B

Step 1: According to Faraday's Law of Electromagnetic Induction, the induced electromotive force (e.m.f.) in a coil is given by:
$$E = -N \frac{d\Phi}{dt}$$
where \( \Phi \) is the magnetic flux through the coil.
Step 2: The magnetic flux \( \Phi \) through the coil can be expressed as:
$$\Phi = B \cdot A \cdot \cos(\theta)$$
where \( \theta \) is the angle between the magnetic field and the normal to the surface of the coil.
Step 3: In this case, as the coil rotates about an axis, \( \theta \) changes with time. Specifically, we can express \( \theta \) as:
$$\theta = \omega t$$
where \( \omega \) is the angular velocity.
Step 4: Substituting \( \theta \) back into the expression for \( \Phi \):
$$\Phi = B \cdot A \cdot \cos(\omega t)$$
Step 5: Now, differentiating \( \Phi \) with respect to time to find \( \frac{d\Phi}{dt} \):
$$\frac{d\Phi}{dt} = -B \cdot A \cdot \omega \sin(\omega t)$$
Step 6: Finally, substituting into Faraday's Law:
$$E = -N \left(-B \cdot A \cdot \omega \sin(\omega t)\right)$$
Thus, we have:
$$E = N B A \omega \, \sin(\omega t)$$
Step 7: Therefore, the induced e.m.f. in the coil is:
- Correct Option: B: NBωsin(ωt)