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I understand that current can't flow through a coil if it is not complete, but surely this means that if a magnet is dropped through the coil, no EMF would be induced either, since work is done when a second field is induced to do work on the magnet opposing its motion, and no second field is induced without current, so what's doing the work here?

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Some transient current can flow in the open coil as there is some (small) capacitance.

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EMF is induced inside metal line even in an open circuit. At the same time, $-\nabla(\phi)$ (the electric field caused by electric scalar potential) is generated, and it cancels EMF so that the total electric field becomes zero. As a result electric current along metal line vanishes.

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In simpler terms to the above comment. In open circuits, whether it be induced EMF or a battery, an initial potential difference is across the ends of the wire. As a result, electrons move against field lines. This causes charge to build up on the ends of of wire, until the electric field produced by this build up of charge cancel with the initial potential difference

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