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: \[ \mathcal{E} = N \cdot B \cdot A \cdot \omega \sin(\omega t) \]