrow$Magnetic field due to $\theta$ angle: $B=\frac{\theta }{2\pi }\times \frac{{\mu }_{o}i}{2R}$

$\Rightarrow B=\frac{\theta {\mu }_{o}i}{4\pi R}$

[$\text{Note:}$ Above result can directly be used for finding magnetic field at centre of an arc subtending an angle $\theta$]

$\text{Step 2: Putting values}$
$\because \theta =\frac{3\pi }{2}$

$\Rightarrow B=\frac{3{\mu }_{o}i}{8R}$

Hence , Option(D) is correct