A wire bent as a parabola `y=ax^(2)` is located in a uniformed magnetic field of induaction `B` , the vector `B` being perpendicular to the plane `x-y` . At moment `t=0` a connector starts sliding translationwise from the parabola apex with a constant acceleration `omega` . Find the emf of electromagnetic induction in the loop thus formed as a function of `y`

Explanation

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,
`chi_(In) = |(d Phi)/(dt)|`
Here, `d Phi = vec(B).d.vec(S) = -2 B x dy`,
and from `y = ax^(2), x = sqrt((y)/(a))`
Hence, `xi_(In) = 2B sqrt((y)/(a)) (dy)/(dt)`
`= By sqrt((8w)/(a))`, using `(dy)/(dt) = sqrt(2 wy)`