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If there is a circular metal loop and a changing magnetic field perpendicular to the loop, there is an induced electric field that causes the electrons to move and cause a current. But my question is why is there even a current? Why don't the electrons just move and rearrange themselves to cancel out the induced electric field, just as they do in case of electrostatic shielding?

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    $\begingroup$ Electrons on the move is what current is made of. Now that you know that, is there still a question? $\endgroup$
    – TimWescott
    Jul 13, 2023 at 4:29

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To cancel an electric field, the charges would have to configure themselves in such a way that they cause the exact same electric field, just with opposite direction. Then the two opposing electric fields add up to zero.

However, electric fields caused by charge configurations do not have loops (this is explicitly included in the Maxwell equations via $\nabla\times E=0$ for any configuration without changing magnetic fields). Those induced by changing magnetic fields do, though. Here specifically, there's a field looping through the metal ring. Such a field can't be cancelled by a charge configuration, because that would require a field looping in the opposite direction, and charges can't generate looping fields.

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There is an induced current if there is a complete conducting circuit.
The induced emf drives the mobile charge carriers around the loop and there is no place where the charges can accumulated.

However consider a conducting loop with a gap in it.
Then the induced emf does drive the mobile charge carriers around the conducting part of the loop but that then results in a redistribution of charges around the gap just like the charges on the plates of a capacitor.
Those charges around the gap produce an electric field in opposition to the induced emf and when a steady state is reached the mobile charge carriers in the conductor experience no net force.
The force the mobile charges experience due to the induced emf is balanced by the force they experience due to the charges at the gap.
There is then no induced current.

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Sometimes they do just move and rearrange themselves. They're known as "eddy currents." The currents generated by the changing magnetic field sort of swirl in place, eventually bleeding off the energy in the form of heat. However, if we have a loop connected to something which can do work with the current, we typically design the hardware to minimize those eddy currents and instead drive most of that current around the loop.

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