# Force between the plates of a plane capacitor

I found this problem while self-studying Electricity and Magnetism, and I want to know if my solution is rigorous.

## Homework Statement

Show that the plates of a plane capacitor attract each other mutually with a force equal to
$$F=\frac{q^2}{2\epsilon_0 A}$$
Obtain this result by calculating the work required to increase the separation between the plates from x to x + dx.

## Homework Equations

Energy stored in a capacitor with electric field E, plate area A and distance x between the plates:
$$U=\frac{1}{2}\epsilon_0 E^2 Ax$$

## The Attempt at a Solution

The energy U stored in a capacitor can be viewed as the work that needs to be done by an external agent to separate its plates by a distance x:
$$U=\frac{1}{2}\epsilon_0 E^2 Ax$$
The electric field E can be rewritten as $q/(\epsilon_0 A)$, where q is the magnitude of the charge on each plate. Thus, the work can be rewritten as:
$$U=\frac{1}{2}\frac{q^2 x}{\epsilon_0 A}$$
The work dU done by the external agent to separate the plates from a separation x to a separation x + dx is:
$$dU=\vec{F}\cdot d\vec{x}$$
where $\vec{F}$ is the force applied by the external agent.
Because in this situation force and displacement are in the same direction, it may be rewritten as:
$$dU=Fdx$$
Isolating the force:
$$F=\frac{dU}{dx}$$
Thus:
$$F=\frac{d}{dx}\frac{1}{2}\frac{q^2 x}{\epsilon_0 A}$$
$$F=\frac{1}{2}\frac{q^2}{\epsilon_0 A}$$

Last edited:
• gracy