A uniform magnetic field of 0.4 T acts perpendicular to a circular copper disc 20 cm in radius. The disc is having a uniform angular velocity of 10 $ \pi $ rad s^-1 about an axis through its centre and perpendicular to the disc. What is the potential difference developed between the axis of the disc and the rim? $(\pi=3.14)$

<p>To determine the potential difference between the center and the rim of the disc (often called a Faraday disc or unipolar generator), we use the formula for motional EMF in a rotating disc:</p> <p>$V = \frac{1}{2} B \omega R^2$</p> <p>where:</p> <p><p>$B$ is the magnetic field strength,</p></p> <p><p>$\omega$ is the angular velocity,</p></p> <p><p>$R$ is the radius of the disc.</p></p> <p>Given:</p> <p><p>$B = 0.4\ \text{T}$,</p></p> <p><p>$\omega = 10\pi\ \text{rad/s}$,</p></p> <p><p>$R = 20\ \text{cm} = 0.2\ \text{m}$,</p></p> <p>we substitute these values into the formula:</p> <p><p>First calculate $R^2$:</p> <p>$R^2 = (0.2\ \text{m})^2 = 0.04\ \text{m}^2$</p></p> <p><p>Substitute into the formula:</p> <p>$V = \frac{1}{2} \times 0.4 \times (10\pi) \times 0.04$</p></p> <p><p>Multiply step-by-step:</p></p> <p><p>$0.4 \times 10\pi = 4\pi$,</p></p> <p><p>Then, $4\pi \times 0.04 = 0.16\pi$,</p></p> <p><p>Finally, multiply by $\frac{1}{2}$: </p> <p>$V = \frac{1}{2} \times 0.16\pi = 0.08\pi$</p></p> <p><p>Substitute $\pi \approx 3.14$:</p> <p>$V \approx 0.08 \times 3.14 = 0.2512\ \text{V}$</p></p> <p>Thus, the potential difference developed between the axis and the rim is approximately $0.2512\ \text{V}$, which corresponds to Option C.</p>