Rate of change of area of a circular loop exiting a uniform magnetic field

Question

The question is to find the variation in induced electromotive force w.r.t to time or put simply $\varepsilon(t)$ for a circular loop being removed from the region of the magnetic field at a constant velocity $v$.
Obviously $$\varepsilon= \frac{d\phi}{dt}$$ Now $$\phi = BA\cos \theta=B(\pi r^2)$$ Now, $$\varepsilon= \frac{d\phi}{dt} = 2\pi rB \frac{dr}{dt}$$ But there's no clear way to determine $\frac{dr}{dt}$. So is there any other way to determine the rate of change of area (how much of the circle is leaving the region per $dt$ time). Any help would be very good right now.

Answer 1

The radius isn't changing, just the portion of the loop within the field.

The flux is $\Phi=2B\int_{-R+c}^R\sqrt{R^2-x^2} \ dx$

Where c = beginning position of vertical line where ring meets field boundary.

$dc/dt $ is the rate of leaving the field.

Using the Fundamental Theorem of Calculus:

$$\frac{d\Phi}{dt}=-2B\frac{dc}{dt}\sqrt{2cR-c^2}$$