TensorTime has moved!

Effective immediately, this blog has moved to

https://tensortime.sticksandshadows.com

so update your bookmarks and follower status. I have already ostensibly migrated all of this blog’s followers to the new blog, and it was spectacularly easy to do thanks to the WordPress tools. If anyone finds they are not still following the new blog, please let me know or just follow the new one as you see fit.

This site, hosted on WordPress.com, will not be updated and will eventually be deleted.

I made the move because I need to be able to embed GlowScript and Trinket programs and this is forbidden in blogs hosted on wordpress.com, and for understandable reasons. The blog is now hosted on a subdomain of my own website. You may find the site slower than wordpress.com and if you do, please let me know and I’ll see what I can do about it.

All LaTeX embedded in posts should render correctly on the new site. After a lot of experimentation, I have found a fantastic LaTeX plugin for WordPress that allows for using external packages and eventually I will implement it retroactively on all existing posts.

Thanks to all my readers and followers for making this project worthwhile to me, and I’ll see you at the new site!

 


A Sordid Little Story…

I want to tell a story. It addresses a taboo topic: workplace issues. It is important to make the community aware of these issues, especially for younger teachers going into the two-year college environment, expecially in certain regions. The landscape is changing in some states, and not in good ways. This story could be set anywhere or nowhere at all. The characters could be anyone, even you. Read this post as though it were a short story, or some other fictionalized account, in the narrator’s voice (I hear it in my mind as an episode of Little House on the Prairie narrated by Laura Ingalls Wilder). I encourage you to read the entire story, though, because I think it has a reasonably happy ending. Do not attack me for telling this story; it’s just a story not necessarily based on reality.

Here are the characters in this sordid little story: the first-person narrator (some random physics instructor at a random two-year college in a random state in a random universe), [the Chair], [previous Dean], [new Dean] (aka [the Dean]), [physics colleague]. Here we go.

It all started in March on a quiet Friday morning. I was in my office, and [physics colleague] knocked on my door and I motioned to come on in. [Physics colleague] asked, “So what do you think of [the Chair’s] idea?” I had no idea what [physics colleague] was talking about so I asked. [The Chair] had just asked if [physics colleague] would be willing to take over the calculus-based courses from me this fall. Not one to say no, but [physics colleague] made it clear that it wasn’t desirable but agreed to out of obligation to obey [the Chair’s] request. Note that my colleage is from another country and was an assistant or associate department chair there and firmly believes that one’s department chair is an ultimate authority not to be crossed. [The Chair] had left the hall so I went to [the Dean] (whom I’ve known for years as a fellow faculty member) and asked about this; nope. Quite a number of days passed before [the Chair] approached me, and it was a strange conversation. First, [the Chair] congratulated me on increasing the enrollment in our astronomy course, which susprised me because I haven’t done anything to increase enrollment becuase I have no control over that. Anyway, [the Chair] then coyly smiled and asked if I would be willing to “do the same” for our conceptual physics course in the fall. Of course I asked why, and why NOW? I got a very mixed response. [The Chair] proceeded to explain that [physics colleague’s] approach to conceptual physics wasn’t appropriate as it used too much math. Given that [the Chair] is not a physicist and has no formal background in physics or physics education I didn’t think such an assessment was appropriate from someone in a non-physics discipline and said so. Well, [the chair] then proceded to express discontent with my own methods in calculus-based physics, stating that I am hurting students’ educations by not requiring formal lab reports and emphasizing computation rather than traditional labs. [The Chair] confidently stated that computation isn’t real science and that I shouldn’t be emphasizing it so much. [The Chair] also claimed that a PhD (not in physics) dwarfs my own knowledge and experience. [The Chair] claimed more extensive knowledge of what prospective physic and engineering majors needed as preparation in the introductory courses, and that it must emphasize formal lab reports. I couldn’t believe I was hearing this, and I really didn’t know how to respond.

A quick digression is in order. In the past, [the Chair] had been nothing but supportive of everything I did and everything my colleague did. Something had changed suddenly and drastically, and I still have no idea what it was or when it was. Micromanaging had been instituted several years ago, with [the Chair] demanding usernames and passwords to all third party websites (including resources like WebAssign). I refused, and had to go to [previous Dean] to get this behavior stopped. Seems [the Chair] thought such micro oversight was necessary to keep an eye on grades and such. [Previous Dean] told me, and this is a quote, “It is not reasonable to assume that people will do the right thing.” and therefore needed micromanaging oversight. However, [previous Dean] also agreed that [the Chair] was going overboard and got much of the micromanaging stopped, thankfully. Then last semester, [the Chair] approached me threatening to cancel second semester astronomy unless I signed a form removing ALL math prerequisities from both astronomy courses. I refused, and [the Chair] grabbed the form from my hands, said my signature wasn’t needed anyway, and stormed off. On Friday of that week, I got an email requesting a meeting, to which I replied positively. A few minutes later, I got a followup email saying the location would be in [the Dean’s] office. WOW! I had no idea what this meeting was about either, and naturally assumed the worst. Sure enough, the subject of the meeting was on “removing barriers to enrollment” in second semester astronomy. Apparently, [the Dean] thought that removing the math prerequisites was MY idea to begin with, and was a little surprised to discover that it wasn’t. I was rather forced, under duress, to agree to removing the prerequisites as “an experiment” this semester and that if necessary, they could be added back in the future. All three of us knew the prerequisites were gone for good, and we knew everyone knew it. Then, a few weeks later after the committee that approves textbook and other curricular changes, at least two people from that committee told me that when the changes for the astronomy courses came up, lots of committee members balked and said that it didn’t sound like something I would request, and they were correct because I didn’t request it. Nevertheless, the “experiment” passed with one department chair voting against it because that person knew what happened. Apparently some other colleagues wondered what was going on.

To continue the story, I began by once again, as I have frequently had to do unnecessarily over the years to administrators with no science background but never to a PhD scientist, quoting findings from PER about the introductory calculus-based course, only to be met with snide remarks like, “That’s YOUR opinion.” and “There are always other ways of doing things.” and “I know better than you.” Again, I was stunned becuase this just didn’t sound like [the Chair]. Furthermore, I staunchly took issue with someone from another discipline claiming to know what’s best for our physics students becuase that’s ostensibly what physics faculty are hired to oversee, and being allowed to suddenly get away with it after years of not doing anything like this. I got exactly nowhere. [Physics colleague] and I met with [the Dean] to express our concern over this situation. [Physics colleague] firmly expressed discontent with having to teach this course for the first time, but also said that there would be no push back out of the desire to not be labeled a troublemaker. In other words, I conveniently slid under an oncoming bus. We asked to form our own physics department, but in a subsequent meeting with me [the Dean] said that would be “like forming a calculus department within a mathematics department.” I smiled politely and didn’t bother to point out the egregiously erroneous analogy. [The Dean] also cited a few other reasons so there was at least some other logic behind the decision and I accepted that.

Let me again digress to say that [the Dean] is only one year into the position and isn’t aware of the history shared by [the Chair] and me. [Previous Dean] promoted micromanagement at the department chair level. When I pushed back, I was called disrespectful, borderline insubordinate, and (my favorite) a “common whistleblower.” I laugh at that now, but I wasn’t laughing when [previous Dean] called me that to my face. Incidentally, because of [previous Dean’s] insistance on promoting micromanaging, two VERY high quality department chairs stepped down to return to regular faculty duties and that was a very unfortunate loss for their respective departments. Fortunately, [previous Dean] retired last May and [new Dean] took the position. Now, I firmly belive [new Dean] supports me and understands my predicament. It took several phone calls and another face to face meeting to fill [the Dean] in on the details, but [the Dean] has openly asserted, in front of both me and [the Chair], that research-based methods are indeed best for our students and the institution overall. [The Dean] has related to me that students have come to her expressing their delight with my classes, and this plays an important role in how this story, or at least this chapter of it, ends. [The Dean] has observed me in my classroom habitat doing my thing, and has heard nothing but positive feedback from students and other persons. Understand that that is not to say that I’m perfect at what I do becuase I am most assuredly not given the imposter syndrome I continually battle and my own internal awareness of every little mistake I make while doing my job.

Now, flash forward to mid to late April. [The Chair] entered my classroom between classes and presented me with two more curriculum change forms to sign. One was to approve my request to remove WebAssign from the astronomy courses (a difficult decision for me to have made, but I had my reasons) and another to approve the textbook changes for the calculus-based courses. Of course I readily signed the former, becuase I had requested that change myself. However, I refused to sign my approval for the latter becuase I had been (intentionally?) kept out of all discussions about text materials for the calculus-based course I had carefully cultivated since the fall semester of 1999. There are places on the form where faculty approving of the changes to sign and those not approving to sign. I refused to sign. [The Chair] demanded several times, “You will sign this now.” I resisted, and then decided to at least have the final word by writing on the form WHY I refused to sign it. As I was writing and explaining again how this is not a good change for students and how I was not included from the deliberations, [the Chair] made comments like “That’s YOUR opinion.” When I said, yes, and as a physicist mine is a more informed opinion in this matter than yours, [the Chair] came back with, “Stop making everything about YOU.” It seemed everything I said was thrown back at me and framed as an expression of arrogance rather than an informed professional position. As I was writing in the latter space that I didn’t approve because I had not been informed or consulted about the proposed changes, [the Chair] grabbed the form from my hands and walked out. Now keep in mind that the ONLY non-discussions about this whole mess that had included me are the ones related in this story. At no time did [the Chair] discuss with with both of the physics faculty. All “discussions” were between [the Chair] and [physics colleague]. I was, I believe, intentionally excluded from these discussions, and it turns out I was right as I would soon discover. Anyway, when [the Dean] distributed the signed curriculum/textbook change forms for the appropriate committee to consider before meeting and holding a final vote, the above mentioned textbook change form was surprisingly missing from the collection. This surprised me, and also surprised another depaertment chair on that committee who was aware of what had happened. I suspect [the Chair] will wait to the very last minute to push the form through, but I know of at least two department heads who know what happened and will probably vote to reject the change, at least on record. It will most likely be approved because it is essentially unheard of for such change to not be approved. At least there will be two voices who know that I had no input into this and one of those will certainly raise that point in the meeting.

Well, I once again related these events to [the Dean], who expressed dismay and borderline incredulity at [the Chair’s] behavior. Then, in one phone conversation with [the Dean], the truth emerged. The TL;DR version is that it’s money. It seems this whole mess began some time ago when [the Dean] and [the Chair] were scheming to find ways to increase enrollments in our conceptual physics course. [The Dean] expressed delight and support for the teaching methods I employ, informed by research findings, contemporary content, use of computation, etc. and suggested that I might do a good job at, for lack of a better word, modernizing our conceptual course. Okay, yeah I can surely take a stab at that, but why didn’t [the Chair] just come out and ask me and [physics colleague] about this plan together rather than talk to us separately, and tell us each a different story for the purpose of the course changes? Why? [The Dean] couldn’t answer that, and seemed troubled by the fact that I’d been kept in the dark. [The Dean] apologized profusely at the way the situation was handled, but I insisted there was no reason to apologize because the responsibility of relaying the information to me lay with [the Chair] and not [the Dean], and the former failed in that responsibility. [The Dean] continued to reassure me that [the Chair] thought highly of my methods, yet in talking with me [the Chair] has made nebulous references to “student complaints” about me, apparently undocumented and therefore unable to be shown to me on paper, that at least partly warranted the changes. Why did [the Chair] tell [physics colleague] that my approach was unsatisfactory and then tell me that [physics colleague’s] approach was unsatisfactory? Why not ask me for textbook suggestions for [physics colleague] to consider in preparing to teach calculus-based physics? Well, I didn’t know the answer at first, but I know now. Read on.

We usually know our fall schedules before the end of the spring semester, mostly because things don’t usually change very much. Well, it wasn’t until last Thursday (the day before final grades were due), that I got an email telling me my fall schedule. Not only was the course change still going through without my input or approval (neither of which was deemed necessary apparently), I was not given an opportunity to consider any textbook changes for the conceptual course which I was assigned whereas [physics colleague] was given free reign to choose new materials for the calculus-based courses and I stronly suspect [the Chair] had a hand in guiding my learned colleague away from the existing textbook but I cannot prove that. Anyway, in responding to this email, which contained a rather minor question about an optional book for my sections, I copied [the Dean] saying that I’d not been given any input into either the course changes or the schedule (which was HIGHLY unusual given procedures from previous years) that I didn’t know whether or not my input on this one minor question was worth my time to give as it would probably be ignored. [The Dean] took the clue and asked for a meeting with me, [the Chair], and [the Dean] to discuss the issues. Incidentally, [the Dean] and I also spoke by phone and [the Dean] assured me that I could indeed request any textbook and ancillary materials I wanted despite [the Chair’s] assumption to the contrary. At that meeting, which took place Friday of last week, [the Dean] once again reaffirmed her support for me and once again declared that the whole plan was intended for me to take what I’ve done in calculus-based physics and do it for conceptual physics with the intent of making it a course all students would want to take, and indeed should take, as a science elective. It seems [the Dean] is concerned that more students take biology and chemistry than physics and wants to change that, and I don’t necessarily think that’s a bad idea. However, as many physics faculty know, this is a cultural phenomenon not easily changed. I’m flattered that [the Dean] has such confidence in me, and I firmly believe [the Dean] is sincere in her confidence because on at least one occasion, there were tears involved from both of us as we both expressed our vision for what physics education could become here, and it turns out we have a shared vision. That’s reassuring, and is part of the happy ending I alluded to earlier. However, in this most recent meeting, [the Chair] adamantly denined having made any of the remarks I had documented. I now wish that I had had a recorder with me at the appropriate times, but you see I never dreamed that such tactics would be necesssary in an academic workplace. I won’t make that mistake again. Oh, and when I asked [the Chair] why I had not been included directly in the discussions, the response was “You’re very difficult to talk to.” I couldn’t help but laugh. Oh, and this definitely confirmed my suspicion that I had indeed been intentionally excluded from discussion about all this. A department chair who is afraid to talk with direct reports probably shouldn’t be a department chair in the first place. The reality is that [the Chair] probably doesn’t like the fact that I push back against bullying and ill-informed policy changes, but I would think that’s expected of informed, educated academics. It particularly saddens me that such assertion of informed opinion is too frequently equated to “you don’t like change.” I love change! I’ve changed so much since I entered this career, and I change every year. Change for the sake of change is almost never good, and uninformed change can be disastrous for everyone involved, especially students. I don’t like bullying from supposed professionals, and I don’t like divisive decision making that excludes affected parties. Furthermore, I resent the implications that students’ concerns must come (and please read the next phrase carefully) AT THE EXPENSE OF FACULTY CONCERNS. We ALL have concerns that must be considered in making decisions and to assert that faculty concerns don’t matter, EVER, is fallacious reasoning and diminishes our earned professional status. We can’t meet students’ needs unless our professional needs are also met. Yet, I get a distinct sensation of being psychologically manipulated into neglecting our own informed concerns in deference to the concerns of others and that this is a newly expected workplace norm. Oh, and it was also during this meeting that [the Chair] proclaimed that the research findings and recommendations by AAPT can’t possibly be “best practices” (I really hate that term by the way) becuase if they were, then everyone would be employing them and the fact that that is not the case is evidence that they must not really be “best practices.” I’ve had high school students who would see through that fallacy and shred it, but I was really too stunned that someone who had, in the past, been open to such findings now dismissed them.

As I mentioned above, all this is ultimately motivated by money. Many public institutions are funded based on enrollment, and if the latter drops then so does the former. The game now is to increase enrollment regardless of student preparation. The preparation issue is important because many two-year colleges now host charter high schools and provide instruction in surrounding high schools either by sending faculty out into the high schools and/or by providing a cohort experience for high school students in which they simultaneously earn a high school diploma and an associate’s degree. As a consequence, a greater percentage of students are not ready for college level work and that presents burdens that I’m afraid we can’t remedy. Fill every classroom to capacity so budgets will be increased in future funding cycles. Okay, but at least be honest about it. In the past,  [the Chair] has ultimately defended every single administrative policy directive by fallacious argumentum ad baculum (appeal to fear) by saying that if we don’t implement this policy (regardless of how badly informed it may be) jobs will be lost due to loss of funding. While that may be a practical reality, I don’t think it’s a particularly good motivator for academics trained to see through propagandistic scare tactics. I also don’t think that it’s good leadership to rely on fear to get something done. I know if I were a department chair I would never do that. I would ensure that I reflected the department’s concerns to administration without dictating to the department what their concerns must or must not be.

So, the whole story, as I currently understand it, is that [the Dean] specifically wants me to channel my informed classroom methods, devotion, and passion for teaching into an effort to try to attract more students to take conceptual physics, with the happy side effect of boosting enrollment and therefore future funding and MOST IMPORTANTLY getting more students into a physics classroom. That makes me feel appreciated and indeed rather proud! It does something else though. [The Dean’s] sincere support and confidence in me makes me WANT to work with [the Dean] to make our shared goal happen. That support motivates me, and there’s something else that really endears me to [the Dean]. In one of our recent face to face conversations, I casually mentioned that after over two decades of teaching, I was tired in every conceivable way and that a sabbatical would be appreciated. [The Dean] took a serious tone and asked what I would do on an ideal (e.g. fully funded, semester long) sabbatical. My honest response was that I would devote it entirely to professional development by traveling around the country to meet with, observe, and hopefully learn from, my fellow physics colleagues who, with permission of course, would allow me to do so. That’s what I would really do. [The Dean] cautiously said something to the effect of “Let me see what I can do.” I appreciated this response, but really didn’t expect anything to come of it. Well, I was shocked when, in the most recent meeting with [the Chair], [the Dean], and me that [the Dean] indicated to me and [the Chair] (who wasn’t aware of the sabbatical discussion) that in lieu of a sabbatical I could have a schedule in which all my classes are M-Th and I could devote all Fridays (in both semesters if I wish) to professional development of my choice! I was floored! I honestly didn’t know how to respond because my previous request for a sabbatical was soundly rejected on the grounds that writing a book for one of my classes was of more value to me personally than to my students and they wouldn’t benefit from my efforts (that really hurt and I considered it a very unprofessional response from an institution that supposedly valued professional development). As always though, everything comes down to “funding” and its lack is used as an excuse, I suspect, even when “funding” is available. I have come to realize that [the Chair] simply doesn’t know, or chooses to not know, how to effectively motivate faculty and for some reason unknown to me has become a workplace bully. On the other hand, [the Dean] clearly does it through sincere appreciation and creating a sense of being valued and by hearing my concerns and responding with something tangible. I respect that, and I want to cultivate a relationship with [the Dean] and be the next department chair. In fact, I have already told [the Dean] that I’m willing to be the next chair and I plan to point that out from time to time as we discuss physics instruction.

I don’t understand why [the Chair] decided unilaterally to take such a ham-handed approach in implementing this strategy and in communicating the strategy to me. I don’t respond well to dictatorial tactics and I don’t think any other reasonable colleague would either and I don’t understand why some people expect me to respond unquestionably and blindly to such behavior. We try to teach our students to be aware of being manipulated like this and yet we’re discouraged from using the very critical thinking that prevents this in the workplace. What an unfortunate dichotomy! I respond quite well to informed decision making and to full disclosure of the underlying reasoning. I stil have some concerns, though, about being “used” like I’m a corporate worker being reassigned to do something new on the spur of the moment with no regard for the lurch in which calculus-based students who have come to expect a certain environment will be left, and will indeed be at a great disadvantage, NOT because [physics colleague] is incompetent (FAR from it!) but becuase the carefully cultivated culture (note the alliteration) of the course will change drastically and the preparation students have heard about from former students and the institutions to which our student transfer will not be gained now. I also don’t understand [the Chair’s] sudden and dramatic dismissal of researched-based methods when in the past [the Chair] has certainly encouraged their use.

[The Chair’s] behavior has changed noticeably over the past few years, and I have not been the only one to notice this. The behavior has become more dictatorial, more unilateral in decision making across the disciplines in the department, more dismissive of the input of colleagues, more bullying in nature, and more assertive of the superiority of [the Chair’s] own discipline as being the paragon of pedagogical excellence. I cannot, and will not, speculate on why [the Chair’s] behavior has changed, but I and one other person can trace at least most of the changes back to the death of a department member a few years ago after a rather long period of declining health.

This is Joe again. Well that’s about it I guess; I hope you enjoyed it. The ending is not as conclusive as I would like it to be, but I plan to develop the characters more and continue the story as necessary. It could very well turn into a series of its own. Remember, this little story is probably fiction, but even in fiction there’s usually a grain of truth. I promise that the next post will return us to vectors and related things I’ve neglected recently.

As always, feedback is welcome.


Visualizing Eigenvectors with GlowScript

I owe this post entirely to my mathematics colleague Ethan Smith and his recent work on visualizing eigenvectors in a plane. This work is based in turn on the paper by Schoenfeld. I’ve taken the visualization to 3D with the help of GlowScript and Trinket.

I was instantly interested in this project becuase I’m looking for ways to bring concepts from linear algebra (e.g. matrices, eigenvalues, tensors, etc.) into introductory claculus-based physics courses. I want to do this in a way that fosters the need for modern computation and visualization of geometric properties.

The concept here is very simple. You create a three dimensional distribution of unit vectors. I chose a spherically symmetric distribution. There’s really no necessity to use unit vectors as far as I can see; it just makes the numbers easier to manage. You then operate on each of these vectors with a linear transformation, represented by a matrix multiplication. The transformed vectors are visualized with arrows with their tails at the tips of the unit vectors. The transformation’s eigenvectors are immediately visible by inspection as the vectors represented by arrows collinear with the arrows representing the original unit vectors.

Unfortunately, I can’t embed trinkets in this blog, so I’ll have to make do by providing a link. When you click the link below, the trinket will open in a new browser window. You should see the GlowScript/VPython code on the left and the visualization on the right. The original unit vectors are white and the transformed vectors are blue (arbitrary color choices). You’re looking for the blue arrows that are collinear with white arrows. Those are the transformation’s eigenvectors. You can experiment with changing the number of unit vectors. More importantly, you’re encouraged to experiment with different transformation to see the effects of different eigenvalues. (I will come back later and include a screenshot here.)

Click this link to open the eigenpictures trinket.

The default matrix has eigenvalues of 1, 2, and 1 and the visualization makes spotting the eigenvectors quite simple.

I think this could have applications in introductory physics for visualizing inertia tensors. They can be represented as ellipsoids whose axes are the eigenvalues. I look forward to exploring this idea, and I thank Ethan Smith for showing me this.

 


Matter & Interactions II, Weeks 13 and 14

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 \mathbf{v}\times\mathbf{B} 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 \hat{n}\,\mathrm{d}A rather than \mathrm{d}\mathbf{A} 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:

    \mathbf{E}\cdot\hat{n}\,\mathrm{d}A=\left\lVert\mathbf{E}\right\rVert\hat{E}\cdot\hat{n}\,\mathrm{d}A=\left\lVert\mathbf{E}\right\rVert\underbrace{\hat{E}\cdot\hat{n}}_1\,\mathrm{d}A=\left\lVert\mathbf{E}\right\rVert\mathrm{d}A

  • Similarly, I like an element of vector length, at least for electromagnetic theory, as \hat{t}\,\mathrm{d}\ell rather than \mathrm{d}\mathbf{\ell} (the \ell 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::

    \mathbf{B}\cdot\hat{t}\,\mathrm{d}\ell=\left\lVert\mathbf{B}\right\rVert\hat{B}\cdot\hat{t}\,\mathrm{d}\ell=\left\lVert\mathbf{B}\right\rVert\underbrace{\hat{B}\cdot\hat{t}}_1\,\mathrm{d}\ell=\left\lVert\mathbf{B}\right\rVert\mathrm{d}\ell

  • 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.


Vector Formalism in Introductory Physics I: Taking the Magnitude of Both Sides

TL;DR: I don’t like the way vectors are presented in calculus-based and algebra-based introductory physics. I think a more formal approach is warranted. This post addresses the problem of taking the magnitude of both sides of simple vector equations. If you want the details, read on.

This is the first post in a new series in which I will present a more formal approach to vectors in introductory physics. It will not have the same flavor as my recently begun series on angular quantities; that series serves a rather different purpose. However, there may be some slight overlap between the two series if it is appropriate.

I am also using this series to commit to paper (screen, really) some thoughts and ideas I have had for some time with the hope of turning them into papers for submission to The Physics Teacher. I’d appreciate any feedback on how useful this may be to the community.

To begin with, I want to address issues in the algebraic manipulation of vectors with an emphasis on coordinate-free methods. I feel that in current introductory physics courses, vectors are not exploited to their full potential. Instead of learning coordinate-free methods, students almost always learn to manipulate coordinate representations of vectors in orthonormal, Cartesian coordinate systems and I think that is unfortunate because it doesn’t always convey the physics in a powerful way. Physics is independent of one’s choice of coordinate system, and I think students should learn to manipulate vectors in a similar way. 

Let’s begin by looking at a presumably simple vector equation:

a\mathbf{A} = -5\mathbf{A}

The object is to solve for a given \mathbf{A}. Don’t be fooled; it’s more difficult than it looks. In my experience, students invariably try to divide both sides by \mathbf{A} but of course this won’t work because vector division isn’t defined in Gibbsian vector analysis. Don’t let students get away with this if they try it! The reasons for not defining vector division will be the topic of a future post.

(UPDATE: Mathematics colleague Drew Lewis asked about solving this equation by combining like terms and factoring, leading to (a+5)=0 and then to a = -5. This is a perfectly valid way of solving the equation and it completely avoids the “division by a vector” issue. I want to call attention to that issue though, because when I show students this problem, they always (at least in my experience) try to solve it by dividing. Also, in future posts I will demonstrate how to solve other kinds of vector equations that must be solved by manipulating both dot products and cross products, each of which carries different geometric information, and I want to get students used to seeing and manipulating dot products. Thanks for asking Drew!)

One could simply say to “take the absolute value of both sides” like this:

\left| a\mathbf{A} \right| = \left| -5\mathbf{A}\right|

but this is problematic for two reasons. First, it destroys the sign on the righthand side. Second, a vector doesn’t have an absolute value because it’s not a number. Vectors have magnitude, not absolute value, which is an entirely different concept from that of absolute value and warrants separate consideration and a separate symbol.

We need to do something to each side to turn it into a scalar because we can divide by a scalar. Let’s try taking the dot product of both sides with the same vector, \mathbf{A}, and proceed as follows:

\begin{aligned} a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{A}\bullet\mathbf{A} && \text{dot both sides with the same vector} \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{A}\rVert^2 && \text{dot products become scalars} \\ \therefore a &= -5 && \text{solve} \end{aligned}

This is a better way to proceed. It’s formal, and indeed even pedantic, but I dare say it’s the best way to go if one wants to include dot products. Of course in this simple example, one can see the solution by inspection, but my goals here are to get students to stop thinking about the concept of dividing by a vector and to manipulate vectors algebraically without referring to a coordinate system.

Let’s now look at another example with a different vector on each side of the equation.

 a\mathbf{A} = -5\mathbf{B}

Once again the object is to solve for a given \mathbf{A} and \mathbf{B}. Note that solving for either \mathbf{A} or \mathbf{B} is obviously trivial so I won’t address it; it’s simply a matter of scalar division. Solving for a is more challenging because we must again suppress the urge to divide by a vector. I will show two possible solutions. Make sure you understand what’s being done in each step.

\begin{aligned} a\mathbf{A} &= -5\mathbf{B} && \text{given equation} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\mathbf{A} && \text{dot both sides with the same vector} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\left(\dfrac{-5}{\hphantom{-}a}\mathbf{B}\right) && \text{substitute from original equality} \\ a\lVert\mathbf{A}\rVert^2 &= \dfrac{25}{a}\lVert\mathbf{B}\rVert^2 && \text{dot products become scalars} \\ a^2 &= 25\dfrac{\lVert\mathbf{B}\rVert^2}{\lVert\mathbf{A}\rVert^2} && \text{rearrange} \\ \therefore a &= \pm 5\dfrac{\lVert\mathbf{B}\rVert}{\lVert\mathbf{A}\rVert} && \text{solve} \end{aligned}

We get two solutions, and they are geometrically opposite each other; that’s the physical implication of the signs. (I suppose we could argue over whether or not to just take the principal square root, but I don’t think we should do that here because it would throw away potentially useful geometric information.) We can find a cleaner solution that accounts for this. Consider the following solution which exploits the concepts of “factoring” a vector into a magnitude and a direction and the properties of the dot product.

\begin{aligned} a\mathbf{A} &= -5\mathbf{B} && \text{given equation} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\mathbf{A} && \text{dot both sides with \textbf{A}} \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{B}\rVert\widehat{\mathbf{B}}\bullet\lVert\mathbf{A}\rVert\widehat{\mathbf{A}} && \text{factor each vector into magnitude and direction}  \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{B}\rVert\lVert\mathbf{A}\rVert\,\widehat{\mathbf{B}}\bullet\widehat{\mathbf{A}} && \text{push magnitude through the dot product} \\ \therefore a &= -5\dfrac{\lVert\mathbf{B}\rVert}{\lVert\mathbf{A}\rVert}\,\widehat{\mathbf{B}}\bullet\widehat{\mathbf{A}} && \text{solve} \end{aligned}

See the geometry? It’s in the factor \widehat{\mathbf B}\bullet\widehat{\mathbf A}. If \mathbf{A} and \mathbf{B} are parallel, this factor is +1 and if they are antiparallel it is -1. Convince yourself that those are the only two options in this case. (HINT: Show that each vector’s direction is a scalar multiple of the other vector’s direction.) This solution won’t work if the two vectors aren’t collinear. If we’re solving for a then both vectors are assumed given and we know their relative geometry.

Let’s look at another example from first semester mechanics, Newton’s law of gravitation,

\mathbf{F} = G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left( -\widehat{\mathbf r}_{12}\right)

where \mathbf{r}_{12} = \mathbf{r}_1 - \mathbf{r}_2 and should be read as “the position of 1 relative to 2.” Let’s “take the magnitude of both sides” by first writing \mathbf{F} in terms of its magnitude and direction, dotting each side with a vector, and dividing both sides by the resulting common factor.

\begin{aligned} \lVert\mathbf{F}\rVert\left(-\widehat{\mathbf{r}}_{12}\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left( -\widehat{\mathbf{r}}_{12}\right) && \text{given equation} \\ \lVert\mathbf{F}\rVert\left(-\widehat{\mathbf{r}}_{12}\bullet\widehat{\mathbf{r}}_{12}\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left(-\widehat{\mathbf{r}}_{12}\bullet\widehat{\mathbf{r}}_{12}\right) && \text{dot both sides with the same vector} \\ \lVert\mathbf{F}\rVert\left(-\lVert\widehat{\mathbf{r}}_{12}\rVert^2\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left(-\lVert\widehat{\mathbf{r}}_{12}\rVert^2 \right) && \text{dot products become scalars} \\ \therefore \lVert\mathbf{F}\rVert &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2} && \text{divide both sides by the same scalar} \end{aligned}

Okay, this isn’t an Earth-shattering result becuase we knew in advance it has to be the answer, but my point is how we formally went about getting this answer. More specifically, the point is how we went about it without dividing by a vector.

Let’s now consider a final example from introductory electromagnetic theory, and this was the example that got me thinking about this entire process of “taking the magnitude of both sides” about a year ago. It’s the expression for the electric force experienced by a charged particle in the presence of an electric field (obviously not its own electric field).

\mathbf{F} = q\mathbf{E}

That one vector is a scalar multiple of another means the two must be collinear, so they must either be parallel or antiparallel. An issue here is that q is a signed quantity. Again, we have a choice about which vector with which to dot both sides; we could use \mathbf{F} or we could use \mathbf{E}. If we use the former, we will eventually need to take the square root of the square of a signed quantity, which may lead us astray. Therefore, I suggest using the latter.

\begin{aligned} \mathbf{F} &= q\mathbf{E} && \text{given equation} \\ \mathbf{F}\bullet\mathbf{E} &= q\mathbf{E}\bullet\mathbf{E} && \text{dot both sides with the same vector} \\ \lVert\mathbf{F}\rVert\widehat{\mathbf{F}}\bullet\lVert\mathbf{E}\rVert\widehat{\mathbf{E}} &= q\lVert\mathbf{E}\rVert^2 && \text{factor LHS, simplify RHS} \\ \lVert\mathbf{F}\rVert\lVert\mathbf{E}\rVert\,\widehat{\mathbf{F}}\bullet\widehat{\mathbf{E}} &= q\lVert\mathbf{E}\rVert^2 && \text{push the magnitude through the dot product} \\ \therefore \lVert\mathbf{F}\rVert &= \dfrac{q}{\widehat{\mathbf{F}}\bullet\widehat{\mathbf{E}}}\lVert\mathbf{E}\rVert && \text{solve} \end{aligned}

This may look overly complicated, but it’s quite logical, and it reflects goemetry. If q is negative, then the dot product will also be negative and the entire quantity will be positive. If q is positive, then the dot product will also be positive and again the entire quantity will be positive. Geometry rescues us again, as it should in physics. We can also rearrange this expression to solve for either q or \lVert\mathbf{E}\rVert with the sign of q properly accounted for by the dot product. By the way, \widehat{\mathbf{F}} and \widehat{\mathbf{E}} can’t be orthogonal becuase then their dot product would vanish and the above expression would blow up. Geometry and symmetry, particularly the latter, preclude this from happening.

In summary, “taking the magnitude of both sides” of a simple vector equation presents some challenges that are mitigated by exploiting geometry, something that is neglected in introductory calculus-based and algebra-based physics courses. I suggest we try to overcome this by showing students how to formally manipulate such equations. One advantage of doing this is students will see how vector algebra works in more detail than usual. Another advantage is that students will learn to exploit geometry in the absence of coordinate systems, which is one of the original purposes of using vectors after all.

Do you think this would make a good paper for The Physics Teacher? Feedback welcome!


Matter & Interactions II, Week 12

We’re hanging out in chapter 19 looking at the properties of capacitors in circuits.

In response to my (chemist) department chair’s accusation that I’m not rigorous enough in my teaching of “the scientific method” as it’s practiced in chemistry, I just had “the talk” about “THE” scientific method with the class and about how it doesn’t exist. I will never forget Dave McComas (IBEX) telling the audience at an invited session I organized at AAPT in Ontario (CA) that we MUST stop presenting “the scientific method” as it is too frequently presented in the textbooks because it simply does not reflect how science works. No one hypothesizes a scientific discovery. Once a prediction is made and experimentally (or observationally in the case of astronomy) verified, that’s not a prediction because the outcome is expected. Even if the prediction isn’t verified, one of the required known possible outcomes is that the prediction is wrong. There’s nothing surprising here. True discoveries happen when we find something we had no reason to expect to be there in the first place. The Higgs boson? Not a discovery, because it was predicted forty years or so ago and we only recently had the technology to test for its presence. I don’t think anyone honestly expected it to not be found, but I think many theoretical particle physicists (not so) secretly hoped it wouldn’t be found because then we would have actually learned something new (namely that the standard model has problems).

The “scientific method” simply doesn’t exist as a finite numbered sequence of steps whose ordering is the same from discipline to discipline. Textbooks need to stop presenting that way. Scientific methodology is more akin to a carousel upon which astronomers, chemists, physicists, geologists, or biologists (and all the others I didn’t specify) jump at different places. Observational astronomers simply don’t begin by “forming an hypothesis” as too many overly simplistic sources may indicate. Practitioners in different disciplines begin the scientific process at different places by the very nature of their disciplines and I don’t think there’s a way to overcome that.

Rather than a rote sequence of steps, scientific methodology should focus on validity through testability and falsifiability. I know there are some people who think that falsifiability has problems, and I acknowledge them. However, within the context of introductory science courses, testability and falsifiability together form a more accurate framework for how science actually works. This is the approach I have been taking for over a decade in my introductory astronomy course. It is not within my purview to decide what is and is not appropriate for other disciplines, like chemistry. My chemist colleagues can present scientific methodology as they see fit. I ask for the same respect in doing so within my disciplines (physics and astronomy).

I now consider “the scientific method” to have been adequately “covered” in my calculus-based physics course.

Feedback welcome as always.

 


Matter & Interactions II, Week 11

More with circuits, and this time capacitors, and the brilliantly simple description M&I provides for their behavior. In chapter 19, we see that traditional textbooks have misled students in a very serious way regarding the behavior of capacitors. Those “other” textbooks neglect fringe fields. Ultimately, and unfortunately, this means that capacitors should not work at all! The reason becomes obvious in chapter 19 of M&I. We see that in a circuit consisting of a charged capacitor and and a resistor, it’s the capacitor’s fringe field that initiates the redistribution of surface charge that, in turn, establishes the electric field inside the wire that drives the current. The fringe field plays the same role that a battery’s field plays in a circuit with a flashlight bulb and battery. It initiates the charge redistribution transient interval. As you may have already guessed, the capacitor’s fringe field is what stops the charging process for an (initially) uncharged capacitor in series with a battery. As the capacitor charges, its fringe field increases and counters the electric field of the redistributed surface charges, thus decreasing the net field with time. If we want functional circuits, we simply cannot neglect fringe fields.

Ultimately, the M&I model for circuits amounts to the reality that a circuit’s behavior is entirely due to surface charge redistributing itself along the circuit’s surface in such a way as to create a steady state or a quasisteady state. It’s just that simple. You don’t need potential difference. You don’t need resistance. You don’t need Ohm’s law. You only need charged particles and electric fields.

One thing keeps bothering me though. Consider one flashlight bulb in series with a battery. The circuit draws a certain current i_1 for example. Now, consider adding nothing but a second, identical flashlight bulb in parallel with the first one. Each bulb’s brightness should be very nearly the same as that of the original bulb. The parallel circuit draws twice the current of the original lone bulb i_2 = 2i_1 but that doubled current is divided equally between the two parallel flashlight bulbs. That’s all perfectly logical, and I can correctly derive this result algebraically. I end up with a factor of 2 multiplying the product of either bulb’s fliament’s electron number density, cross sectional area, and electron mobility.

i_2 \propto 2nAu

My uneasiness is over the quantity to which we should assign the factor of 2. A desktop experiment in chapter 18 that establishes we get a greater current in a wire when the wire’s cross sectional area increases. Good. However, in putting two bulbs in parallel is it really obvious that the effective cross sectional area of the entire circuit has doubled? It’s not so obvious to me because the cross sectional area can possibly only double by virtue of adding an identical flashlight bulb in parallel with the first one. Unlike the experiment I mentioned, nothing about the wires in the circuit change. Adding a second bulb surely doesn’t change the wire’s mobile electron number density; that’s silly. Adding a second bulb also surely doesn’t change the wire’s electron mobility; that’s equally silly. Well, that leaves the cross sectional area to which we could assign the factor of 2, but it’s not obvious to me that this is so obvious. One student pointed out that the factor of 2 probably shouldn’t be thought of as “assigned to” any particular variable but rather to the quantity nAu as a whole. This immediately reminded me of the relativistic expression for a particle’s momentum \vec{p} = \gamma m \vec{v} where, despite stubborn authors who refuse to actually read Einstein’s work, the \gamma applies to the quantity as a whole and not merely to the mass.

So, my question boils down to whether or not there is an obvious way to “assign” the factor of 2 to the cross sectional area. I welcome comments, discussion, and feedback.