This was the first full week of classes and I feel as though we’re dreadfully behind where we need to be.
This week was about chapter 14, electric fields and matter. I love this chapter becuase it’s mostly experimental. The experiments use plastic tape (aka sticky tape or Scotch tape). I try to let students devote as much class time as possible to these experiments becuase they can be quite engaging and lead to some excellent physics understanding.
This chapter also addresses polarization in conductors and insulators and I think this is some of the most interesting elementary physics in the course. In conductors polarization is all about mobile electrons, while in insulators it’s all about molecules. Charge distribution matters and representing these distributions with diagrams seems rather difficult for most students and I’m not sure why.
My favorite part of the chapter is the derivation of an expression for the interaction (force) experienced by a neutral atom and a charged particle. This interaction is inversely proportional to separation to the fifth power, something no one could expect (at least I couldn’t have expected it). It’s interesting to start making a catalog of separation dependencies for various electric interactions. A particle-particle (both charged) interaction depends on separation squared. A (fixed) dipole-particle (charged) interaction depends on separation cubed. Now we see a (neutral) atom-particle (charged) interaction varies inversely as separation to the fifth power. Something’s missing! Where’s the dependence upon fourth power? Is there even such an interaction? Indeed there is, and it’s quietly tucked away in problem P63 at the end of chapter 13! The interaction experienced by two fixed dipoles varies inversely with separation to the fourth power. This is a nice little progression of dependencies to be aware of and to remember. It’s a good foundation for things like the Lennard-Jones interaction, where dependencies are to the seventh and thirteenth power (that’s for the force, not the potential). In the next chapter, students will see some charge distributions that don’t vary with separation at all! How can that be? Well, one way to frame it is with symmetry. I’ll say more about that in a future post.
Oh, there’s one more interesting connection students can make in this chapter, and that in their chemistry courses they learned about various subatomic forces. Well, these forces can be traced to interactions involving dipoles and have different names depending upon the discoverer or other associated person. I think it’s important to point these connections out.
I still struggle with (lack of) student motivation, and I still feel rather stymied by it. I wish there were a way to physicall force students to engage outside of the classroom but no such thing exists. Ah well…
Feedback is welcomed as always.
(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!