A parallel-plate capacitor has charge q and plate area A ....

A parallel-plate capacitor has charge $q$ and plate area $A .$ (a) By finding the work needed to increase the plate separation from $x$ to $x+d x,$ determine the force between the plates. (Hint: See Eq. $8-22 .$ ) (b) Then show that the force per unit area (the electrostatic stress) acting on either plate is equal to the energy density $\varepsilon_{0} E^{2} / 2$ between the plates.

A parallel-plate capacitor has charge $q$ and plate area $A .$
(a) By finding the work needed to increase the plate separation
from $x$ to $x+d x,$ determine the force between the plates. (Hint: See Eq. $8-22 .$ ) (b) Then show that the force per unit area (the electrostatic stress) acting on either plate is equal to the energy density $\varepsilon_{0} E^{2} / 2$ between the plates.

Key Concepts

Parallel?Plate Capacitor

This concept refers to a simple electrical component consisting of two conductive plates separated by a small distance, which stores electrical energy in the electric field established between the plates. It is a fundamental system in electromagnetism, used to illustrate principles of charge, electric field, and energy storage.

Work?Energy Principle in Electrostatics

This principle involves calculating the work done on a system when a physical parameter (such as the distance between capacitor plates) is changed, and relates that work to the change in the stored energy. By determining the infinitesimal work required to alter the configuration, one can derive forces acting within the system.

Energy Density in an Electric Field

Energy density describes how much energy is stored per unit volume in an electric field. It is quantified by the expression (1/2)??E² in a vacuum, linking the intensity of the electric field to the energy stored in the field, which in turn can be related to mechanical forces acting on charged conductors.

Electrostatic Stress (Pressure)

Electrostatic stress refers to the force per unit area (pressure) exerted on the surfaces within an electric field. This concept shows that the pressure on the plates of a capacitor is equivalent to the energy density of the electric field between them, illustrating the interplay between electromagnetic energy and mechanical forces.

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