Symmetry is part of the foundation of contemporary physics, but it is seldom emphasized in introductory physics in proportion to its significance. There may be some value in discussing how symmetry applies to otherwise traditional introductory problems rather than just replicating numerical examples from a textbook (even a good textbook). These questions illustrate symmetry in electromagnetic theory, but could trivially be adapted to gravitational interactions in mechanics.
Assume space is isotropic. Using only symmetry, argue (no numbers, no equations, only words) that
(a) the electric force between two charged particles must lie along the line connecting the two particles.
(b) the electric field of a very large (so large that its size need not matter) uniformly charged disk must be perpendicular to the disk and must not vary in magnitude with respect to distance from the disk.
(c) the electric field of a particle must be radially toward (or away from) the particle and if it varies in magnitude, must only do so with respect to distance from the particle.
(d) the electric field of a very long (so long that its length need not matter) uniformly charged rod must be perpendicular to the rod and if it varies in magnitude, must only do so with respect to perpendicular distance from the rod.
Here’s a hint. A symmetry implies some transformation that leaves some property unchanged. In each case, think of a change (perhaps a rotation and/or translation) that leaves the system (in this case, a charge distribution) unchanged and then look at any consequences that follow.
As usual, let me know if you present these questions to your students. I’m always interested in the the results.
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