I love teaching undergraduate electromagnetism. It has such an elegance, logic, and completeness about it. It introduces such a host of powerful techniques and ideas to the student, taking them across the threshold into maturity in their physics studies: Once you’ve done electromagnetism, you don’t usually think about large chunks of Physics in quite the same way ever again.
Today saw me give one of my favourite lectures, in any subject. It’s always a thrill. Summarize all that has gone before in their studies of electromagnetism – Gauss’ Law for the electric field produced by charges, the statement of the absence of magnetic monopoles (the Law with No Name), Faraday’s Law for the electric field produced by changing magnetic fields (induction:- another really fun set of lectures there), and Ampère’s Law for the magnetic fields produced by a current density. Write them all down next to each other and stare at them. Realize that they are not internally consistent, in general, as Maxwell did (he was motivated slightly differently, but in an essentially equivalent way). The culprit is Ampère, and the problem is fixed by Maxwell’s realization of the existence of the displacement current term. Ah… symmetry. Changing electric fields produce magnetic fields. All is well. Do some nice examples to show how it all works in concrete terms.
The resulting beautiful and consistent set of equations sent a shiver down my spine when I first saw and appreciated them as an undergraduate. They still send a shiver down my spine, and I hope your spine shivers too. Maxwell’s equations:
[tex]
\begin{eqnarray}
\nabla\cdot \mathbf{E}\, &=&\,\frac{\rho}{\epsilon_0} \ ;\nonumber \\
\nabla\cdot \mathbf{B}\, &=&\, 0 \ ;\nonumber \\
\nabla\times\mathbf{E}\, &=&\, -\frac{\partial \mathbf{B}}{\partial t}\ ;\nonumber \\
\nabla\times\mathbf{B}\, &=&\, \mu_0\mathbf{J} +\mu_0\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ .\nonumber
\end{eqnarray}
[/tex]
After the shiver, a lovely warm feeling. From here to light, radiation, Relativity, and beyond…
Every time, I especially love giving this lecture. It never gets old.
-cvj