Induced Voltage

$\Large{x^{2}+y^{2} = t^{2}}$

A conducting loop in the $xy$-plane takes the form of the curve given above. The parameter $t$ denotes time. Suppose that there is a uniform magnetic flux density $B$ which is normal to the $xy$-plane.

If the magnitude of the voltage induced in the loop at time $t = 2$ can be expressed as $\alpha \pi B$, determine the value of $\alpha$.

Details and Assumptions:

• Neglect units and assume that all physical expressions take their simplest forms.