Question

A flexible wire loop in the shape of a circle has a radius that grows linearly with time. There is a magnetic field perpendicular to the plane of the loop that has a magnitude inversely proportional to the radius of the loop, $B\left(r\right)\propto \frac{1}{r}$. How does the e.m.f, $ϵ$ vary with time?

• A. $ϵ\propto t^2$
• B. $ϵ\propto t$
• C. $ϵ\propto \sqrt{t}$
• D. $ϵisconstant$

Answer 1

The correct option is D $ϵisconstant$

Here, $B\propto \frac{1}{r}=\frac{c}{r}$
$A=\pi r^2$
$\varphi =BA=\frac{c}{r}.\pi r^2=\pi cr$
$\therefore ϵ=\frac{d\phi }{dt}=\pi c\frac{dr}{dt}=\pi ck[\because r_0+kt]$
So, $ϵ$ is constant.