Ampère’s law

Tomorrow I’m covering we’re working on Ampere’s law in my calc-based general physics course. In preparation I was looking around at different ways to present it, and I realized that I was getting crabby about most of the presentations I found in the various texts that I looked at. Authors were saying “trust me, this is true so let’s use it”, “See how it works for infinite lines? It always works” or “Here’s a whole different truth about magnetic fields so let’s use it!” Ok, I’m still crabby as you can tell.

Since the last time I taught this course I’ve taught our advanced electromagnetism course and I work hard to get the students to understand the connections between current and magnetic fields. In that class we do all the ugly multivariable calculus to show that Ampere’s law follows directly from the Biot-Savart law. In other words, it’s not any new physics, it’s just a restatement that helps out in very symmetric situations.

That last point about symmetry gives a great connection with electric fields. Students learn how to calculate any field using Coulomb’s law and then are later shown how the structure of electric fields is such that Gauss’s law can come in handy in symmetric situations. I want to do the same thing tomorrow, but there’s a problem with extending the analogy. In electrostatics, the notion that Gauss’s law is the same as Coulomb’s law is easy for a single charge, and then I work with the students to see how any charge distribution is really just a combination of little charges. Therefore Gauss’s law always works. But in magnetostatics (I hate that word), while it’s easy to show that Ampere’s law works for an infinite straight wire, I don’t see an easy way to convince students that any current setup can be thought of as a collection of infinite (parallel – since that’s all our book does) straight lines of current. I’m so nervous that a student will say “but what if the wires bend just above the surface we’re considering?”

So, what to do? I went over and talked with my colleague who teaches multivariable calculus this semester and asked what he thought about me doing a little of what I do in our advanced electromagnetism course. He said they haven’t quite gotten to curls and they’re a long way away from Stoke’s theorem, but he figured it’d be good if I gave a good hand-wavy try at it along with saying something like “and you’ll see how this works in math later this semester.” So I thought I’d get some thoughts down here about how I might try to do this tomorrow.

One thing I could do is show them a simulation of a couple of nearby infinite, parallel currents. I’ll make one of them have an adjustable current and I want to get the students to talk about what happens to the overall field as you change the current. I guess I’m going for comments about circularity, but I’m willing to see where that goes.

Another idea, playing off something we did yesterday, is to continue to look at the analogy between electrostatics and magnetostatics (I still hate that word). If Coulomb’s law leads to Gauss’s law, what could the Biot-Savart law lead to? Magnetic field lines never begin or end, so a straight Gauss’s law isn’t very useful (unless there were magnetic monopoles – oh well). Electric field lines poke out, so let’s see how many poke out (Gauss’s law). Maybe something like: Magnetic fields go around in circles, let’s see what happens if you follow them around (weak, I know).

Big idea: local (non-infinite) currents cause magnetic field circularity. So a measure of the circularity should tell you about the local currents. If we add up all the curl in an area, we’d have all the current poking through that area. But the fundamental theorem of calculus tells us that we ought to be able to get interior information from the border. Maybe just adding up the circulating field at the edge will tell us about the current poking through:

\oint \vec{B}\cdot \vec{dl}=\mu_0 I_\text{poke}

Another approach (less likely, though). What is the curl of the Biot-Savart law? This is tough because you need at least one vector identity and the use of at least one delta function. I guess I’m not sure what it means to handwave this.

And, of course, I want them to see that this’ll only be useful in symmetric situations, where the integral above is a near write-me-down.

Thoughts? Some starters for you

  • Why do you hate the word magnetostatics? I love it and here’s why . . .
  • I’m in this class and I’m super excited for tomorrow. Mostly I can’t wait to . . .
  • I’m in this class but I’m skipping tomorrow because this seems dumb.
  • If the students are going to eventually take advanced electromagnetism, why do you care?
  • Just give them the formula and show them the three cases that it works for. Then they’ll be all set to do tons of meaningless homework problems.
  • I love Ampere’s law. My favorite thing to do when teaching it is . . .
  • I hate Ampere’s law. I avoid teaching it by . . .
  • Whatever you do, make sure you give them 17 more right-hand-rules to memorize.
  • Why didn’t you put the accent in everywhere you typed Ampère?
Posted in physics, syllabus creation | 3 Comments

Google moderator use in class

I’ve talked before about how I like to use Google Moderator to have students crowd-source the questions for me in class. I wanted to get down some notes about the conversation I had with my current class about our current implementation.

Every Friday I spend at most 30 minutes going over the questions that they’ve been submitting and voting on all week. I start at the top and go down from there. Most weeks I’ve only gotten to about a third of the submitted questions, but I’ve been fine with that given that they decided which ones were most important.

But I’ve noticed something. The questions at the top are always about the material we cover early in the week. So I asked the students this past Friday whether the early questions get more votes because they’re seen by more students. They all thought that was likely the case. So we brainstormed some ways to fix that. At first I wondered if it was worth it to try to get more students to submit things early, but we really focused instead on holding off the voting until the bulk of the questions were in. A student asked if I could set Google Moderator to accept questions but not votes for a while but I’m pretty sure that’s not possible. But I said we could certainly make that our policy, so we focused on when the voting should start. I threw out ideas like Wednesday or Thursday night, but they said that it seemed most questions come in late Thursday and early Friday. I asked if they could do it before class, but a lot of the students have class all morning.

Then I had an epiphany! We could just use the first 5 minutes of class on Friday to do the voting. I asked if GM worked well with phones and a few said it worked fine. I then asked how many thought they could do it on their phones in class and most raised their hands.

So we’re going to give that a try this week. I’m hopeful that it’ll make me more confident in the vote, especially given that I don’t get to the rest of the questions.

The other thing we talked about was what I should do about the other questions. I offered to make screencasts dealing with them, but I was surprised by the vocal negative reaction to that. I asked why and a few said that the low vote ones really don’t need screencasts. I pointed out that people didn’t have to watch them, but it was still interesting that they thought that might be a waste of time. Some thought maybe we could have a vote total threshold where if I didn’t get to such a question, I’d still make a screencast. One student talked about how she experienced at least once some disappointment that I didn’t get to just one more question (they can see what’s coming because I project all the questions). I wondered aloud whether the new plan would make that problem go away, but I think we’ll still keep a close eye on that. I did tell them that I’m always happy to do a screencast on demand (via email or whatever) so that if they’re still dying to know the answer to something they could always do that.

Your thoughts? Here are some starters for you:

  • I’m in this class and I’m glad we talked about this. What I like most about the new approach is . . .
  • I’m in this class and I wish we had also addressed . . .
  • Can you let the rest of us see these awesome questions? [right now I've made it limited to people in the Hamline google world but I'd love to hear some thoughts about that]
  • I work for Google and I’d respond about ways to limit the time when voting occurs, but I notice that you’ve switched from Blogger to WordPress so you’re dead to me.
  • You still use Google Moderator for this? You fool! Instead you should use . . .
  • What’s wrong, Mathematica can’t do this for you? I bet python could, loser.
  • What if you found that it was always the same students who get voted down? What would you do to help them?
  • How do you get students to actively vote down someone’s questions, as opposed to just skipping it and only doing “up-votes”? [I know I'm supposed to put that question mark inside the quotes, but I always feel weird doing that]
Posted in syllabus creation | 1 Comment

I hate Kirchhoff’s loop law

Sorry for the incendiary title, but it does express my feelings pretty well. When analyzing DC circuits, students are often encouraged to use the two Kirchhoff’s laws:

  1. [sometimes called the node law] all current flowing into a node must flow out
  2. [sometimes called the loop law] the change in voltage around any closed loop is zero

Using those two, you can analyze any DC circuit that has batteries and resistors in it (and of course other elements if you appropriately define the impedance). However, while the first of the two is a no-brainer for most students, the loop law gives students all kinds of fits, mostly due to trying to imagine walking a charged particle all the way around a loop, keeping track of any voltage changes. If you go through a battery the normal way, that’s a gain. If you go through a resistor the way all other (positive – damn you Ben Franklin) charges go, that’s a voltage loss. Those two aren’t really that big of a deal. However, in any circuit that really needs this approach, you often have to imaging taking a charge through a resistor the other way, against the flow of current. You say it gains voltage during that. Students seem to have a hard time with that. Except the rule-followers, those students don’t seem to have much trouble with this. Ask them “why,” though, and you’ll regret it.

Here’s the other problem with this whole approach: at the end of the analysis, you’re usually staring at a bunch of equations with a bunch of unknowns. Often the equations outnumber the unknowns, but we try to show the students that if you use all the node equations or all the loop equations, you’re being a little redundant. Whatever. The problem is that as soon as it’s more than two unknowns, you’re really testing their math abilities much more so than their physics abilities.

So what to do instead? Here’s what I do: I give them a complex complicated circuit with a bunch of batteries and resistors and then I do the unthinkable: I give them one of the currents for free. That’s right, I tell them one of the answers. Then I ask them to figure out the rest of the currents by building on that information. Here’s an example. I give them this circuit:

Circuit that I give the students to analyze

Circuit that I give the students to analyze

and then I tell them that the current flowing up through the center 1 ohm resistor is 6 amps. From there they try to reason out the rest, usually in this order:

  1. The voltage at the top is 10 volts because you lose 6 of the 16 volts of the central battery going up to the top.
  2. The voltage drop on the left leg is also 10, so the voltage drop across the top left resistor is 2 volts. Therefore the current through that one is 2 amp downward.
  3. That means that there’s 4 amps flowing down the right leg.
  4. The top right resistor burns 4 volts, meaning that the voltage drop across the parallel portion is 10-4=6 volts.
  5. The current through the right leg of the parallel section must be 6 amps to account for step (4)
  6. The current in the left leg of the parallel section must then by 2 amps upward so that the node law works at the top.

None of that is solving 4 equations for 4 unknowns. Rather, it’s demonstrating an understanding of ohms law and batteries all at a local level.

I like this approach, and I use it on quizzes/exams a lot. I certainly don’t want to test their linear algebra ability in those situations. Now, in the old days I used to give them a complex circuit and ask for enough loop and node equations that would, in principle, enable them to solve everything. I like this new way better.

But what about the student who asks how I got that first unknown in the first place? I’ve been thinking about that a lot lately and I think I’ve come up with a cool way to talk about that. Instead of showing them the n equations and n unknowns approach (assuming that they really understand the underpinnings of some powerful linear algebra), now I think I’ll have them tackle problems like that by just simply guessing one of the currents. Then they can follow similar steps as above until they run into a discrepancy. For example, imagine if you thought that the middle current above was 1 amp upward:

  1. then the voltage at the top would be 15V
  2. the voltage difference across the upper left resistor would be 7 V so you’d have 7 amps flowing down there.
  3. That would mean the right portion would have 6 amps flowing up.
  4. That would put the voltage at the top of the parallel section at 15+6=21 V
  5. That would make the current in the lower right portion 21 amps down
  6. That would make the current in the other portion of the parallel section 27 amps up
  7. BUT THAT WOULD MAKE THE VOLTAGE DROP OF THAT LEG WRONG

So you see we’d have a discrepancy. What I’ve been playing with today is to see how that discrepancy (in this case, a voltage discrepancy) varies with the initial guess of one amp flowing up in the middle portion. It turns out, unsurprisingly given the linear nature of all of this physics, that it’s a straight line if you plot the discrepancy versus the current guess. Then what you’re looking for is the current that would give you zero discrepancy. That’s the same as asking for the x-intercept of that straight line.

Now, I think students could do that, especially since never would they do linear algebra. However, I would still give them freebies in quiz/exam situations because I think they just need to do this exercise I mentioned once to see how I got the freebie. Note that to fully solve a circuit this way you have to do the 7 steps above twice (to get two points for the straight line) and then a third time with the right value. That’s a lot of work, but every step is explainable.

Your thoughts? Here’s some starters for you:

  • I like this. I too hate the Kirchhoff loop law, but my issue was always . . .
  • What’s wrong with the loop law?! It’s awesome, especially because . . .
  • Why are you even mentioning the loop law when we all know that circuits with induction show that it’s wrong?
  • Don’t you think the “linear discrepancy theory” is just as hard to explain as imagining a charge going against the current?
  • Why do you think our positive charge convention is the fault of my beloved Ben Franklin?
  • What happens when you teach this and your colleagues teaching lab assume the students know the loop law?
  • What happens when the discrepancy is with a current instead of a voltage?
Posted in physics, teaching | 11 Comments

String resonance

My friend Will posted a cool animation today:

It got me thinking about the lab we do with vibrating strings and I learned a couple of cool things.

First of all, his animation didn’t seem like what the lab looks like if you slowly increase the frequency of the driving speaker (which is tied to one end of the string). Will’s animation looked more like that end was a free end, so he and I had a nice twitter conversation about it. I finally got what he was doing when he posted

So I sent him a link to some stuff I’ve done in the past, but thought I’d try to make an animation where the speaker slowly increases its frequency. First I had to think about the mathematical function for the speaker that would represent a sinusoidal motion that speeds up. Here’s where my work in grad school with all kinds of temperal/fourier transform functions came in handy. It turns out that the instantaneous frequency for any function that can be written Sin/Cos/Exp[i … is given by the time derivative of the argument of the trig function. So, if I wanted a function that linearly increased the frequency, I just need to integrate my goal and I’d have the argument to feed into the code. So, here’s the code with some annotation

The Mathematica code for solving the wave equation with a stretched string. The boundary conditions are highlighted in yellow and the initial conditions in blue.

The Mathematica code for solving the wave equation with a stretched string. The boundary conditions are highlighted in yellow and the initial conditions in blue.

You can see that for the moment the code has no friction (because the effective \beta is zero. That was my first thought to make this animation, but here’s the somewhat disappointing results

animation of the string with no friction. n is the mode number, so cool things should happen at the integers

animation of the string with no friction. n is the mode number, so cool things should happen at the integers

That’s when I realized that I needed to add some friction so that the energy stored in the earlier modes would be dissipated by the time the next mode comes around. That’s how I got this animation:

now a little friction is added

now a little friction is added

Of course, both suffer from a sampling problem, but you get the gist. I think it looks a lot like the lab, so that’s cool.

Thoughts? Here’s some starters for you:

  • This is cool! What is the Mathematica command to make the animation?
  • This is dumb. The strings in my lab don’t look like that at all. Instead . . .
  • This is cool! Can you do the same boundary conditions as Will?
  • This is dumb. The jerkyness of the animated gifs really bothers me. Couldn’t you just upload, I don’t know, the 3MB file with better temporal resolution? (ok fine, see below)
This one has 10 times as many frames

This one has 10 times as many frames

Posted in mathematica, physics, twitter | 6 Comments

Averages vs histograms

With graphics being so easy to add to documents these days, why don’t we show more histograms in place of the typical approach of representing very complicated data with one or two numbers (eg average and standard deviation)? Sure, if your data is normally distributed, then those two numbers really are a great distillation of the data. However, lots of things aren’t normally distributed, and I’m lobbying for more use of histograms instead of (or, I suppose, in conjunction with) the numeric characteristics of the data set.

Here’s the example that got me thinking about this today. At my school student evaluations of instructors are very important. We use a seven-point Likert scale on questions such as “The instructor encourages me to learn actively” and “This course was a valuable learning experience.” Quite often reviews of faculty are peppered with means and occasionally standard deviations of evaluation data for the reviewed faculty member. However, the data is not normally distributed at all! It can be bimodal (some hate me, some love me), or highly skewed in other ways. I’ve been working lately to provide an interface for our evaluations to help people on the tenure and promotion committee make wise recommendations. Instead of having to click through to each course, I’ve made a nice table that shows the average for the class on each question. The table rows are the various courses the faculty member has taught. But while thinking about the notion of showing histograms in addition to averages, I hit upon using PHP to dynamically create SVG’s with the histograms. Here’s what it looks like:

5 courses for an anonymous faculty member. Each column is a different question on our standard evaluation.

5 courses for an anonymous faculty member. Each column is a different question on our standard evaluation.

I feel like you learn a lot by looking at the (tiny) histograms. Take the three “4.44”s that are in the third class. The middle one is much more bimodal than the other two.

What am I lobbying for? I’d love it if many more reports/journal articles/newspaper stories did this kind of thing. The graphics generation and inclusion is really not that hard, and I think it communicates the whole story, not just a distilled version.

One downside is the inability to describe the data very easily. I was showing this to my partner and I was trying to say “this one is different than that one” and I had to point to them. I couldn’t easily describe them. So I resorted to saying “the 4.44 one . . .” etc. I suppose this is backing up my point that the data sets are complex and resist easy description, but I know my colleagues on the tenure and promotion committee like to really discuss these evaluations a lot.

Here’s another interesting point from a friend of mine (who’ll remain anonymous):

Averages and SDs are **NOT** appropriate for categorical data. They assume the “distance” between each category is equal, as if the numerical choices were locations on a spatial scale. They are not. You’ve got two choices: Report number of responses in each bin (as you’re playing with); or turn to Rasch analysis, which is designed for exactly this problem. But it’s not for the faint of heart…

Interesting, huh?

Your thoughts? Here are some starters for you:

  • This is great. I totally agree that representing all of the data is much better than any distillations. I would even go further by suggesting . . .
  • This is dumb. We use the distillations for several very good reasons . . .
  • Why do you use evaluation data at all? They’ve clearly been shown to be problematic.
  • Why a 7-point Likert scale? How about a 2-point Love-ert scale?
  • How did you make those SVG histograms in PHP?
  • PHP?!!? I’m never reading this blog again.
  • Wait, I thought you only knew how to use Mathematica.
Posted in math | Tagged , | 4 Comments

Wave equation first

This semester I kick off my general physics 2 course with waves. I really want the early focus to be on what waves are, and, more specifically, what the wave equation means. The reason I want to do this is because the wave equation is, for me, the biggest hallmark of a system being able to support waves. If you look at a system and find any parameter whose spatial curvature is proportional to its acceleration, you know it can support a wave and you can determine the speed of the wave from the proportionality constant.

When teaching about the speed of a wave on a string, which is often the first example that texts “cover,” I get frustrated with the various approaches to determining the speed of the wave, since often it’s a very contrived situation. Typically it’s considering a situation of moving one end up at a constant speed. That leads to a non-calculus calculation of the speed, but I’ve always thought it felt too contrived. I much prefer convincing the students that the wave equation is what to watch for and to simply look at the proportionality constant.

What I thought I’d do is start by asking my students about examples of waves. I don’t really want to get bogged down in SHM stuff, so really my hope is that we land at something like “disturbances that travel through the medium.” Then I want to see how far we can get talking about why the propagation happens. For a string, that’ll be talking about portions of the string where there’s a curvature (or, really, a change in slope). The idea would be to realize that a portion of the string that’s a constant slope will not have a force pulling the string up (or down) since the (constant) tension will balance out. However, if there’s a change in slope, then there will be a net force up (or down). So, a change in the slope will lead to an acceleration. The wave equation!

To figure out that the proportionality constant is related to the speed of the wave, I really want them to have a sense that any function f(x) simply turns in to f(x-vt). I’ll try to get to that by getting the students to comment on the similarities and differences of the motion of different portions of a string that has a disturbance traveling down (for now without friction). I’ll be hoping for thoughts like “they all do the same thing, only delayed” and “they’re delayed by how long it takes the disturbance to get there.” 

If I can do both of those last paragraphs, then getting that v=\sqrt{T/\mu} should be relatively straightforward.

So what about using this approach with other systems? Sound in air, for example? Well, what causes sound. Again you have to have a change in the pressure distribution or nothing happens. Again you’d have a second order spatial change proportional to an acceleration (of the air particles, say). I think it could work, but it’s not as clear cut as the string example.

Where this approach will come in handy is when we get to Maxwell’s equations and light. You futz with Maxwell’s equations and you end at a proportionality between the second spatial derivative of E (or B) and the second temporal derivative of E (or B) with the permeability and permitivity of free space sitting there as the proportionality constant.

Your thoughts?

  • Why do you put the word “cover” in quotes?
  • I really like the “one end moving up at a constant speed” approach. I like it better than this because . . .
  • I, too, don’t really like the “one end moving up at a constant speed” approach. What I do instead (which is MUCH better than your plan) is . . .
  • I don’t get all this. Just give them the equation and assign some homework.
  • With friction the notion that “everyone does the same thing, only delayed” doesn’t work. How do you deal with that?
  • In multiple dimensions the notion that “everyone does the same thing, only delayed” doesn’t work. How do you deal with that?
  • I like this approach a lot. Please let us know how it goes!
  • With sound in air, here’s a good way to get students to think about a second spatial derivative needed . . .
  • For light, it’s not an acceleration. Will your students be ok with that?
Posted in physics, teaching | Leave a comment

What if they don’t do it?

I’m still refining yesterday’s idea of having one day a week for assessment/review/integration in my General Physics II course. As often happens, I’m narrowing in on an idea/approach that I think could really help students learn, but there’s always the fear that they won’t do what’s necessary to get enough out of it.

An example of that is my traditional flipped class approach from a few years ago. I carefully crafted some screencasts trying to elucidate some of the harder issues in the text and came into class hoping we’d all be ready to hit the ground running. Unfortunately I never seemed to have more than about 1/3 of the class ready. Looking at viewing statistics backed that up, though it was interesting to note that the vids were viewed a ton leading up to any assessments.

So here’s my pie-in-the-sky thoughts about this weekly assessment day. First I would respond to the top 5 few questions from the weekly Google Moderator series. Those would be questions entered throughout the week on the content of the week. That would take 15 minutes. Then we’d do a 15 minute quiz. Then the students would collectively identify the additional resources they’d need to help them understand the material better for future reassessments (note that the whole Google Moderator series would likely help in that conversation).

Why do I call it pie-in-the-sky? Because it really relies on students doing a number of things throughout the week

  • Use class time to explore the landscape of the material and request useful resources that I would provide before the end of that class day’s day (that sounds confusing).
  • Utilize the resources I provide to learn more of the details of the content. This should be heavily interspersed with working as many problems as they can to get ready for the quiz
  • Both submit and vote on questions/issues in the Google Moderator series.
  • Working together in the last 30 minutes of the assessment day to figure out what they need to improve their understanding of the material
  • Commit to doing reassessments (these can be videos, office visits, virtual office visits, etc)

If some or all of those break down, the assessment day is a big waste of time (except the 15 minute quiz, I suppose). I’m very cognizant of that because I’m proposing doing 18 chapters using only 28 days of instruction time.

The hope is that the students will see the value of all of the proposed activities. They’ll see how all class days are really opportunities for them to request specific resources that will help them (worked example problems are surely going to be very popular). They’ll work problems with an eye toward doing well on the quiz (oh, and to learn, I suppose). They’ll use Google Moderator to make sure that class time isn’t “wasted” on less important issues (note: in student evaluations in the past I’ve had students complain that I answered every question and wasted class time working on what were really easy ideas).

One concern of course is that they’ll do everything and ace the quiz, making the second half of the reassessment day a waste of time. I’m not really concerned about that, but I could imagine stronger students thinking that way.

My good friend Bret Benesh commented on my last post about the usefulness of giving students lots of time to understand material. I thought about maybe pushing the quiz on this week’s material off to next week. But, in my experience, that just means the students will push off to that week the work needed to do well on that quiz. That could work, of course, especially if we pushed off the google moderator series too, but I think I want all of that happening in the week when they’re seeing all of this in class.

So, what could I do if the students don’t do what’s best for their learning? Here are some starters for you:

  • I’m in this class and I’m intrigued. What happens if . . .
  • I’m in this class and I’m worried. What about when . . .
  • If the students don’t do the work, it’s their problem. You should plow ahead and . . .
  • If the students don’t do the work, you’ll have to change the approach toward . . .
  • I like Bret’s idea, here’s how you could make that work better . . .
  • They’ll never do any homework problems is you don’t collect them. Here’s an idea that’ll make that work in your Standards-Based Grading system . . .
Posted in sbar, sbg, syllabus creation, teaching | 7 Comments