DC current through capacitor
Solution 1
DC current does pass through a capacitor. If you connect an ideal capacitor to an ideal current source, the current will flow through the capacitor forever (click for simulation):
But note that the voltage across this ideal capacitor is continually increasing. Obviously this is not possible in the real world, as something will break down and/or arc when the voltage gets too high.
The reason is that current can pass through the capacitor, but charges cannot jump from one plate to the other. Electric charge is still moving into one side of the capacitor, and moving out of the other side (a current is flowing), but no particles are actually crossing the gap; they are building up on one plate and depleting off of the other plate, causing the voltage to rise.
In an AC circuit, the current is constantly switching directions, and so the voltage goes back and forth and never exceeds the breakdown voltage of the capacitor.
In typical DC circuits, you don't have a current source driving the capacitor, you have a voltage source through some resistance. This means the voltage across the capacitor can only rise as high as the source's voltage. When this happens, the current stops.
Read this: ELECTRICITY MISCONCEPTIONS: Capacitor
Solution 2
DC current does not pass through because there is no conducting path from one side of the capacitor to the other.
AC current "passes through" in a different sense than conduction through the device. As much charge enters one side as leaves the other. But this cannot be maintained forever; eventually one side will run out of electrons. Charge can alternately enter and leave each side, so AC "conduction" is possible.
Time rub
Updated on February 02, 2020Comments

Time rub almost 4 years
AC current does pass from a capacitor. But DC current does not pass through it why?

Yuriy S almost 6 yearsTo the OP, why did you completely change the content of the question instead of asking a new one?

garyp almost 6 years@YuriyS I see what you mean. When you directed me to check the history, I checked the history of the Wikipedia entry. Why someone would use edit to completely change a question is beyond me.

Time rub almost 6 yearsActually I was trying to ask a new question. But stack exchange says "we are not accepting more questions from this account" so I edited it completely

Kyle Kanos almost 6 years@AccidentalFourierTransform: you do realize that your rolling back to v2 now makes the answers (one of which being accepted) completely invalid, right? I'm rolling it back to the previous version (where the answers will actually make sense) & hope that you'll pay attention to answers when deciding whether you should roll back edits or not.


endolith almost 6 yearsDC current passes through the capacitor when it's charging and discharging

Yuriy S almost 6 years@endolith, it's not a DC current then, it varies in time

endolith almost 6 years@YuriyS An ideal capacitor can pass a constant DC current forever.

Yuriy S almost 6 years@endolith, it has infinite resistance at zero frequency.

Yuriy S almost 6 yearsThe explanation and the link are quite interesting, thank you. Though I would consider the first "DC" case completely unphysical, since this kind of "ideal current source" is as impossible as "ideal capacitor" and this is in fact just an infinite "charging" process, but with infinite energy source available. (I absolutely agree with the link you offer, of cource capacitors store voltage, or electric energy, not charge

garyp almost 6 years@YuriyS I think his ideal capacitor has an infinite supply of charge.

endolith almost 6 years@YuriyS but this is exactly what happens in real life with real current sources and real capacitors, up until the point at which the capacitor explodes. The important thing is to understand that it is happening. Capacitors do pass current, whether DC or AC.