Symmetry

In theory, we now know how to calculate the electric field of any number of source charges. However, most materials are made up of many, many charges, 1023 or more of them, and adding together that many vectors can be difficult. In the most general cases, we can make use of integration to calculate the electric field, but in certain nice cases we can learn much about the electric field of a charge distribution by determining its symmetries.

Definition

You presumably have some familiarity with the notion of symmetry; you may have discussed bilateral symmetry in biology class, for instance. For the purposes of this chapter I will define symmetry as follows:

An object is symmetric under an action if the object's position, orientation, and other properties are not changed by performing that action on it.

This sphere is rotating around an axis which is not through its center. The sphere does not have rotational symmetry around this axis because you can tell it is moving.

This sphere could be rotating right now and you could never tell, because the sphere has rotational symmetry.

For example, if I rotate a sphere around some axis through its center, the sphere will look exactly the same afterwards: no one would be able to tell whether or not I rotated the sphere unless they actually watched me do it. The sphere is symmetric under rotation around an axis through its center, or we might say it has rotational symmetry around that axis. Note that symmetry requires that the object maintain the same position and orientation, not just that it looks the same. So for instance, the sphere does not have rotational symmetry around the axis in the animation shown here, even though the sphere itself looks identical.

Symmetry can be defined for any action. (For example, your understanding of the Peloponnesian War is probably symmetric under the action of reading this textbook.) However, there are three types of symmetry which will be of particular interest to us.

Rotational symmetry

As suggested above, rotational symmetry is always defined in terms of an axis which the object rotates around. It can be broken down into two categories:

Note that every object has 360° discrete rotational symmetry around any axis; this trivial symmetry is usually ignored.

One useful thing to remember: a point lying on an axis always has rotational symmetry around that axis.

Reflection symmetry
Interactive 5.3.1

Reflection symmetry refers to the action of reflecting all the points of an object through a mirror—which, in this case, is a perfectly thin, flat plane which is reflective on both sides. To reflect a point, move it along a line perpendicular to the mirror until it is just as far behind the mirror as it was in front of the mirror; points lying on the mirror do not move at all, and automatically have reflection symmetry. The figure to the right will probably explain better than words can. Reflection symmetry is sometimes referred to as bilateral symmetry, especially in biology: many organisms have approximate reflection symmetry across the mirror which runs straight down their body.

Reflection symmetry is always defined in terms of a plane which represents the mirror. For example, would you believe that the words on this page have reflection symmetry? It's true! If you consider your monitor screen to be the mirror, then every word written here lies directly on top of the mirror, and so is its own reflection.

Translation symmetry

In this context, the word translate means to move in space from one place to another without rotation, and so an object has translation symmetry if you can move it from one spot to another, and no one would be able to tell that you moved it. Sounds impossible, and it is for any real object: only infinite objects can have translation symmetry. One example is an infinite cylinder, which is like a fireman's pole which never ends: this has translation symmetry along its axis. An infinite sheet (a flat surface which continues forever) has translation symmetry in two directions.

If you're a practical sort, you may wonder why we would even bother talking about "infinite" objects, since they clearly do not exist. Infinite objects are mathematical abstractions, but they can provide useful approximations for very large objects: a table looks pretty much like an infinite plane to a bug. Calculations are often much simpler when we can take the limit as the size of a particular object goes to infinity, and while the result may not be exactly correct when applied to a finite object, the error of the approximation gets smaller as the object becomes larger, to the point where the error becomes negligible compared to other approximations in your analysis.

Symmetry and Electric Fields

Symmetry plays an important role in calculating the electric field of a collection of charges for the following reason:

An electric field has the same symmetries as the source charges that created it.

We can see this in some of the fields we've already investigated:
Interactive 5.3.2

In certain circumstances, we can use this principle to predict what the electric field will be without doing any calculations. For example, if we have six identical charges spread evenly around a circle, their net electric field at the center of the circle must be zero, because an electric field pointing in any direction at the center would not have the 60° rotational symmetry around the center that the charges have.


The points on this piece of paper don't look very symmetric, but they do have reflection symmetry, if we consider the piece of paper to be a mirror.
Another example: a bunch of charges which lie on the XY-plane (like this screen) have reflection symmetry across the XY-plane. Thus the electric field at any point on the plane cannot have a Z-component: it must point along the plane. If the electric field did point out of the screen, then we could reflect everything across the plane: now the charges are the same, but the field at that point now points into the screen, and we'd have a contradiction.