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If I have a simple setup of a open circuit with a battery (constant battery) , resistor, and uncharged capacitor, and then I close the switch of circuit, and keep it closed until the capacitor is fully charged, what will happen with I open the circuit after a long time? Will charge be stored on the capacitor even after I open the switch on the circuit?

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Theoretically: The charge would stay on the capacitor forever and never drain off.

Reality: The capacitor's charge will eventually dissipate due to the internal parasitic resistance inside the capacitor. It may take days (heck, maybe even weeks), but the capacitor's charge would eventually drain off.

Hope that helps!

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Will charge be stored on the capacitor even after I open the switch on the circuit?

The capacitor will be charged when you open the switch and will ideally remain charged.

However, (electric) charge will not be stored on the capacitor before or after you open the switch. There will be charge $Q = CV_{BAT}$ on one plate and charge $-Q$ on the other plate and so no net charge is stored, only separated.

There is an electrostatic energy $W = \frac{1}{2}CV^2_{BAT}$ associated with this separated charge and so a charged capacitor stores energy and not charge.

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