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(D) The magnetic flux $\phi$ through the loop is given by $\phi = \vec{B} \cdot \vec{A}$.For $0 \leq t \leq \frac{\pi}{\omega}$,the loop is in the reg

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(D) The magnetic flux $\phi$ through the loop is given by $\phi = \vec{B} \cdot \vec{A}$.For $0 \leq t \leq \frac{\pi}{\omega}$,the loop is in the region $y \geq 0$. The area vector $\vec{A}$ makes an angle $	heta = \omega t$ with the $Z$-axis. Thus,$\phi = B_0(\cos \omega t) A \cos(\omega t) = B_0 A \cos^2(\omega t)$.However,the projection of the area in the $XY$ plane is $A \sin(\omega t)$. The magnetic field is along the $Z$-axis. The flux is $\phi = B_0(\cos \omega t) A \sin(\omega t) = \frac{B_0 A}{2} \sin(2\omega t)$.The induced e.m.f. is $\varepsilon = -\frac{d\phi}{dt} = -\frac{d}{dt} \left( \frac{B_0 A}{2} \sin(2\omega t) ight) = -B_0 A \omega \cos(2\omega t)$ for $0 \leq t \leq \frac{\pi}{\omega}$.For $\frac{\pi}{\omega} \leq t \leq \frac{2\pi}{\omega}$,the loop is in the region $y \leq 0$ where the magnetic field is zero,so $\phi = 0$ and $\varepsilon = 0$.This pattern repeats,matching the plot in option $D$.

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