Given how little we assumed about Figure 24.1—that the charge distribution is cylindrically symmetric and that electric fields point away from positive charges—we’ve been able to deduce a great deal about the electric field. In particular, we’ve deduced the shape of the electric field.
Now, shape is not everything. We’ve learned nothing about the strength of the field or how strength changes with distance. Is E constant? Does it decrease like or We don’t yet have a complete description of the field, but knowing what shape the field has to have will make finding the field strength a much easier task.
That’s the good of symmetry. Symmetry arguments allow us to rule out many conceivable field shapes as simply being incompatible with the symmetry of the charge distribution. Knowing what doesn’t happen, or can’t happen, is often as useful as knowing what does happen. By the process of elimination, we’re led to the one and only shape the field can possibly have. Reasoning on the basis of symmetry is a sometimes subtle but always powerful means of reasoning.