This week was all about calculating electric fields for continuous charge distributions. This is usually students’ first exposure to what they think of as “calculus-based” physics because they are explicitly setting up and doing integrals. There’s lots going on behind the scenes though.
In calculus class, students are used to manipulating functions by taking their derivatives, indefinite integrals, and definite integrals. In physics, however, these ready made functions don’t exist. When we write dQ, there is no function Q() for which we calculate a differential. The symbol dQ represents a small quantity of charge, a “chunk” as I usually call it. That’s is. There’s nothing more. Similarly, dm represents a small “chunk” of mass rather than the differential of a function m(). The progress usually begins with uniform linear charge distributions and progresses to angular (i.e. linear charge distributions bent into arcs of varying extents), then area, then volume charge distributions (Are “area” and “volume” adjectives?). One cool thing is how each type of distribution can be constructed from a previous one. You can make a cylinder of charge out of lines of charge. You can make a loop of charge out of a line of charge. You can make a plane of charge out of lines of charge. You can make a sphere of charge out of loops of charge. Beautiful! Lots of ways to approach setting up the integral that sweeps through the charge distribution to get the net field.
It’s interesting to ponder the effect of changing the coordinate origin. Consider a charge rod. If rod’s left end is at the origin, the limits of integration are 0 and L (the rod’s length). If the rod’s center is at the origin, the limits of integration are -L/2 and +L/2. The integrand looks slightly different, but the resulting definite integral is the same in both cases! Trivial? No! It’s yet another indication that Nature doesn’t care about coordinate systems; they’re a human invention and subject to our desire for mathematical convenience. This is also a good time to recall even (f(-x) = f(x)) and odd (f(-x) = -f(x)) functions becuase then one can look at an integral and its limits and predict whether or not the integral must vanish and this connects with symmetry arguments from geometry. This, to me, is one of the very definitions of mathematical beauty. A given charge distribution’s electric field is independent of the coordinate system used to derive it. The forthcoming chapter on Gauss’s law and Ampère’s law relies on symmetries to predict electric and magnetic field structures for calculating flux and circulation and that’s foreshadowed in this chapter.
This is a lot to convey to students and from their point of view it’s a lot to understand. I hope I can do better at getting it all across to them than was done for me.
Feedback welcome as always.