Q.A stiff semi-circular wire of radius R is rotated in a uniform magnetic field B about an axis passing through its ends. If the frequency of rotation of the wire is f, calculate the amplitude of the alternating emf induced in the wire.

An AC generator (dynamo) is a device based on electromagnetic induction: rotating a coil in a magnetic field converts the mechanical work of rotation into an induced alternating EMF.

An AC generator has a coil that is mechanically rotated (e.g. by a turbine, water flow, or hand-crank) inside a magnetic field $B$. As the coil rotates, the magnetic flux through it changes, and by Faraday's law an EMF is induced:

$\varepsilon = -\dfrac{d\Phi_B}{dt} = NBA\omega\sin\omega t = \varepsilon_0 \sin\omega t$

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Induced emf in a rotating coil is $\varepsilon=NAB\omega\sin(\omega t)$, which is maximum when the coil's plane is parallel to the field (i.e. the rate of change of flux is greatest).

For a coil of $N$ turns, area $A$, rotating with angular velocity $\omega$ in a uniform magnetic field $B$, the flux through it is $\phi = NAB\cos\omega t$ (taking $t=0$ when the plane is perpendicular to $B$, i.e. normal parallel to $B$), and the induced emf is

$\varepsilon = -\frac{d\phi}{dt} = NAB\omega\sin\omega t$

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