As we know that from, Faraday law that a varying current will induce voltage in a coil. When a current passes through a coil it will generate its own magnetic field and this field cuts the same coil producing an emf across it by self induction (L=Coefficient of self induction).
The current which produces flux which further induce a voltage decides the behavior of inductor. If the current is DC then no varying flux is produced so no voltage is induced in the inductor, if the current is varying (AC) then a varying flux is produces which induces some voltage in the coil.
Inductor opposes the change in current, in DC there is no change so inductor has nothing to do. The di/dt is Zero in case of DC so the voltage across the inductor is 0 i.e. it works as short circuit.
Here by the formula we see that,
- When current which increases with time is applied then the inductor try to oppose it by applying a +ve potential to it i.e. di/dt is +ve here so V is +ve.
- When current which decreases with time is applied then the inductor tries to support the supply by applying a -ve supporting potential i.e. di/dt is -ve here so V is -ve.
- When DC is applied, there is no change in voltage or current so di/dt is zero so the voltage across the inductor is zero.