Matter & Interactions II, Week 7

This week, I was away at the winter AAPT meeting in Atlanta. Students began working on the experiments from chapter 17, which serve to introduce magnetic fields.

I want to emphasize some really cool things about the mathematical expression for a particle’s magnetic field:

\vec{B}_{\mbox{\tiny particle}}=\dfrac{\mu_o}{4\pi}\dfrac{Q\vec{v}\times\hat{r}}{\lVert\vec{r}\rVert^2}

This is really a single particle form of the Biot-Savart law. I’m going to morph it into something really interesting. I’m going to make use of the fact that c^2 = \frac{1}{\mu_o\epsilon_o}, which I assert to students will be derived in a later chapter.

\vec{B}_{\mbox{\tiny particle}}=\dfrac{\mu_o}{4\pi}\dfrac{Q\vec{v}\times\hat{r}}{\lVert\vec{r}\rVert^2}

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{4\pi\epsilon_oc^2}\dfrac{Q}{\lVert\vec{r}\rVert^2}\vec{v}\times\hat{r}

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{c^2}\vec{v}\times\left(\dfrac{1}{4\pi\epsilon_o}\dfrac{Q}{\lVert\vec{r}\rVert^2}\hat{r}\right)

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{c^2}\vec{v}\times\vec{E}_{\mbox{\tiny particle}}

THIS IS AMAZING! This demonstrates that this new thing called magnetic field is kind of like a velocity dependent electric field. That’s an oversimplification, but it hints that something deep is revealing itself here. Velocity connotes reference frame, and we see a big hint here that magnetic field depends on one’s reference frame. This is foreshadowing special relativity! We can show something else with one more slight rearrangement.

c\vec{B}_{\mbox{\tiny particle}}=\dfrac{\vec{v}}{c}\times\vec{E}_{\mbox{\tiny particle}}

This means that if we express velocity in fractions of c, then the quantity c\vec{B} has the same dimensions as \vec{E} and can thus be expressed in the same unit as electric field! This conceptualization allows for some beautiful symmetry to show itself later on when we get to the Maxwell equations. In some ways, electric fields and magnetic fields are interchangeable. Again, this is a hint of some underlying unification of the two, the electromagnetic field tensor, which I’m working hard to find a way to introduce into the introductory course. If students can understand simple Lorentz transformations, then they should be able to understand how the electromagnetic field tensor transforms from one frame to another within the framework of special relativity and we can show some beautiful physics. I realize I’m in the minority when it comes to something like this becuase we tend to think of our students as not being mathematically prepared. I’ve come to realize that perhaps…just perhaps…that is our perception only becuase we aren’t giving them the best mathematical foundations upon which to prepare for physics. Maybe it’s our fault. Maybe.

Anyway, these ruminations are things I want, and hope, students see on their own but all too often I find they, at least in my case, have difficulty even engaging at a minimal level. I struggle with this, and like to think and hope that maybe it’s because they don’t see the beauty. That’s why I nudge them in these new and different directions. Like I said above, it may very well be our fault.

Feedback is welcome as always.


Matter & Interactions II, Week 6

I’m writing this a whole week late due, in part, to having been away at an AAPT meeting and having to plan and execute a large regional meeting of amateur astronomers.

This week was all about the concept of electric potential and how it relates to electric field. I love telling students that this topic is “potentially confusing” becuase the word “potential” comes up in two different contexts. The first is in the context of potential energy. Potential energy, which I try very hard to call interaction energy, is a property of a system, not of an individual entity. There must be at least two interacting entities to correctly speak of interaction energy. Following Hecht [reference needed], I like to think of energy, and thus interaction energy, as a way of describing change in a system using scalars rather than vectors. Conservative forces, like gravitational and electric forces, can be described with scalar energies and fortunately, these forces play a central role in introductory physics. The second context is that of electric potential, a new quantity that is the quotient of a change in electric potential energy and the amount of charge that gets moved around as a result of an interaction. The distinction between the two contexts is subtle but very important.

Oh and speaking of potential or interacting energy, Matter & Interactions is the only textbook I know of that correctly shows the origins of The World’s Most Annoying Negative Sign (TWMANS) and how it relates to potential energy. When you write the total change in your system’s energy, you can attribute it to work done by internal forces and work done by external forces. When you rearrange this expression to put all the internal terms on the lefthand side and all the external terms on the righthand side, you pick a negative sign that goes on to become TWMANS. This term with the negative sign, which is nothing more than the oppositve of the work done by forces internal to the system, is DEFINED to be the change in potential energy for the system. It’s just that simple, but this little negative sign caused me so much grief in both undergrad and graduate courses. Some authors explicitly included it, other didn’t, and instead flipped the integration limits on integrals to account for it. Chabay and Sherwood include it explicitly and consistently and there should be no trouble in knowing when and where it’s needed.

There is also some interesting mathematics in this chapter. Line integrals and gradients are everywhere and we see they are intimately related. In fact, they are inverses of each other. I want to talk about one mathematical issue in particular, though, and that is within the context of the following problem statement:

Given a region of space where there is a uniform electric field \vec{E} and a potential difference \Delta V between two points separated by displacement \Delta \vec{r}, calculate the magnitude of the electric field \lVert \vec{E}\rVert.

This problem amounts to “unwrapping” a dot product (in this case \Delta V = -\vec{E}\bullet\Delta\vec{r} ), something the textbooks, to my knowledge, never demonstrate how to do. My experience is that student inevitably treat the dot product as scalar multiplication and attempt to divide by \Delta\vec{r} and of course dividing by a vector isn’t defined in Gibbsian vector analysis. I think the only permanent cure for this problem is to take a more formal approach to introducing vectors and dot products earlier in the course but I tend to think I’m in the minority on that, and I don’t really care. The problem needs to be addressed one way or the other. Solving either a dot product or a cross product for an unknown vector requires knowledge of two quantites (the unknown’s dot product with a known vector and the unknown’s cross product with a known vector OR the unknown’s divergence and curl) as constraints on the solution. Fortunately, at this point in the course we’re dealing with static electric fields, which have no curl (\nabla\times\vec{E}=0) or equivalently (I think) \vec{E} is collinear with \nabla V (differeing in signs because gradient points in the direction of increasing potential (I don’t like saying that for some reason…) and electric field points in the direction of decreasing potential) so we can find something about \vec{E} from just a dot product alone. So, students need to solve \Delta V = -\vec{E}\bullet\Delta\vec{r} for \lVert\vec{E}\rVert. Here’s the beginning of the solution. The first trick is to express the righthand side in terms of scalars.

\Delta V = -\vec{E}\bullet\Delta\vec{r}

\Delta V = -\lVert\vec{E}\rVert \lVert\Delta\vec{r}\rVert\cos\theta

\lVert\vec{E}\rVert = -\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}

We have a slight problem, and that is the lefthand side is a vector magnitude and thus is always positive. We must ensure that the righthand side is always positive. I see two ways to do this. If \vec{E} and \Delta\vec{r} are parallel (\theta=0) then \Delta V must represent a negative number and TWMANS will ensure that we get a positive value for the righthand side, and thus also for the lefthand side. If \vec{E} and \Delta\vec{r} are antiparallel (\theta=\pi) then \Delta V must represent a positive number and TWMANS, along with the trig function, will ensure that we get a positive value for the righthand side, and again also for the lefthand side. I want to install this kind of deep, geometric reasoning in my students but I’m finding that it’s rather difficult. Their approach is to simply take the absolute value of the righthand side.

\lVert\vec{E}\rVert = \left\lvert-\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}\right\rvert

It works numerically of course, but bypasses the physics in my opinion. There’s one more thing I want students to see here, and that is the connection to the concept of gradient. Somehow, they need to see

\lVert\vec{E}\rVert = -\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}

as

E_x = -\dfrac{\partial V}{\partial x}

and I think this can be done if we think about the role of the trig function here, which tells us how much of \Delta\vec{r} is parallel to \vec{E}, and remembering that the component label x is really just an arbitrary label for a particular direction. We could just as well use y, z, or any other label. We must be careful about signs here too, because the sign of E_x must be consistent with the geometry relative to the displacement.

As an aside, it kinda irks me that position vectors seem to be the only vectors for which we label the components with coordinates. I don’t know why that bothers me so much, but it does. Seems to me we should use r_x rather than just x and there’s probably a deep reason for this, but I’ve yet to stumble onto it. Perhaps it’s just as simple as noticing that a position’s components are coordinates. Is it that simple?

As always, feedback is welcome.


My Most Frequent LaTeX Problems and Fixes

I’ve spent a lot of time lately updating my mandi LaTeX package and I decided to eliminate every warning I saw when building the package. TeX and LaTeX errors and warnings are notorious for being vague so I had to do a lot of Googling to get solutions for them. Some errors and warnings have specific causes and need specific fixes but many do not. I will share the ones I encountered and for which I found fixes.

  • Any warnings about font sizes or font shapes “not availble” and some kind of substitution being made: Include \usepackage{anyfontsize} in your preamble and these warnings should go away.
  • Persistent underfull hbox warnings: Look for and remove any lines ending with \\ followed by a blank line. Remove the \\ (which tells LaTeX to immediately begin a new line) and find a more graceful way to handle the new line. Paragraphs should not end with \\.
  • Persistent overfull hbox warnings: Look for long words LaTeX doesn’t know how to hypenate. Tell LaTeX how to hypenate such words with something like al\-ter\-ca\-tion as you type the word.
  • Warnings related to labels when resetting the equation counter inside an environment like align or align*: There are two ways to fix this. One is to load the hyperref package with the hypertexnames option set to false as in \usepackage[hypertexnames=false]{hyperref}. The other is to load hyperref, then use \hypersetup{hypertexnames=false} to apply the new setting. (By the way, the hyperref package is a black box to me with seemingly infinitely any settings, none of which is adequately documented and most seem to be present only when specifically searched for. I hate it!) I found another fix for this, but it’s not intuitive and requires a different, and harder, way to reset the equation counter within the environment so I elected not to use it.
  • You have requested package ‘somepackage’ but the package provides ‘somepackage’: This one took some time to solve but I think I know what’s going on and how to fix it. When I build my package, I test it from the folder where it’s built. My workflow included a shell script that copies the necessary files to their appropriate places in my local texmf tree (~/Library/texmf/ and so forth on my system). This is where LaTeX looks for document classes and packages and other associated files. Files in my local texmf tree are found before files in the master locations used by the TeXLive distribution (/use/localtexlive/2016/texmf-dist/ and so forth on my system). Apparently when you say \usepackage{somepackage} LaTeX gets confused if the package exists simultaneously in the current folder and in another folder in LaTeX’s PATH and this causes this warning when compiling. The package in the current folder always takes precedent though and is the one that gets loaded as far as I can tell. Compiling the documentation from the dtx file necessarily requires the package to be loaded, and the copy in the current folder is the version that gets loaded of course, so there’s really no good way to avoid this warning when building the package and documentation from scratch. However, before I understood what was going on, I found a fix. In the dtx file, find the \ProvidesPackage command. located between the <package> guards. This command provides not only the package’s name, but also its version and build date, which are stored in \fileversion and \filedate respectively. The fix is to comment out the line with the \ProvidesPackage command and to explicitly define a new command specifying the version and build date. In my case this was easy because my package defines a \mandiversion command containing both the version and build date in one command. So I just used \mandiversion to print the version and build date on the title page rather than letting LaTeX define them from the \ProvidesPackage command. I have no idea why, but this supresses the warning. Now that I know about the other issue I will go back to using the \ProvidesPackage command by uncommenting that line. As long as the package file isn’t in the same folder as the document I’m compiling this warning should never appear again.
  • Any warning about \headheight being too small: Add \setlength{\headheight}{14pt}, with whatever font size is appropriate, to your preamble and the error should go away.

I will add to this list as I accumulate more of these fixes.

 


Matter & Interactions II, Week 5

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.


Matter & Interactions II, Week 4

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.

Feedback welcome.