Question

A rectangular coil of $N$ turns and area of cross-section $A$ is rotated at a steady angular speed $\omega$ in a uniform magnetic field. Obtain an expression for the emf induced in the coil at any instant of time.

Hint:

The maximum emf is induced when the plane of the coil is perpendicular to the magnetic field, i.e., when $\sin(\omega t) = 1$. This occurs twice during each complete rotation, once in each half-cycle.

Answer 1

Step 1: Derive the formula for induced emf. The magnetic flux $\Phi$ through the coil at any time $t$ is given by: $$\Phi = B \cdot A \cdot \cos(\omega t)$$ where $B$ is the magnetic field strength, $A$ is the area of the coil, and $\omega t$ is the angle made by the normal to the coil with the magnetic field due to its rotation. 

Step 2: Apply Faraday's Law of Electromagnetic Induction. Faraday's law states that the induced emf ($\mathcal{E}$) in the coil is equal to the negative rate of change of magnetic flux through the coil: $$\mathcal{E} = -N \frac{d\Phi}{dt}$$ Differentiating the flux equation with respect to time gives: $$\frac{d\Phi}{dt} = -B \cdot A \cdot \omega \sin(\omega t)$$ Therefore, the induced emf is: $$\mathcal{E} = N \cdot B \cdot A \cdot \omega \sin(\omega t)$$