When the battery is connected, the voltage is fixed, so inserting the dielectric increases the capacitance and thus the charge stored. When disconnected, the charge is fixed, so the capacitance increase causes the voltage to decrease. This distinction is crucial for solving JEE problems on energy change. My old letterpress machine is like that disconnected capacitor—once I've inked the type, the impression is fixed, and adding more pressure (like the dielectric) just changes how deep it goes, not the design itself.
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