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a current can be induced by changing the area of a coil in a constant magnetic field. By Faraday's Law: the induced current is proportional to the rate of the change of flux in a loop of wire. With magnetic flux being defined as the product of the magnitude of the magnetic field and the area of the loop. The direction of the current is found from Len's Law: The induced current produces an induced magnetic field that opposes the change of flux causing the current.

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You don't induce a current, you induce a voltage. And Faraday's Law states that the induced voltage, not current, is proportional to the rate of change of flux! If the coil is open circuited, a voltage is still induced into the coil but no current will flow. For current to flow, the coil must be connected to a load (or short circuited), and this current is dependent upon the values of the induced voltage and the resistance of the load.

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