Question

The magnetic field at the center of an equilateral triangular loop of side $2L$ and carrying a current $i$ is:

• A. $\frac{{9\mu }_{0}i}{4\pi L}$
• B. $\frac{{3\sqrt{3}\mu }_{0}i}{4\pi L}$
• C. $\frac{{2\sqrt{3}\mu }_{0}i}{\pi L}$
• D. $\frac{{3\mu }_{0}i}{4\pi L}$

Answer 1

The correct option is A $\frac{{9\mu }_{0}i}{4\pi L}$

Magnetic field at $O$ is:

$B=3\left[\frac{{\mu }_0}{4\pi }\frac{i}{r}(\sin {60}^o+\sin {60}^o)\right]$

$=3\left(\frac{{\mu }_0}{4\pi }\right)\left(i\right)(2\times \frac{\sqrt{3}}{2})\frac{\left(2\sqrt{3}\right)}{2L}$
$=\frac{9{\mu }_{0}i}{2\pi \times 2L}=\frac{9{\mu }_{0}I}{4\pi L}$