This question came to me while I was planning for this semester’s introductory calculus-based e&m course (using Matter & Interactions of course). My overall desire and plan is to move away from the traditional number crunching type of problems, where all students really do is manipulate coordinate components of vectors or perhaps vector magnitudes, all without any genuine concern for the underlying geometrical implications. I completely understand the importance of this skill, but with computing having become rather ubiquitous I think such number crunching can be relegated to computational activities and labs. To change the status quo, I want to build a library of conceptual questions and problems that go as far beyond number crunching as I can get. I want students to think about the assumptions we make in physics and about how those assumptions are formulated. I want students to be able to, as Cliff Swartz once said, know the answer to a problem before calculating it. (I’ll link to the reference for that paraphrased quote once I dig it up.)
This particular question addresses two things, one that I never questioned as a student and one I only recently thought about as a teacher. It also addresses my continual search for ways to introduce symmetry arguments into introductory physics as early as possible. See what you think. You may find this question intimately related to this post.
(a) Formulate an explanation for why the electric field of a particle, or any other finite charge distribution, must decrease, as opposed to increase or remain constant, as distance from the charge distribution increases.
(b) Formulate an explanation for why the electric field of an infinite (keeping in mind that true infinite charge distributions don’t exist) charge distribution must remain constant, as opposed to increase or decrease, as distance from the charge distribution increases. (It may help to consider an Aronsonian operational definition of “infinite charge distribution.” In other words, if a charge distribution can’t be truly infinite then what precisely do we really mean by “infinite charge distribution” in the first place?)
By the way, as always these questions are framed within the context of introductory calculus-based physics. I hope I have made correct assumptions about the physics of the situations. If not, please feel free to let me know. Oh, and yes, you could probably use gravitational or magnetic fields instead of electric fields in this question.