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Induced voltage happens through induction. When a wire goes through a magnetic field a voltage is induced across its ends.

The basic equation is Ampere's law: Curl E = dB/dt at every point, where E is the electric field and B the magnetic flux density.

For a loop of wire Stokes's theorem can be applied to this: Integral E.dl = d/dt of the surface integral of B.dS which says that the induced voltage along the loop equals the rate of change of the magnetic flux linking the area of the loop.

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