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.


Conceptual Understanding in Introductory Physics XVIII

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!