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(D) According to Faraday's law of electromagnetic induction,the induced e.m.f. $E$ is given by the negative rate of change of magnetic flux $\phi$ wit

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(D) According to Faraday's law of electromagnetic induction,the induced e.m.f. $E$ is given by the negative rate of change of magnetic flux $\phi$ with respect to time $t$:$E = -\frac{d\phi}{dt}$From the given diagram,the magnetic flux $\phi$ varies sinusoidally with time $t$,which can be represented as:$\phi = \phi_0 \sin(\omega t)$Now,differentiating this expression with respect to time $t$ to find the induced e.m.f. $E$:$E = -\frac{d}{dt} [\phi_0 \sin(\omega t)]$$E = -\phi_0 \omega \cos(\omega t)$Using the trigonometric identity $-\cos(	heta) = \sin(	heta - \frac{\pi}{2})$,we get:$E = \phi_0 \omega \sin(\omega t - \frac{\pi}{2})$This shows that the induced e.m.f. $E$ also varies sinusoidally with time $t$,but it has a phase shift of $-\frac{\pi}{2}$ relative to the magnetic flux $\phi$. At $t = 0$,the flux $\phi = 0$,while the induced e.m.f. $E = -\phi_0 \omega$,which is a negative maximum value. Among the given options,the graph that starts at a negative maximum value and follows a sinusoidal pattern is represented by option $(D)$.

The magnetic flux $\phi$ through a coil varies with time $t$ as shown in the diagram. Which graph best represents the variation of the induced electromotive force (e.m.f.) $E$ in the coil with time $t$?

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