I’m combining two weeks in this post.
The first week, we dealt with magnetic forces. One thing that I have never thought much about is the fact that the quantity is effectively an electric field, but one that depends on velocity. When velocity is involved, reference frames are involved, and that of course means Einstein is talking to us again. M&I addresses the fact that what we detect as an electric field and/or a magnetic field depends on our reference frame. This is fundamental material that I feel should be included in every introductory electromagnetic theory course. There’s really no good reason to omit it given that special relativity is a foundation of all contemporary physics. It’s sad to think that beginning next fall, our students won’t be exposed to this material any more.
The second week gets us into chapter 21, which presents Gauss’s law and Ampére’s law. There are many fine points and details to present here. I’ll try to list as many as I can think of.
- I use the words pierciness, flowiness, spreadingoutness, and swirliness to introduce the concepts of flux, circulation, divergence, and curl respectively.
- We have the term flux for the quantity given by surface integrals, but we rarely if ever see the term circulation for line integrals. I recommend introducing the term, primarly because it forms the basis for the definition of curl.
- The distinction between an open surface and a closed surface is very important.
- I, like M&I, prefer to write vector area as rather than because it allows for introducing a “sneaky one” into the calculation of flux that lets a dot product become a product of scalars when the field is parallel to the surface’s unit normal:
- Similarly, I like an element of vector length, at least for electromagnetic theory, as rather than (the is supposed to be bold but it doesn’t look bold to me). I don’t think I have ever seen this notation in an introductory course before, but I like it because students have seen unit tangents in calculus and this notation closely parallels that for vector area as described above. Plus, it also allows for a “sneaky one” into the calculation of circulation when the field is parallel to the path’s unit tangent::
- After this chapter, we can finally write Maxwell’s equations for the first time. I show them as both integral equations and as differential equations. One of my usual final exam questions is to write each of the four equations as both an integral equation and a differential equation and to provide a one sentence interpretation of each form of each equation.
That’s about it for these two chapters. I thought there was something else I wanted to talk about, but it seems to have escaped me and I’ll update this post if and when I remember it.
Feedback welcome as always.
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.
This week was entirely flipped in that class time was devoted to letting students do whatever they needed to do to practice with the material in chapters 13 and 14.
Until now, no one has touched the WebAssign problem sets or much programming. In an administrative environment where “teaching” is defined as lecturing from a textbook and “learning” is defined as “taking notes and working reams of repetitive problems” I feel tremendous guilt at letting student try to learn at their own pace and for using precious class time for such individual efforts. I feel so guilty about it, in fact, that I sometimes consider it borderline incompetence on my part.
However, there is an upside. When students begin to ask, out loud, questions like
- “So what is vector r?”
- “How do I calculate a dipole’s field?”
- “How do we get r cubed in the demoninator?”
- “How do I estimate the amount of charge on a piece of tape?”
- “What does epsilon_zero stand for?”
- “How would I code this in VPython?”
then I know that the time is well spent because most of the students are actually engaging for the very first time, which is what learning is all about. They’re finally asking the questions that should arise while they read the corresponding textbook chapters. This helps me pinpoint where they have not yet gained understanding, where the need to go from here, and most importantly…where they are right now. That last thing is a perpetual problem in my environment becuase with very few exceptions, students lack motivation and expect ME to do most of the motivating and drag them along as we progress. I am trying very hard…sometimes I think not hard enough and sometimes I think too hard…to put that onus on them. These are questions that should have been raised and addressed over the past two or three weeks.
So despite my guilt, I think there is really good justification for using class time like this. The downside is that it’s probably frowned upon in the stuffy, old fashioned, traditional, ineffective models that still exist at many institutions and, probably, in too many community colleges across the country. I hope I’m wrong.
(a) In first semester physics, you learned that mass is necessary to calculate the gravitational force shared by two interacting entities. What are the physical implications of mass always being positive?
(b) In second semester physics, you just learned that charge is necessary to calculate the electric force shared by two interacting entities. What are the physical implications of charge bring both positive and negative?
This week was yet another partial week. Between weather and holidays, we’ve not yet had a full week of classes. Such is life I guess.
This week, we looked at the electric field of a static particle and the electric field of a fixed dipole on the dipole’s axis and on the perpendicular bisector of the axis. I really with introductory textbooks would introduce the full expression, in coordinate-free form of course, for a dipole field. I think it would go along way toward reinforcing introductory understanding of vectors. We already present a particle’s field in coordinate free form, but why not a dipole’s field? No one that I know of has taken the plunge. That includes me unfortunately. Maybe someday.
We spent all of Thursday (the course meets M-Th 10:00 a.m. -11:20 a.m.) working with GlowScript, our main programming environment this semester. I demonstrated how to define a new function, sgn() in this case. I’m rather surprised that it’s not internally defined by default, but it’s trivial to add to one’s program.
There’s not much else to say about this week. It’s all about laying a good foundation for the coming chapters. That’s important, but alas not always exciting.
This week was supposed to begin on Monday, but we lost both Monday and Tuesday to snow and icy roads so this week was effectively just a two day week.
On Wednesday, I demonstrated Jupyter notebooks and informed the class that effective this semester, we’re moving away from Classic VPython. From this point on, we will only use GlowScript and Jupyter VPython. Using the latter is very important because it allows for file I/O whereas there’s no easy way to do that (that I’m aware of) with GlowScript. We will also continue using LaTeX (via Overleaf) for writing solutions.
On Thursday, I gave an overview of chapter 13 on electric force and the electric field of a particle. It’s interesting to note that the denominators of both expressions contains an area, specifically the area of a sphere. What might that be related to? I teased the class with this question in anticipation of the chapter on Gauss’s law.
Note the presence of absolute value bars and the sgn() function in each expression. Charge, unlike mass, can be positive or negative. A vector’s magnitude, however, must always be positive without exception, at least if we are going to stick with the fundamental definitions from first semester physics. That means that we must use the absolute value of charge to calculate the magnitude of an electric force or electric field. We could always sidestep this issue by instead defining the signed magnitude to be the scalar part of the vector, but this isn’t consistent with a vector being the product of a magnitude and a direction. In the expression for electric force, note that we could also take the absolute value of the product of the two charges, which might be a better way to write it. I’ll have to think about that.
Anyway, the sgn() function is necessary computationally. A person can work out the correct directions for force and field by physical and geometric reasoning, but a computer must be told explicitly how to do it, and that’s the purpose of the sgn() function here. It assures the correct geometry based on the signs of the charges. I’ve never seen this use in any textbook, but it seems quite necessary to me in order to maintain the fundamental definition of a vector’s magnitude. Thus, I include it.
Also note that we use double bars for vector magnitudes and single bars for absolute values. These are two conceptually different things and thus I feel they warrant different symbols. It is also consistent with what my students see in their calculus textbook and I try to maintain some sense of consistency between their math and physics texts.
UPDATE: Oh, one more thing. Every textbook I know of freely switches between Q and q for chcarge, even for the same expression and sometimes even for the same expression in the same chapter. This is confusing. To eliminate this confusion, I consistently use Q for a source charge (a charge associated with the creation (I don’t like that word) of an electric field) and q for an experiential charge (a charge that experiences an electric field created by another charge).
UPDATE: In the fourth edition of Matter & Interactions, Chabay and Sherwood deal with the sign issue by treating everything to the left of the unit vector in the above expressions as a signed scalar quantity and mention on page 520 that one should take the absolute value of this quantity to get the magnitude of the associated vector. Computationally, they calculate a particle’s electric field in one expression, without separately calculating the magnitude and direction, and this is fine. I think students should be aware of different sign conventions and their implications, but I also think foundational definition should be sacrosanct. If the foundation is variable, it isn’t a foundation after all.
UPDATE: After much thought, I have decided that I am okay with defining the magnitude of a particle’s electric field to be the absolute value of the quantity preceding the direction and excluding the sgn() function. The resulting caveat is that without taking the absolute value, we must not call this quantity a magnitude; it is a signed scalar.
I ended Thursday’s class with a question:
WHY must the electric force shared by two charged particles lie along the line connecting them?
This question can be answered with no numerical calculation or computation at all, but with physical reasoning using symmetry, specifically the fact that space is isotropic. The logic goes something like this:
- Define a system to consist of two charged particles with charges Q and q, isolated from all other influences.
- Assume that the force on q due to Q has a component that is NOT along the line connecting them, and draw an arrow representing this force with its tail on q.
- Rotate the system around an axis coinciding with the line connecting q and Q by 180 degrees, and draw the new system.
- Note that the rotated system is indistinguishable from the original system. This is important, because if nothing about the system changed, then we should expect there to be no change in the force on q due to Q.
- However, since we assumed that the force on q has a component perpendicular to the line connecting q and Q, the force “looks different” for the rotated system compared to the original system. A uniqueness theorem guarantees that for every charge distribution, there is one and only one net force on each particle. Thus, there cannot be more than one “correct” net force on q due to Q.
- If space is indeed isotropic, then if a change to the system causes the system to “look the same” then it cannot be the case that the force on q due to Q can have a component perpendicular to the line connecting q and Q.
- Therefore, the force on q must be such that is has no component perpendicular to the line connecting q and Q, and thus it must lie along that line, and we have used a simple proof by contradiction.
This type of powerful reasoning, appealing to symmetry, has many uses in electromagnetic theory, specifically in the introductory course where students need to ascertain the directions of electric fields due to certain charge distributions. Symmetry plays a role in setting up the integrals necessary for such calculuations. I think it is important to introduce reasoning by symmetry as early as possible. Note that this reasoning can also be applied to the geometry of the gravitational force from introductory mechanics.
Feedback is always welcome!
This question was inspired by chapters 13, 14, and 15 of Matter & Interactions and would, I think, make a good final exam question even in courses where M&I isn’t used. The story line in those chapters makes a wonderful progression through different charge distributions and their fields and interactions with other similar charge distributions. The rather obvious patterns in this progression are worth emphasizing. They seem to be a consequence of superposition, which is one of most conceptually astounding ideas in physics.
Make a table giving at least one charge distribution, or combination of distributions, that gives rise to an electric field or electric interaction (force) that varies as 1/(r^n) where n = 0, 1, 2, …, 9. It may be the case that not all values of n are represented in the table.
I can think of at least one example where a double digit value of n is needed, but most courses don’t deal with that situation.