MHTCET 15 May 2024 Shift 2 Paper

Concept:This is a unipolar generator (Faraday's disc), where rotation of a conducting disc in a constant magnetic field induces an EMF between the axis and the rim.Explanation:The magnetic field $B$ acts perpendicular to the plane of the disc and remains constant.As the disc rotates with angular velocity $\omega$, the radius sweeps out area continuously, so the magnetic flux linked with the disc changes with time.In one complete revolution, the area swept by a radius of the disc is:$$dA = \pi R^{2}$$The time taken for one complete revolution is:$$dt = \frac{2\pi}{\omega}$$The change in magnetic flux during this time is:$$d\Phi = B \cdot dA = B \pi R^{2}$$Using Faraday's law, the magnitude of the induced EMF is:$$|e| = \left|-\frac{d\Phi}{dt}\right| = \frac{B \pi R^{2}}{2\pi / \omega}$$$$\therefore |e| = \frac{B R^{2} \omega}{2}$$The negative sign in Faraday's law only indicates the direction of the induced EMF (Lenz's law), not its magnitude.Answer:The induced EMF is $\frac{B R^{2} \omega}{2}$.Hence, the correct option is D: $\frac{BR^{2}\omega}{2}$.