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!