This is a very quick post addressing a frequently asked conceptual question. Maybe it my heightened awareness, but I’ve also seen this question asked a lot on various physics Q&A sites lately. It’s a question that gets to the heart of how vectors are often defined, loosely and incorrectly, in introductory physics. Here’s the question.
Given that current (either electron current or conventional current) has both magnitude and direction, why is it not, and indeed cannot be, defined as a vector quantity whereas the closely related quantity current density is defined as a vector quantity.
The answer, I think, lies in one single simple property of vectors that does not apply to current. What do your students say?
With little advanced notice, I was assigned a section of conceptual physics to teach this semester (because second semester astronomy was cancelled due to low enrollment and replaced with a section of conceptual physics with the same low enrollment..go figure). After teaching Matter & Interactions for over a decade and after much enlightenment I have completely changed my views on what the focus of introductory courses should be, I decided to modernize the content by beginning with special relativity and particle physics as presented in the corresponding chapters in Hewitt. I also decided not to present content with the traditional chapter-by-chapter approach and instead to tell a story, with the chapters from Hewitt to be used as reference reading. For special relativity, I gave the class the same handout from Arons that I use in my calculus-based course, but only for the historical information and for the superbly articulated operational definitions of simultaneity and measurement.
So after a non-traditional introduction, we then turned to more conventional material. I knew I wanted to take a systems approach a la Matter & Interactions, but I also felt weirdly obligated to do something with motion, and we did. However, this time, we used Vernier motion sensors and handheld LabQuests (dammit Vernier…you’re changing my attitude toward experimental physics!) and I dare say I’ve never seen a class have such fun with matching position and velocity curves! That by itself was a breakthrough for me.
But then, with only a few weeks left in the semester (this was a late start course), I had to decide how to bring chapters two through ten in Hewitt together coherently, and I decided to do something I’d thought about for years: introducing Noether’s theorem as the foundation for introductory physics. If you don’t know what Noether’s theorem is all about, start with this paper by Hill and Lederman (for an overview) and this paper by Hanc, Tuleja, and Hancova (for undergraduate intro level approach).
So I began by defining symmetry as it’s used in physics: something that remains unchanged when Nature allows some process to happen (that definition is of course subject to improvement). We discussed reflection symmetry, of which most students are already aware and then I demonstrated rotational symmetry with a nondescript blue beach ball. The ball looked the same regardless of how I rotated it. But then I said some really strange things (as I’m known to frequently do). I noted that the motion experiments we did previously could be done today, a different day, and should we expect the results to change just because we did the experiments on a Thursday compared to a Tuesday? Everyone agreed that we should expect the same results. I then noted that when we did the motion experiments, there were three groups, each working in a different part of the room, and each group doing the same experiment with the same instructions and with the same apparatus. Should we expect the results to vary just because the groups are working in different locations? Everyone agreed that this is nonsense. Finally, I noted that our class meets in a room with walls that face each of the four cardinal directions but we could just as well hold class in a building or in a room for which the corners, not the walls, face the four cardinal directions. Should we expect different results from our motion experiments in a room that’s rotated compared to our current room? Everyone once again agreed that that would be a silly expectation. Then I hit them with this: each of these three scenarios displays a symmetry of nature. Experiments are independent of the time, location, and classroom orientation. It’s that simple. Read that again because it really is just that simple. Read it once again to make sure you see how simple this is.
Okay, so what? Well, Noether’s theorem formalizes this into a simple statement:
For each symmetry, there exists a corresponding conserved quantity.
And there it is. Getting the same experimental results at different times means energy is conserved. Getting the same experimental results in different locations means momentum is conserved. Getting the same experimental results in differently oriented rooms means angular momentum is conserved. Ack! What do these words mean?
Then at this point, rather than focus on energy, momentum, and angular momentum, I focused on invariant, constant, and conserved. Beginning a course with special relativity inevitably introduces the concept of invariance (something being the same in different reference frames), but be careful! Invariant, constant, and conserved have different meanings. Constant simply means “having a value that doesn’t change.” Conserved means “having the same value before and after something happens, like an interaction.” I mentioned this in a previous post. There is still one piece missing, and that is the concept of “system.” I defined system to be “the part of the Universe that you conceptually isolate for study.”
Okay, so then we sat down in the middle of the room with four cue balls (purchased in a clearance sale for one dollar each at a local sporting goods store). I defined the system to be “two balls” and I rolled them toward each other. We’d previously discussed vectors (the language of the arrows as I call it) and everyone got the idea of a zero initial momentum (we’d previously defined momentum too). With very little effort on my part, everyone saw that for this system (both balls, oppositely directed initial momenta with equal magnitudes) had the same total momentum throughout the interaction (collision). Of course we ignored friction from the floor. Then I redefined the system to be just one of the cue balls, and everyone agreed that the new system’s momentum wasn’t conserved. BOOM! Newton’s second law appears! Then we rolled the cue balls into a bowling ball and looked at what momentum conservation looks for unequal masses. Great! Then I pulled out my rotating platform, rolled the bowling ball toward it, and we saw that angular momentum (“spinniness” for rotational angular momentum and “aroundness” for translational angular momentum) were conserved for certain choices of system just like ordinary (translational) momentum. I was asking various check questions all along, and I couldn’t stump the class. Next week, we’ll address energy conservation and with that, we will have stitched together all the ideas in Hewitt’s chapters two through ten with one single, but exceedingly powerful, foundational concept (this is a CONCEPTUAL course for crying out loud, right?) that exists in all aspects of contemporary physics. No inclined planes were necessary.
Not surprisingly, momentum, energy, and angular momentum are concepts that all predate Noether’s theorem. These quantities were well defined decades (centuries even) ago and can certainly be operationally defined independently of each other, and that’s basically how classical physics treats them. But here we see quite explicitly that these quantities are the “Big Three” for a reason, a deep underlying reason that forms the bedrock of all contemporary physics but one that students rarely ever hear of. There’s a reason Matter & Interactions focuses on the Momentum Principle, the Energy Principle, and the Angular Momentum Principle (okay, and the Fundamental Assumption of Statistical Mechanics, but whatever…). That reason is Noether’s theorem.
I intend to use this same approach next year if I’m assigned a section of conceptual physics (as I’ve requested), but here’s the thing. This could also be done in the introductory calculus-based course too! Noether’s theorem makes a superb preface to Matter & Interactions and to Six Ideas That Shaped Physics (Moore’s innovative text). And yes, it even gives us a reason to bring Lagrangians in to the intro course! Just do it!
The Maxwell equations contain everything mentioned in an introductory calculus-based electromagnetic theory course, and then some. They contain detailed mathematical structure and deep insight into electromagnetic fields. They are a magnificent playground for learning the various theorems of vector calculus, and applying these theorems to the fields of particles naturally leads to things like divergence, curl, flux, circulation, and the Dirac delta function, something no introductory course I know of incorporates (Matter & Interactions mentions it, as it does the differential vector operators). The Maxwell equations also make a great starting point for alternative ways of thinking about vector fields, like geometric algebra, differential forms and tensors. Of course the culmination of any introductory electromagnetic theory course is “solving for light” or more correctly, solving the Maxwell equations in free space and showing how the solution leads to light as a pulse of electromagnetic energy. If you start your mechanics course or electromagnetic theory course with special relativity, you can come full circle and demonstrate how the Maxwell equations are relativistically rigorous under a Lorentz transformation, bringing students back to the topic that prompted Einstein to think about relativity in the first place.
I’m saving the best motivation for last though. The Maxwell equations are the best place in introductory physics to let students practice with symmetry and duality. Mere visual inspection of the equations, especially when written for free space, leads to adding terms that make the equations fully symmetric under a duality transformation (see here and here) and predicting the existence of magnetic monopoles. As one of my students said just last week, “The symmetry of Maxwell’s equations is so beautiful that magnetic monopoles just have to exist!” I don’t know that that argument will hold up to current experimental tests, but it’s great to see students understand how symmetry leads to predictions that lead to experiments. That’s meaningful.
So here are some questions on the Maxwell equations, some of which I’ve used at the end of second semester calculus-based physics.
(a) Write Maxwell’s equations in integral form. Notation must be consistent and correct.
(b) For each integral equation, write a one sentence interpretation of that equation. Do not simply write a verbal translation of the equation; interpret the equation’s meaning. There is probably more than one correct interpretation for each equation.
(c) Transform each equation into a differential equation, showing each step clearly. Notation must be consistent and correct.
(d) For each differential equation, write a one sentence interpretation of that equation. Do not simply write a verbal translation of the equation; interpret the equation’s meaning. Again, there is probably more than one correct interpretation for each equation.
(e) For each differential or integral equation, give one specific application of that equation from this course. Do not cite trivial applications.
(f) Using symmetry arguments, rewrite each set of equations (the integral set and the differential set) to account for the existence of magnetic monopoles.
(g) Write each set of equations for free space. Comment on how the free space equations would look if magnetic monopoles are found.
These questions themselves could conceivably serve as an entire final exam. Give it a try!
This question is straight to the point. It emphasizes terminology and associated conceptual understanding.
Without using equations or numerical examples, describe the connections and differences among the following quantities or concepts: electric potential energy, electric potential, change in electric potential energy, change in electric potential, potential difference, and electromotive force.
Electromotive force and potential difference are especially confusing because they have the same unit. They both are related to circuits too, but they have very different origins. I like to tell students that keeping all these quantities straight in their minds is potentially confusing. It was for me when I was a student anyway. See how your students do with this question.
Superposition is a powerful principle in physics, especially in introductory electromagnetic theory. A charged particle’s electric field exists independently of the fields of any other particles. A thorough understanding of superposition can prevent the frequent misunderstanding that fields can somehow be “blocked,” which is subtly different from saying that the net field at a point can be zero. In other cases, superposition can help in calculating the net field of continuous sources. Superposition is closely related to symmetry in that both can be used to justify certain geometric properties of fields, as the following questions illustrate.
(a) Using superposition (no numbers, no mathematical symbols), explain why the electric field of a uniformly charged, infinitely large, planar charge distribution must be perpendicular to the plane. Using a similar argument, explain why the electric field’s magnitude must be independent of distance from the plane.
(b) Construct a similar line of reasoning for the electric field of an infinitely long, uniformly charged linear charged distribution.
The concept of solid angle may make the first question easier to tackle, but solid angle is rarely mentioned in the introductory course. I’ll have a question about this in a future post.
Symmetry is part of the foundation of contemporary physics, but it is seldom emphasized in introductory physics in proportion to its significance. There may be some value in discussing how symmetry applies to otherwise traditional introductory problems rather than just replicating numerical examples from a textbook (even a good textbook). These questions illustrate symmetry in electromagnetic theory, but could trivially be adapted to gravitational interactions in mechanics.
Assume space is isotropic. Using only symmetry, argue (no numbers, no equations, only words) that
(a) the electric force between two charged particles must lie along the line connecting the two particles.
(b) the electric field of a very large (so large that its size need not matter) uniformly charged disk must be perpendicular to the disk and must not vary in magnitude with respect to distance from the disk.
(c) the electric field of a particle must be radially toward (or away from) the particle and if it varies in magnitude, must only do so with respect to distance from the particle.
(d) the electric field of a very long (so long that its length need not matter) uniformly charged rod must be perpendicular to the rod and if it varies in magnitude, must only do so with respect to perpendicular distance from the rod.
Here’s a hint. A symmetry implies some transformation that leaves some property unchanged. In each case, think of a change (perhaps a rotation and/or translation) that leaves the system (in this case, a charge distribution) unchanged and then look at any consequences that follow.
As usual, let me know if you present these questions to your students. I’m always interested in the the results.
This post continues this series into second semester introductory calculus-based physics, usually electromagnetic theory. This question addresses basic DC circuits. In a traditional introductory e&m course, circuits are presented with so many idealizations that according to such treatments simple circuits shouldn’t work at all! Two of the most important idealizations are that potential differences along wires are neglected so there are effectively no wires, and that fringe fields associated with real capacitors are neglected so capacitors effectively don’t work. These same treatments are also based on the concepts of potential difference and resistance, which is fine for an engineering perspective. However, all of DC circuit analysis can be done more realistically by adopting a surface charge gradient model that emphasizes some deceptively fundamental physics: charged particles act in response to local electric fields. That’s the whole story really.
Here are some conceptual questions that may be quite difficult for students of the traditional approach to circuits.
(a) Two circuits have identical wires, identical batteries, and identical light bulbs. The only difference between the circuits is that one has two bulbs in series with the battery and the other has only one bulb in series with the battery. Why does the single bulb glow brighter than either of the two bulbs in the other circuit? Do not invoke the concept of resistance (traditional treatments say that adding a second identical bulb doubles the circuit’s effective resistance, but do not use this concept in your explanation).
(b) Two circuits have identical wires, identical batteries, and identical light bulbs. The only difference between the circuits is that one has two bulbs in parallel with the battery and the other has only one bulb in parallel (really in series I suppose) with the battery. Why does the single bulb glow with the same brightness as either of the two bulbs in the other circuit? Do not invoke the concept of resistance (traditional treatments say that adding a second identical bulb doubles the circuits effective resistance, but do not use this concept in your explanation).
(c) What effect on a circuit does adding two identical bulbs in series, compared to having just one bulb in series, have?
(d) What effect on a circuit does adding two identical bulbs in parallel, compared to having just one bulb, have?
(e) What effect on a circuit does adding two identical capacitors in series, compared to having just one capacitor in series, have?
(f) What effect on a circuit does adding two identical capacitors in parallel, compared to having just one capacitor, have?