What we know as electric current is rate of flow of charge per unit time what we express as dq/dt. What happens when we apply a voltage across the electrodes of a capacitor is, the dipoles present in the dielectic media is get polarised and this in the form of displacement current is established in the circuit. Even when a DC voltage is applied to a capacitor which is not charged a current will flow till the capacitor is fully charged as in the process of charging there exist dq/dt. once it is fully charged no additional charge is pumped in or out of the capacitor and it blocks current. In case of AC because of change in polarity of the applied voltage there will be continuous polarisation and depolarisation in each cycle causing rate of change of charge stored in the capacitor and hence flow of current with a 90 degree phase shift with the applied voltage. Please feel free if you need further clarifications. Thank you
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A capacitor has two plates of conducting material kept at a distance from each other.  They are not in contact. A resister or an inductor is a continuous medium of a conductor.

   So a capacitor does not pass current in the normal sense.  Due to proximity of the two parallel plates, electrostatic induction takes place.  Suppose a battery is connected to a capacitor.  The potentials on the plates are initially 0.  As the potentials at the positive and negative terminals of the battery are different, current flows and then the charges accumulate on the two plates.  Due to electrostatic attraction between positive and negative charges on the plates, they remain there.

   The charging process continues until a limit on the amount of charge is reached.  Then the potential across the capacitor becomes same as that of battery. Then there is no more flow of electrons. and so no more current.

   When an AC voltage source is connected to a capacitor, then there is a current continuously that varies according to the voltage.  The capacitor gets charged and discharged repeatedly.
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