Q.Describe the construction and working of a simple AC generator, and derive the expression $e=e_0\sin\omega t$ for the emf induced in its rotating coil.

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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