According to Faraday's Law of Induction, the magnitude of the induced electromotive force ($|\varepsilon|$) is equal to the rate of change of magnetic flux ($\Phi_B$).

1. Calculate Magnetic Flux ($\Phi_B$): The flux through the loop is given by $\Phi_B = \mathbf{B} \cdot \mathbf{A} = BA \cos(\theta)$. Since the loop is placed perpendicular to the magnetic field, the area vector is parallel to the field lines (angle $\theta = 0^\circ$), so $\cos(0^\circ) = 1$. Given: $A = 2.5 \times 10^{-3} \text{ m}^2$ $B = 0.5 \sin(100\pi t) \text{ T}$

$\Phi_B = (0.5 \sin(100\pi t)) \times (2.5 \times 10^{-3})$ $\Phi_B = 1.25 \times 10^{-3} \sin(100\pi t) \text{ Wb}$

2. Differentiate Flux with respect to time ($t$): $|\varepsilon| = \left| \frac{d\Phi_B}{dt} \right| = \frac{d}{dt} \left[ 1.25 \times 10^{-3} \sin(100\pi t) \right]$ $|\varepsilon| = 1.25 \times 10^{-3} \times (100\pi) \times \cos(100\pi t)$ $|\varepsilon| = 0.125\pi \cos(100\pi t) \text{ V}$

3. Calculate EMF at $t = 0$: At $t = 0$, $\cos(0) = 1$. $|\varepsilon|_{t=0} = 0.125\pi \text{ V}$

4. Convert Units: To convert Volts (V) to millivolts (mV), multiply by 1000: $0.125\pi \text{ V} = 0.125\pi \times 1000 \text{ mV} = 125\pi \text{ mV}$.