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Current through a capacitor is proportional to the rate of change of voltage across it. But in the case of dc supply, the rate of change of voltage is zero. Hence the capacitor current is also zero.

Capacitive reactance -- the "resistance" of a capacitor to current flow -- is found by 1/(2*pi*f*C). For high frequencies, this "resistance" is low, allowing current flow. The lower the frequency, the higher this "resistance" is. DC or direct current is consider "zero" Hertz frequency and the capacitive reactance ("resistance") at this point is infinite, allowing no current flow at all. So, the higher the frequency, the easier it is for the signal to pass through the capacitor; the lower the frequency, the more difficult it is for current to pass and impossible for DC.

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