A couple days before I started teaching recitation sessions for Physics 208 (the E&M half of calc-based intro physics) this summer, I found out that in fact I was not teaching the second summer session, I’m teaching for the second half of the full summer session. Turns out there is a difference! For some bizarre reason, this means that the summer Physics 208 class is actually taught by one professor for the first half of the summer and another professor for the second half. Each 208 section switches TAs midstream as well - I’m therefore kind of a relief pitcher, I guess. Why is it set up this way? I have no idea, but if I had to guess it’s so professors have time off during the summer to go to more conferences in Hawaii and get more research done without interruption. Oh well, makes no difference to me, other than hoping the previous TA hasn’t been sloppy in his teaching. (It in a “he” in this case, though there are plenty of female TAs as well!)
This week we’re doing magnetic fields. They’re surprisingly weird. The fields encountered so far in introductory classes are gravitational and electric fields, in which force points along the field direction and a nice simple scalar potential exists. Neither of these are true with magnetic fields. The force on a particle with charge q in a magnetic field B is
We have a vector cross product involved (What’s a cross product? This.), and weirder still it involves the velocity v of the particle. A stationary particle feels no magnetic force, and a faster moving particle experiences a greater force. Since it’s a cross product the angle matters too. A particle moving parallel to the field experiences no force; a particle moving perpendicular to the field experiences the greatest force.
For a long time electricity and magnetism were treated as two completely different things before James Maxwell found a theory which could describe the relationship between the interactions. Einstein took this even farther and discovered that electricity and magnetism weren’t just related halves of the same theory, they are literally the same thing under a particular coordinate transformation. We don’t worry about the relativistic description in an intro class obviously, we’re just interested in communicating to students what magnetic fields do and where they come from. A little unusually, the question I always get asked is why is “B” used for the magnetic field instead of the more obvious “M”. The answer is that M is already taken by a quantity called magnetization.
The equation above is called the Lorentz force law, and theme and variations on that one equation make up the entire week’s worth of discussion. Let me quote the simple quiz I gave them (modified from one of the homework problems); working through it will help bring home the concepts.
A helium nucleus (charge +2e) moving horizontally from west to east with a speed of 1000.0 km/s experiences a magnetic force of 0.000500 nN vertically downward. What is the magnitude and direction of the weakest magnetic field required to produce this force? How could this same force originate from a stronger field?
Let’s take it from the top. The force from the cross product is going to have to be perpendicular to both the velocity and the field, so since the force is downward and the velocity is west-east, the field has to also be parallel to the ground. The cross product is a maximum where the velocity and the field are perpendicular, so the strongest force from that weak field will be when the field is pointing directly south. That’s the direction.
This reasoning also answers the second question: a stronger field parallel to the ground but not directly pointing south could produce the same force in the same direction as the weaker field pointing exactly in the right direction.
But what’s the magnitude? Well, we have (since we know v and B are perpendicular)
Solve for B.
Plugging in the values in the problem, I get around 1.56 Tesla. This is a pretty huge magnetic field, roughly comparable to that of an MRI.
That’s about as bare-bones of an introduction as you can get, but I hope it’s a good place to start for students who happen across this site. Next week is sources of magnetic field.
3 responses so far ↓
1 Uncle Al // Jul 3, 2008 at 10:15 am
A strong magnetic field causes vacuum dichroism, blurring photons and pair production. Alas for untenured experimental faculty, at least 10^12 teslas required in theory (re SGR 1806-20).
Classical gravitation can be written as curvature (metric; pseudo-Riemannian spacetime) or torsion (teleparallel; Weitzenböck spacetime). There is but one observed reality and both treatments share identical predictions. However… spacetime torsion transforms as Lorentz force in electrodynamics. Spacetime torsion has a disjoint excess of prediction that may be physical.
Lorentz force is demonstrably chiral - railguns to cyclotrons, A(X)B = -[B(X)A]. Spacetime torsion allows a chiral vacuum background - alsotestable (pdf). How much fun would that be if it happened? Somebody should look.
2 CCPhysicist // Jul 4, 2008 at 10:16 am
Hmmmm. Don’t forget H and A when talking about B (or D when talking about E), as well as P and M. I also seem to recall that Maxwell’s original treatise used triplets of capital letters for the components of the field, since the Gibbs notation for vectors was a circa 1900 invention, but no longer recall if he used ABC for “B”.
What fraction of the students correctly answered part c by giving a different direction in the horizontal plane rather than just “different direction”?
By the way, 1.5 T is big but easily achieved over large volumes (e.g. a 1 m radius cyclotron magnet) with normal magnets, not just in the small volume of an MRI, so this problem has good design features. Ditto for keeping v/c small. But why didn’t you use 0.5 pN or a simple power of 10 to at least minimally respect SI?
Also, FYI, it was Voigt who first noticed that Maxwell’s equations were invariant under the Lorentz transformation. Unfortunately, he did not quite make the connection to the general concept of invariance and/or was cowed by famous physicists and problems with the first Michelson experiment into not using it to predict “relativity” about 20 years before Einstein. It is historical details like this that illustrate how SR was a paradigm shift (giving primacy to Maxwell) and not an entirely new mathematical approach like GR was.
3 CCPhysicist // Jul 4, 2008 at 10:22 am
I was wondering how you could teach all of Physics 2 in only about 6 weeks.
Splitting lectures is unusual, but not unknown. I had that in a freshman-level organic class taught in a quarter system during the regular school year: 5 weeks of one guy, 5 weeks of the other.
Splitting TAs is bizarre, but might be motivated by the same economics of trying to make teaching and research money go further during the summer. Faculty usually take somewhere between 6 and 8 weeks (the max for most grants) of “summer salary” from a grant, often choosing the smaller number to put more money into student funding or other resources. That, and travel, can make it hard to get a good regular prof in the lecture hall in summer since research is such a high priority in a university.
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