Matter & Interactions II, Week 7

This week, I was away at the winter AAPT meeting in Atlanta. Students began working on the experiments from chapter 17, which serve to introduce magnetic fields.

I want to emphasize some really cool things about the mathematical expression for a particle’s magnetic field:

\vec{B}_{\mbox{\tiny particle}}=\dfrac{\mu_o}{4\pi}\dfrac{Q\vec{v}\times\hat{r}}{\lVert\vec{r}\rVert^2}

This is really a single particle form of the Biot-Savart law. I’m going to morph it into something really interesting. I’m going to make use of the fact that c^2 = \frac{1}{\mu_o\epsilon_o}, which I assert to students will be derived in a later chapter.

\vec{B}_{\mbox{\tiny particle}}=\dfrac{\mu_o}{4\pi}\dfrac{Q\vec{v}\times\hat{r}}{\lVert\vec{r}\rVert^2}

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{4\pi\epsilon_oc^2}\dfrac{Q}{\lVert\vec{r}\rVert^2}\vec{v}\times\hat{r}

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{c^2}\vec{v}\times\left(\dfrac{1}{4\pi\epsilon_o}\dfrac{Q}{\lVert\vec{r}\rVert^2}\hat{r}\right)

\vec{B}_{\mbox{\tiny particle}}=\dfrac{1}{c^2}\vec{v}\times\vec{E}_{\mbox{\tiny particle}}

THIS IS AMAZING! This demonstrates that this new thing called magnetic field is kind of like a velocity dependent electric field. That’s an oversimplification, but it hints that something deep is revealing itself here. Velocity connotes reference frame, and we see a big hint here that magnetic field depends on one’s reference frame. This is foreshadowing special relativity! We can show something else with one more slight rearrangement.

c\vec{B}_{\mbox{\tiny particle}}=\dfrac{\vec{v}}{c}\times\vec{E}_{\mbox{\tiny particle}}

This means that if we express velocity in fractions of c, then the quantity c\vec{B} has the same dimensions as \vec{E} and can thus be expressed in the same unit as electric field! This conceptualization allows for some beautiful symmetry to show itself later on when we get to the Maxwell equations. In some ways, electric fields and magnetic fields are interchangeable. Again, this is a hint of some underlying unification of the two, the electromagnetic field tensor, which I’m working hard to find a way to introduce into the introductory course. If students can understand simple Lorentz transformations, then they should be able to understand how the electromagnetic field tensor transforms from one frame to another within the framework of special relativity and we can show some beautiful physics. I realize I’m in the minority when it comes to something like this becuase we tend to think of our students as not being mathematically prepared. I’ve come to realize that perhaps…just perhaps…that is our perception only becuase we aren’t giving them the best mathematical foundations upon which to prepare for physics. Maybe it’s our fault. Maybe.

Anyway, these ruminations are things I want, and hope, students see on their own but all too often I find they, at least in my case, have difficulty even engaging at a minimal level. I struggle with this, and like to think and hope that maybe it’s because they don’t see the beauty. That’s why I nudge them in these new and different directions. Like I said above, it may very well be our fault.

Feedback is welcome as always.

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