Correct way to represent the inductive reactance ? XL = ωL or XL = jωL

Faraday's Law deals with induced EMF in a conductor (inductive elements) when exposed to a changing magnetic field. When the term back EMF is used, it implies a conductor rotating in a magnetic field, such as in a motor.

Capacitive elements are not part of Faraday's Law.

The statement by the lecturer about back EMF does imply a motor. In single phase AC motors, a capacitor is used in conjunction with a start winding to generate a torque in the motor in a specific direction so the motor starts rotating in the desired direction.

Maybe the lecturer wasn't clear on how the capacitor works in a motor, or its function in the motor.

If the value of resistance tends to zero, the induced emf becomes equal and opposite to applied emf resulting in no net current.

No, not remotely so. You're caught in the trap of thinking that an equal and opposite induced emf to the applied emf implies zero current but that does not logically follow.

What's required here is clarity of thought.

Carefully consider the consequences of the following conditions for an ideal inductor:

• there is an emf, proportional to the rate of change of current through, that tends to oppose the change in current
• there is an applied voltage that is precisely opposite the emf

Logically, if the above hold (which they must for an ideal inductor), the rate of change of current through the inductor is proportional to the applied voltage across.

If the voltage across an ideal inductor is a constant, non-zero value, it is illogical to conclude that the current is zero. The logical conclusion is that the rate of change of current is constant.

If the inductor current were constantly zero, the induced emf would be zero but that would be inconsistent with a non-zero applied voltage since it would imply 'infinite' current for an inductor with vanishing resistance.

Put another way, if the current is to be finite for a non-zero applied voltage across, there must be an emf precisely opposite the applied voltage and that implies a non-zero rate of change of current.