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When you apply DC directly to a capacitor, it charges to the value of the DC potential, and then there is (nearly1) zero current flow through the capacitor. If the capacitance is large enough, though, and the DC source has a low enough impedance, the current flow can be quite substantial, damaging things.

The reason the equilibrium current is zero is that a capacitor resists a change in voltage, proportional to current and inversely proportional to capacitance...

dv/dt = i/c

... which makes the capacitor essentially a high pass filter, and a DC blocker.

1 The equilibrium current is "nearly" zero because, in our non-ideal world, every capacitor has some leakage current. Practically, the current is zero - from a purist perspective, it is not.

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