A coil of cross-sectional area A having n turns is placed in a uniform magnetic field B. When it is rotated with an angular velocity $$\omega$$, the maximum e.m.f. induced in the coil will be:

We begin by recalling the statement of Faraday’s law of electromagnetic induction. It says that the e.m.f. $$\mathcal{E}$$ induced in a coil is equal to the negative time-rate of change of magnetic flux through the coil. Symbolically, the magnitude is

$$\mathcal{E}=n\left|\frac{d\Phi}{dt}\right|,$$

where $$n$$ is the number of turns and $$\Phi$$ is the magnetic flux linked with one turn.

Now we write the expression for the instantaneous flux. The coil of cross-sectional area $$A$$ is kept in a uniform magnetic field $$B$$. If the normal to the plane of the coil makes an angle $$\theta$$ with the magnetic field, the flux through one turn is

$$\Phi = BA\cos\theta.$$

The coil is rotating with a constant angular velocity $$\omega$$, so the angle changes with time as

$$\theta = \omega t.$$

Substituting this time-dependent angle into the flux expression, we get

$$\Phi(t)=BA\cos(\omega t).$$

We now differentiate this flux with respect to time to find the induced e.m.f. Using the derivative of the cosine function, $$\dfrac{d}{dt}\cos(\omega t) = -\omega\sin(\omega t),$$ we obtain

$$\frac{d\Phi}{dt}=BA\left[-\omega\sin(\omega t)\right] = -BA\omega\sin(\omega t).$$

Putting this into Faraday’s law and multiplying by the number of turns $$n$$, the instantaneous e.m.f. becomes

$$\mathcal{E}(t)=n\left|\,\frac{d\Phi}{dt}\,\right| = n\left| -BA\omega\sin(\omega t) \right| = nBA\omega\left| \sin(\omega t) \right|.$$

The function $$|\sin(\omega t)|$$ reaches its maximum value of $$1$$ whenever $$\sin(\omega t)=\pm1$$, that is, when $$\omega t=\dfrac{\pi}{2},\dfrac{3\pi}{2},\ldots$$ Therefore the maximum possible value of $$\mathcal{E}(t)$$ is

$$\mathcal{E}_{\text{max}} = nBA\omega \times 1 = nBA\omega.$$

Thus the maximum induced e.m.f. is directly proportional to the number of turns, the magnetic field, the area of the coil, and the angular velocity.

Hence, the correct answer is Option C.