A circular metal plate of radius R is rotatin | vedclass

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(D) Consider a small radial element of length $dx$ at a distance $x$ from the centre of the circular plate.As the plate rotates with angular velocity

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$\frac{\omega B R^2}{2}$

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(D) Consider a small radial element of length $dx$ at a distance $x$ from the centre of the circular plate.As the plate rotates with angular velocity $\omega$,the linear velocity of this element is $v = \omega x$.The motional emf $d\varepsilon$ induced across this small element of length $dx$ in a magnetic field $B$ is given by $d\varepsilon = B v dx$.Substituting $v = \omega x$,we get $d\varepsilon = B (\omega x) dx$.To find the total emf $\varepsilon$ developed between the centre $(x=0)$ and the rim $(x=R)$,we integrate the expression:$\varepsilon = \int_{0}^{R} B \omega x dx = B \omega \int_{0}^{R} x dx$.$\varepsilon = B \omega \left[ \frac{x^2}{2} \right]_{0}^{R} = \frac{B \omega R^2}{2}$.

$A$ circular metal plate of radius $R$ is rotating with a uniform angular velocity $\omega$ with its plane perpendicular to a uniform magnetic field $B$. Then the emf developed between the centre and the rim of the plate is

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