A parallel-plate capacitor is charged and then disconnected from a battery. By what fraction doe

A parallel-plate capacitor is charged and then disconnected...

Key Concepts

Capacitance and Plate Separation

In a parallel-plate capacitor, the capacitance is determined by the geometry of the plates and the distance between them. The relationship is given by C = ??A/d, meaning that capacitance is inversely proportional to the plate separation. Doubling the distance halves the capacitance, which directly affects the stored energy when other factors remain constant.

Stored Energy and Its Dependence on Capacitance

The energy stored in a capacitor when charged is given by U = Q²/(2C) for a capacitor isolated from any voltage source. This equation shows that the stored energy is inversely proportional to the capacitance. Therefore, if the capacitance decreases due to an increase in plate separation while the charge remains fixed, the stored energy increases proportionally.

Charge Conservation in an Isolated Capacitor

When a capacitor is disconnected from a battery after being charged, the charge on the capacitor remains constant. This constraint means that any change in the capacitor's physical parameters, such as an increase in plate separation, will adjust the energy stored because the same amount of charge is now distributed over a different capacitance.

Fractional Change in Stored Energy

Given that the energy stored in an isolated capacitor is U = Q²/(2C), if the capacitance is halved (as happens when the plate separation is doubled), the new energy becomes U' = Q²/(2(C/2)) = 2U. This corresponds to a fractional change (increase) of 100%, showing the direct effect of changing the capacitor's geometry on its stored energy.

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