24.1 Symmetry
To continue our exploration of electric fields, suppose we knew only two things:
The field points away from positive charges, toward negative charges, and
An electric field exerts a force on a charged particle.
From this information alone, what can we deduce about the electric field of the infinitely long charged cylinder shown in FIGURE 24.1?
FIGURE 24.1 A charge distribution with cylindrical symmetry.
We don’t know if the cylinder’s diameter is large or small. We don’t know if the charge density is the same at the outer edge as along the axis. All we know is that the charge is positive and the charge distribution has cylindrical symmetry. We say that a charge distribution is symmetric if there is a group of geometric transformations that don’t cause any physical change.
To make this idea concrete, suppose you close your eyes while a friend transforms a charge distribution in one of the following three ways. He or she can
Translate (that is, displace) the charge parallel to an axis,
Rotate the charge about an axis, or
Reflect the charge in a mirror.
When you open your eyes, will you be able to tell if the charge distribution has been changed? You might tell by observing a visual difference in the distribution. Or the results of an experiment with charged particles could reveal that the distribution has changed. If nothing you can see or do reveals any change, then we say that the charge distribution is symmetric under that particular transformation.
FIGURE 24.2 shows that the charge distribution of Figure 24.1 is symmetric with respect to
Translation parallel to the cylinder axis. Shifting an infinitely long cylinder by 1 mm or 1000 m makes no noticeable or measurable change.
Rotation by any angle about the cylinder axis. Turning a cylinder about its axis by or makes no detectable change.
Reflections in any plane containing or perpendicular to the cylinder axis. Exchanging top and bottom, front and back, or left and right makes no detectable change.
FIGURE 24.2 Transformations that don’t change an infinite cylinder of charge.
A charge distribution that is symmetric under these three groups of geometric transformations is said to be cylindrically symmetric. Other charge distributions have other types of symmetries. Some charge distributions have no symmetry at all.
Our interest in symmetry can be summed up in a single statement:
The symmetry of the electric field must match the symmetry of the charge distribution.
If this were not true, you could use the electric field to test whether the charge distribution had undergone a transformation.
Now we’re ready to see what we can learn about the electric field in Figure 24.1. Could the field look like FIGURE 24.3a? (Imagine this picture rotated about the axis.) That is, is this a possible field? This field looks the same if it’s translated parallel to the cylinder axis, if up and down are exchanged by reflecting the field in a plane coming out of the page, or if you rotate the cylinder about its axis.
FIGURE 24.3 Could the field of a cylindrical charge distribution look like this?
But the proposed field fails one test: reflection in a plane perpendicular to the axis, a reflection that exchanges left and right. This reflection, which would not make any change in the charge distribution itself, produces the field shown in FIGURE 24.3b. This change in the field is detectable because a positively charged particle would now have a component of motion to the left instead of to the right.
The field of Figure 24.3a, which makes a distinction between left and right, is not cylindrically symmetric and thus is not a possible field. In general, the electric field of a cylindrically symmetric charge distribution cannot have a component parallel to the cylinder axis.
Well then, what about the electric field shown in FIGURE 24.4a? Here we’re looking down the axis of the cylinder. The electric field vectors are restricted to planes perpendicular to the cylinder and thus do not have any component parallel to the cylinder axis. This field is symmetric for rotations about the axis, but it’s not symmetric for a reflection in a plane containing the axis.
FIGURE 24.4 Or might the field of a cylindrical charge distribution look like this?
The field of FIGURE 24.4b, after this reflection, is easily distinguishable from the field of Figure 24.4a. Thus the electric field of a cylindrically symmetric charge distribution cannot have a component tangent to the circular cross section.
FIGURE 24.5 shows the only remaining possible field shape. The electric field is radial, pointing straight out from the cylinder like the bristles on a bottle brush. This is the one electric field shape matching the symmetry of the charge distribution.
FIGURE 24.5 This is the only shape for the electric field that matches the symmetry of the charge distribution.
