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Ok, I've seen this question on this site before and other forums as well.

A similar argument can be made if a resistor is connected to a voltage source through wires. The voltage across the resistor should be equal and opposite to the applied voltage so no current should flow.

But a real voltage source has an internal resistance and the wire has a small resistance too so the voltage across the resistor and applied voltage is never the same that's why current flows.

I want to see if we take an ideal case (applied source resistance=0; wire resistance =0) how does the current flow in an inductor if it's back emf is equal and opposite to the applied voltage ? Same argument for voltage source connected to a resistor with ideal conditions

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The voltage across the resistor should be equal and opposite to the applied voltage so no current should flow

I think that this is the heart of your misconception. You seem to think that there is a requirement that there should be some leftover voltage for current to flow. This is not the case, in fact, this would violate Kirchoffs voltage law. There is no leftover voltage needed to drive current.

Instead, each circuit element defines a relationship between voltage across that element and current through it. Let’s discuss only ideal circuit elements for now. For a resistor current and voltage are proportional, meaning that through a resistor you need a voltage for a current to flow. That is not the case for other circuit elements. For an inductor you can have a steady current of any amount with 0 voltage, and for a wire there is always 0 volts regardless of the current, and for a capacitor you can have 0 current with any steady voltage.

So the basic principle is that each circuit element has its own current-voltage relationship, and for many of them a current does not require a voltage. Not all devices are resistors, so you shouldn’t think of their behavior as though they were resistors.

how does the current flow in an inductor if it's back emf is equal and opposite to the applied voltage ?

This question is a little upside-down. A back emf is due to a changing current, so how could current not flow (except instantaneously) while there is a back emf. Since the current is changing, by definition, then even if it starts at zero it will immediately change to some non-zero value.

A back emf is related to the change in the current, the larger the change in the current the larger the back emf. But again, remember that it does not require any voltage to drive a steady current through an inductor, so the back emf poses no restriction on the amount of current, only on its rate of change.

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The is the same question as to why does something move when we push it, given that Newton's third law says that it pushes back on us with a force equal and opposite. The answer is the also the same. So understand mechanical pushing and you understand the answer to your question.

The answer to the pushing paradox is found in the concept of the "free body diagram." Draw the objects and the forces that act on them, then all is clear.

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