$A$ circular coil of radius $10\; cm$,$500$ turns,and resistance $2\; \Omega$ is placed with its plane perpendicular to the horizontal component of the Earth's magnetic field. It is rotated about its vertical diameter through $180^{\circ}$ in $0.25\; s$. Estimate the magnitudes of the emf and current induced in the coil. The horizontal component of the Earth's magnetic field at the place is $3.0 \times 10^{-5}\; T$.

(N/A) Given: Radius $r = 0.1\; m$,Area $A = \pi r^2 = \pi \times (0.1)^2 = \pi \times 10^{-2}\; m^2$,Number of turns $N = 500$,Resistance $R = 2\; \Omega$,Time $\Delta t = 0.25\; s$,Magnetic field $B = 3.0 \times 10^{-5}\; T$.
Initial flux through the coil,$\Phi_{\text{initial}} = N B A \cos 0^{\circ} = 500 \times 3.0 \times 10^{-5} \times \pi \times 10^{-2} = 1.5 \pi \times 10^{-4}\; Wb$.
Final flux after $180^{\circ}$ rotation,$\Phi_{\text{final}} = N B A \cos 180^{\circ} = -1.5 \pi \times 10^{-4}\; Wb$.
Change in flux,$\Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = -3.0 \pi \times 10^{-4}\; Wb$.
Induced emf,$\varepsilon = -\frac{\Delta \Phi}{\Delta t} = -\frac{-3.0 \pi \times 10^{-4}}{0.25} = 12 \pi \times 10^{-4} \approx 3.77 \times 10^{-3}\; V$.
Induced current,$I = \frac{\varepsilon}{R} = \frac{3.77 \times 10^{-3}}{2} \approx 1.88 \times 10^{-3}\; A$.