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Potential difference across an inductor (ie p.d. between current exit point and current entry point) is given as

$$V= -L\frac{di}{dt} -iR,$$

where $L$ is the inductance of inductor and $R$ is its resistance.

So, if a current is produced in an inductor, then the current in inductor has to increase ($di/dt > 0$). This would imply that a negative p.d. develops across the inductor. This would imply that no further current should pass through it, as (as far as I know) current can't flow from a low potential point to a high potential point unless there is an EMF device (which can do some positive work on the charges) between the points.

This is because the charge carriers incur several collisions with the metal lattice, time difference between subsequent collisions being very low, and on each collision they lose their drift speed. To gain drift speed again they require an electric field to be present. In general, electric field is directed from high potential point to low potential point. So, without any external work done on the charge, the charge can only move from high potential point to low potential point to constitute a current.

Inductor can be called an emf device, but it does negative work on charges if current (if any) through it is increasing.

Therefore it should not be possible to even develop any current in an inductor in the first place.

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Well, the negative current that you are talking about is not the net current! The net current is still in the direction according to the emf source.

Inductor produces a potential difference across itself as $$V=-Ldi/dt$$ Please be careful regarding the minus sign.

Cases:

  1. (di/dt<0) Its not the current that is negative it is just that the amount of current passing through the inductor is decreasing. In this case the inductor increases the potential difference in the same direction as that of the source.
  2. (di/dt>0) the current is increasing with the time. In this case the inductor opposes the source i.e. it produces negative potential difference across the source or we can also say it produces potential difference in the opposite direction as that of the source.

You can understand the working of the inductor better with the analogy that inductor plays the same role that inertia plays in that of mechanics.

If you try to decrease the speed it will resist that and if you try to increase the speed then it will resist that too! It simply doesn't like changes!

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Inductors try and stop the current changing but an inductor can never win and so cannot totally stop the current changing. The best an inductor can do is reduce the rate of change of current.

A reductio ad absurdum argument goes as follows:

Suppose the inductor did win and there was no current.
If there is no current then the current is not changing.
So the inductor is not trying to stop the current changing.
If the inductor is not trying to stop the current changing the current will change.

The point is that the opposition to the changing current is only there when the current is changing.
Stop the change and you stop the opposition.

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