Question

An equilateral triangular loop having a resistance $R$ and length of each side $l$ is placed in a magnetic field which is varying at $\frac{dB}{dt}=1\text{T/s}$. The induced current in the loop will be

• A. $\frac{\sqrt{3}}{4}\frac{l^2}{R}$
• B. $\frac{4}{\sqrt{3}}\frac{l^2}{R}$
• C. $\frac{\sqrt{3}}{4}\frac{R}{l^2}$
• D. $\frac{4}{\sqrt{3}}\frac{R}{l^2}$

Answer 1

The correct option is A $\frac{\sqrt{3}}{4}\frac{l^2}{R}$

Magnetic flux $\varphi =NBA\cos \theta$
$\varphi =\left(1\right)B\times \frac{1}{2}\left(l\right)\left(\frac{\sqrt{3}l}{2}\right)\cos 0^{\circ }$
$\varphi =\frac{\sqrt{3}B{l}^2}{4}$

Emf generated,
$\left|ϵ\right|=|\frac{d\varphi }{dt}|$
$\left|ϵ\right|=\frac{\sqrt{3}{l}^2}{4}\frac{dB}{dt}$
$\left|ϵ\right|=\frac{\sqrt{3}{l}^2}{4}$ $(\because \frac{dB}{dt}=1)$

Current generated,
$i=\frac{\left|ϵ\right|}{R}$
$i=\frac{\sqrt{3}{l}^2}{4R}$