24.1.2 Three Fundamental Symmetries

Three fundamental symmetries appear frequently in electrostatics. The first row of FIGURE 24.6 shows the simplest form of each symmetry. The second row shows a more complex, but more realistic, situation with the same symmetry.

FIGURE 24.6 Three fundamental symmetries.

The figure illustrates the Planar symmetry of a positively charged horizontal infinite plane.
The figure illustrates the Cylindrical symmetry of a positively charged infinite cylinder.
The figure illustrates the Spherical symmetry positively charged sphere.
The figure illustrates two oppositely charged infinite parallel-plates of a capacitor.
The figure illustrates coaxial cylinders. A thin solid cylinder is shown at the center of a bigger hollow cylinder.
The figure illustrates concentric spheres. A solid sphere is shown at the center of a bigger hollow sphere.

NOTE   Figures must be finite in extent, but the planes and cylinders in Figure 24.6 are assumed to be infinite.

Objects do exist that are extremely close to being perfect spheres, but no real cylinder or plane can be infinite in extent. Even so, the fields of infinite planes and cylinders are good models for the fields of finite planes and cylinders at points not too close to an edge or an end. The fields that we’ll study in this chapter, even if idealized, have many important applications.