A parallel-plate vacuum capacitor with plate area A and...

A parallel-plate vacuum capacitor with plate area $A$ and separation $x$ has charges $+Q$ and $-Q$ on its plates. The capacitor is disconnected from the source of charge, so the charge on each plate remains fixed. (a) What is the total energy stored in the capacitor? (b) The plates are pulled apart an additional distance $d x$ . What is the change in the stored energy? (c) If $F$ is the force with which the plates attract each other, then the change in the stored energy must equal the work $d W=F d x$ done in pulling the plates apart. Find an expression for $F .(\mathrm{d})$ Explain why $F$ is not equal to $Q E,$ where $E$ is the electric field between the plates.

A parallel-plate vacuum capacitor with plate area $A$ and separation $x$ has charges $+Q$ and $-Q$ on its plates. The capacitor is disconnected from the source of charge, so the charge on each plate remains fixed. (a) What is the total energy stored in the capacitor? (b) The plates are pulled apart an additional distance $d x$ . What is the change in the stored energy? (c) If $F$ is the force with which the plates attract each other, then the change in the stored energy must equal the work $d W=F d x$ done in pulling the plates apart. Find an expression for $F .(\mathrm{d})$ Explain why $F$ is not equal to
$Q E,$ where $E$ is the electric field between the plates.

Key Concepts

Capacitance of a Parallel?Plate Capacitor

This concept refers to the ability of a parallel?plate capacitor to store charge, which is determined by the geometry of the capacitor—in particular the area of the plates and their separation. The capacitance is defined as C = ??A/x for a vacuum capacitor, where ?? is the permittivity of free space, and it forms the basis for how much energy the capacitor can store when charged.

Energy Stored in a Capacitor

The energy stored in a capacitor is a measure of the work required to charge it and is given by the formula U = Q²/(2C) when the charge is constant. This relation ties together the stored energy with the charge and the capacitance, providing insight into how changes in capacitance (due to changes in plate separation, for instance) affect the energy storage.

Mechanical Work and Energy Change in Variable Capacitors

When the physical parameters of a capacitor (like the plate separation) are changed, the capacitance changes, which in turn changes the energy stored in the capacitor. The change in stored energy corresponds to the mechanical work done on or by the system. This concept is key to understanding problems where mechanical motions, such as pulling the capacitor plates apart, alter the electrostatic energy.

Electrostatic Force Derived from Energy Considerations

The force between the plates of a capacitor can be determined by considering the derivative of the energy with respect to the displacement. By relating the change in stored energy to the work done (F dx), one can derive an expression for the force that reflects the system's tendency to minimize its energy. This approach emphasizes that the force is an intrinsic property derived from the energy landscape of the capacitor.

Electric Field Contribution to Force and the Q·E Fallacy

While the electric field between the plates is crucial in generating the force, the total force on a plate is not correctly given by the naive product Q·E because the electric field acting on a plate must be that due to the opposite plate alone, not the net field between the plates. This distinction arises because each plate contributes to the overall field, but when calculating the force, you must consider only the field from the other plate.

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