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As written in my book and as I saw on some videos about inductors that in an ac circuit if an inductor is connected with an a.c. source then the induced emf would be the same voltage as the source that causes it, but with opposite poles of the source which causes it.
why the induced should have opposite poles? According to Lenz's law, when current increases the induced emf poles should be opposite of the source but when the current decreases the flux inside the coil decreases. As the emf wants to stop this change hence the emf would have the same poles as source.

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    $\begingroup$ Hint: Kirchoff's Voltage Law $\endgroup$ – The Photon Feb 27 '18 at 17:17
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Let us analyse the process of variation of AC voltage and current in an inductive circuit.

A change in value of current in a inductor coil (either a rise or a fall) causes a corresponding change of the magnetic flux around the coil.

And the rate of change of flux is in tune with rate of change of current.

The Induced e.m.f. is directly proportional to the rate of change of flux, of course with a negative sign. The opposite sign is the Lenz's Law signature.

In A.C. Circuits the driving voltage oscillates and the current is maximum when the voltage is at zero passing through time axis. So a phase difference of 90 degree exists through out.

As the current changes at its maximum rate when it is going through its zero value at PI/2 when a.c. voltage is at its maximum and at 3.pi/2, the flux change is also the greatest at those times.

Thereby , the self-induced e.m.f. in the coil is at its maximum (or minimum) value at the above points and its opposed to the driving voltage.

Because the current is not changing at the point when it is going through its peak value at 0°, pi, and 2.pi the flux change is zero at those times.

Therefore, the self induced e.m.f. in the coil is at its zero value at these points.

The above explanation will be more clear if a plot is made of voltage, current and induced e.m.f. on the same graph.

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You are talking about the phenomenon of self-induction. The magnetic field produced by the time changing current through the loop induces an EMF in the loop which opposes the time change of the current. Thus when you apply an ac voltage, the induced EMF is always opposed to the applied voltage. This leads to a very high impedance of the coil, which reduces the amplitude of the current through the coil. This effect of self-inductance is fully consistent with Lenz's law.

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