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

1. Formula: The magnetic flux through the loop is $\Phi_B = B \cdot A = B(\pi r^2)$, where $B$ is the magnetic field and $r$ is the radius. Differentiating with respect to time $t$: $|\varepsilon| = \left| \frac{d\Phi_B}{dt} \right| = \left| \frac{d}{dt} (B \pi r^2) \right|$ Since $B$ is constant, we use the chain rule for the changing radius: $|\varepsilon| = B \pi \frac{d(r^2)}{dt} = B \pi (2r) \left| \frac{dr}{dt} \right|$

2. Given Values: Magnetic Field $B = 0.04 \text{ T} = 4 \times 10^{-2} \text{ T}$ Radius $r = 2 \text{ cm} = 2 \times 10^{-2} \text{ m}$

• Rate of shrinking $\left| \frac{dr}{dt} \right| = 2 \text{ mm/s} = 2 \times 10^{-3} \text{ m/s}$

3. Calculation: Substituting the values into the derived equation: $|\varepsilon| = (4 \times 10^{-2}) \times \pi \times 2(2 \times 10^{-2}) \times (2 \times 10^{-3})$ $|\varepsilon| = (4 \times 2 \times 2 \times 2) \pi \times 10^{-2-2-3}$ $|\varepsilon| = 32 \pi \times 10^{-7} \text{ V}$ $|\varepsilon| = 3.2 \pi \times 10^{-6} \text{ V}$ Since $10^{-6} \text{ V} = 1 \mu\text{V}$: $|\varepsilon| = 3.2\pi \mu\text{V}$