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Does current flow through a capacitor ?
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Imagine a pipe with a flexible diaphram in the middle separating two haves of the pipe.

Now the pipe is filled with air.

If you apply a pressure to one end the diaphram will deform in the direction of the non pressurised end. And will transfer that pressure to the other end, even though theres no connection of the air from each end.

Now apply a steadily reversing pressure / suction. Pressure / suction, to one end. (Like AC )

What happens on the other end ? A pressure / suction, a reproduction of the alternating input pressure. But theres no connection !

A capacitor is just like that. It has no electrical connection through the capacitor.. But it will pass the effects of the input signal. So you apply an ac signal to a capacitor, the effects on electrons will be recreated across the boundary. But theres no electrical connection...
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