Question

A wire bent as a parabola $y=k{c}^2$ is located in a uniform magnetic field of induction $B$, the vector $B$ being perpendicular to the plane $x,y$. At the moment $t=0$ a connector starts sliding translationwise from the parabola apex with a constant acceleration $w$ (Fig.). Find the emf of electromagnetic induction in the loop thus formed as a function of $y$.

Answer 1

Obviously, from Lenz's law, the induced current and hence the induced e.m.f. in the loop is anticlockwise.
From Faraday's law of electromagnetic induction,
${\xi }_{in}=\left|\frac{d\Phi }{dt}\right|$
Here, $d\Phi =\overrightarrow{B}\cdot d\overrightarrow{S}=-2Bxdy$,
and from $y=a{x}^2,x=\sqrt{\frac{y}{a}}$
Hence, ${\xi }_{in}=2B\sqrt{\frac{y}{a}}\frac{dy}{dt}$
$=By\sqrt{\frac{8w}{a}}$, using $\frac{dy}{dt}=\sqrt{2wy}$.