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(Here's my understanding just in case if I misunderstood something)

I understand that for order to induce a voltage on the other side of the transformer or in the secondary coil even though there's no battery over there, a varying magnetic field is needed to be cut repeatedly (lenz's law).

This can be done using an AC generator which produces a varying current, both in direction and value. And since a current in a wire produces its own magnetic field and the current is varrying the magnetic field will be varrying too.

Then these magnetic field travel through the soft iron core which, I guess, happens because the primary coil rotates around the core, they are touching, so it's basically touching a magnet and getting magnetised by induction (when a magnetic material touches a permanent magnet and becomes a magnet itself too my book call this magnetisation by induction).

However the current doesn't travel through it too, even though iron must be a good conductor of electricity, because the coil must be coated with an insulator.

My question is what will happen if the insulating coat of some wires fell off and current can travel through the soft iron core? The wires doesn't have to be next to each other because I know they may create a short circuit maybe they're separated by other wires which are still insulated.

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First: the iron core of a transformer is not made of a solid iron , but is composed of thin sheets of iron, isolated against each other. Otherwise you would have circular currents in the core. But even so, if just e very short piece of your wire has no insulation, there would not be a current trough the core, since the second pole is mission, so just the little uninsulated piece would be short-circuited , the same effect as making it thicker. It would be different if one pole of the AC generator , for example earth, would be connected to th core, than you would have a short circuit.

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what will happen if the insulating coat of some wires fell off and current can travel through the soft iron core?

If the core is ungrounded then the core will just be energized to the voltage level of which ever winding (primary or secondary) became shorted to it (neglecting electrostatics etc.).

If the core is grounded (likely) then current can flow and will burn/damage the iron and laminations. The current that will flow depends on how low or high in the winding the short occurs (e.g. at the top of the coil or at the bottom), the source configuration and grounding, and overall impedance of the fault circuit.

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