Capacitor blocks DC but allows AC to flow

Why Capacitor blocks DC but allows AC to flow?

 We can explain this fact mainly in two ways:

• We know,

Capacitive reactance, X_c = 1/ 2ΠfC

For DC supply, frequency f is zero. So the reactance becomes infinity. As a result it blocks DC. But for AC supply the frequency f is not zero. So the reactance is not infinity and it allows some AC to flow.

• We know,

Current through capacitor, I = C dv/dt

As the voltage does not change during DC supply,

dv/dt = 0

So Current through capacitor, I = 0

It means for DC supply capacitor blocks it. But for AC supply dv/dt is not equal to zero. So surely some certain value of I passes through the capacitor.