They Seemed to Like It

What? The final exam I set on Friday. I spent a lot of time trying to get this final exam right. The problem is that I tend to decide at some point that I want to set an “interesting” exam, and then this usually ends up being more work for me than for the students, since I not only have to think of the questions and make them the right level of difficulty (made harder by being open book and, in this case, a take home -well, take to where ever for 4 hours), but then endlessly debug to see that it has no mistakes (since I won’t be present to answer questions). This time, I spent a lot of time on units, since I wanted to set an exam that kept all the factors of c and mu and epsilon present in all the relativistic notation and right down to the final unpacked Maxwell equations and all the quantities they computed in various examples. I decided to have them explore a little non-linear electrodynamics, since everything they’d seen was mostly linear. You start with the familiar (Maxwell) form of the equations:

[tex]
\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\ , \qquad \nabla\cdot\mathbf{B}=0\ , \\
\nabla\cdot \mathbf{D}=\rho\ , \qquad \nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}\ ,
[/tex]

where [tex]\mathbf{D}[/tex] and [tex]\mathbf{H}[/tex] are related to [tex]\mathbf{E}[/tex] and [tex]\mathbf{B}[/tex] using the “constitutive equations”:
[tex]
\mathbf{D}=\mathbf{D}(\mathbf{E},\mathbf{B})\ , \qquad \mathbf{H}=\mathbf{H}(\mathbf{E},\mathbf{B})\ ,
[/tex]
in order to fully define the electric and magnetic fields. Those constitutive relations are usually linear in most stuff we’ve met. I thought it would be fun to play with something that goes beyond the usual setup generated by the Maxwell Lagrangian density [tex]
{\cal L}_{\rm M}=\frac{1}{2\mu}(E^2/c^2-B^2)[/tex] and explore the system that comes from this one:

[tex] {\cal L}=\frac{b^2}{\mu}\left(1-\sqrt{1-\frac{E^2/c^2-B^2}{b^2}-\frac{(\mathbf{E}\cdot\mathbf{B})^2}{b^4c^2}}\right)\ ,[/tex] where [tex]b[/tex] is a constant. This system (actually called the Born-Infeld electrodynamics) gives non-linear constitutive equations, but reduces to Maxwell for small fields, and it has a wealth of interesting properties, which the students explored in the exam. (It has actually become rather important in the context of string theory, but that was not the prime motivator here – it is just a rather splendid model overall.) I was tempted to make the entire exam about the BI system and examine other things like dispersion, properties of BI wave propagation in waveguides, etc., but had to cut things short due to lack of time, and also for variety put in a fun radiation problem (in ordinary electrodynamics).

This might seem like weird torture since, you might say, we know that Maxwell is the right physics and Born Infeld is not. But that’s not the point. A great way of demonstrating your understanding of a physical system is to study and compare and contrast alternatives to it. In that way, often you can appreciate a system’s properties more. Maybe properties of the Maxwell electrodynamics are more firmly cemented in their minds now? Time will perhaps tell…

There was a concern that they might just crash and burn under the unfamiliar- the worry that somehow students don’t want “interesting” exams but rather straightforward ones with no new ideas to discover while doing the exam. But I trusted this group. I’ve now seen from the class’ solutions that they seemed to get it, and it looks like it will be enjoyable to grade. Must start on that very soon…

-cvj