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Fri, 07 Dec 2007

Freshman electromagnetism questions
As I haven't quite managed to mention here before, I have occasionally been sitting in on one of Penn's first-year physics classes, about electricity and magnetism. I took pretty much the same class myself during my freshman year of college, so all the material is quite familiar to me.

But, as I keep saying here, I do not understand physics very well, and I don't know much about it. And every time I go to a freshman physics lecture I come out feeling like I understand it less than I went in.

I've started writing down my questions in class, even though I don't really have anyone to ask them to. (I don't want to take up the professor's time, since she presumably has her hands full taking care of the paying customers.) When I ask people I know who claim to understand physics, they usually can't give me plausible answers.

Maybe I should mutter something here under my breath about how mathematicians and mathematics students are expected to have a better grasp on fundamental matters.

The last time this came up for me I was trying to understand the phenomenon of dissolving. Specifically, why does it usually happen that substances usually dissolve faster and more thoroughly in warmer solutions than in cooler solutions? I asked a whole bunch of people about this, up to and including a full professor of physical chemistry, and never got a decent answer.

The most common answer, in fact, was incredibly crappy: "the warm solution has higher entropy". This is a virtus dormitiva if ever there was one. There's a scene in a play by Molière in which a candidate for a medical degree is asked by the examiners why opium puts people to sleep. His answer, which is applauded by the examiners, is that it puts people to sleep because it has a virtus dormitiva. That is, a sleep-producing power. Saying that warm solutions dissolve things better than cold ones because they have more entropy is not much better than saying that it is because they have a virtus dormitiva.

The entropy is not a real thing; it is a reification of the power that warmer substances have to (among other things) dissolve solutes more effectively than cooler ones. Whether you ascribe a higher entropy to the warm solution, or a virtus dissolva to it, comes to the same thing, and explains nothing. I was somewhat disgusted that I kept getting this non-answer. (See my explanation of why we put salt on sidewalks when it snows to see what sort of answer I would have preferred. Probably there is some equally useless answer one could have given to that question in terms of entropy.)

(Perhaps my position will seem less crackpottish if I a make an analogy with the concept of "center of gravity". In mechanics, many physical properties can be most easily understood in terms of the center of gravity of some object. For example, the gravitational effect of small objects far apart from one another can be conveniently approximated by supposing that all the mass of each object is concentrated at its center of gravity. A force on an object can be conveniently treated mathematically as a component acting toward the center of gravity, which tends to change the object's linear velocity, and a component acting perpendicular to that, which tends to change its angular velocity. But nobody ever makes the mistake of supposing that the center of gravity has any objective reality in the physical universe. Everyone understands that it is merely a mathematical fiction. I am considering the possibility that energy should be understood to be a mathematical fiction in the same sort of way. From the little I know about physics and physicists, it seems to me that physicists do not think of energy in this way. But I am really not sure.)

Anyway, none of this philosophizing is what I was hoping to discuss in this article. Today I wrote up some of the questions I jotted down in freshman physics class.

What are the physical interpretations of μ₀ and ε₀, the magnetic permeability and electric permittivity of vacuum? Can these be directly measured? How?
Consider a simple circuit with a battery, a switch, and a capacitor. When the switch is closed, the battery will suck electrons out of one plate of the capacitor and pump them into the other plate, so the capacitor will charge up.
When we open the switch, the current will stop flowing, and the capacitor will stop charging up.
But why? Suppose the switch is between the capacitor and the positive terminal of the battery. Then the negative terminal is still connected to the capacitor even when the switch is open. Why doesn't the negative terminal of the battery continue to pump electrons into the capacitor, continuing to charge it up, although perhaps less than it would be if the switch were closed?
Any beam of light has a time-varying electric field, perpendicular to the direction that the light is travelling. If I shine a light on an electron, why doesn't the electron vibrate up and down in the varying electric field? Or does it?
[ Addendum 20080629: I figured out the answer to this one. ]
Suppose I take a beam of polarized light whose electric field is in the x direction. I split it in two, delay one of the beams by exactly half a wavelength, and merge it with the other beam. The electric fields are exactly out of phase and exactly cancel out. What happens? Where did the light go? What about conservation of energy?
Suppose I have two beams of light whose wavelengths are close but not exactly the same, say λ and (λ+dλ). I superimpose these. The electric fields will interfere, and sometimes will be in phase and sometimes out of phase. There will be regions where the electric field varies rapidly from the maximum to almost zero, of length on the order of dλ. If I look at the beam of light only over one of these brief intervals, it should look just like very high frequency light of wavelength dλ. But it doesn't. Or does it?
An electron in a varying magnetic field experiences an electromotive force. In particular, an electron near a wire that carries a varying current will move around as the current in the wire varies.
Now suppose we have one electron A in space near a wire. We will put a very small current into the wire for a moment; this causes electron A to move a little bit.
Let's suppose that the current in the wire is as small as it can be. In fact, let's imagine that the wire is carrying precisely one electron, which we'll call B. We can calculate the amount of current we can attribute to the wire just from B. (Current in amperes is just coulombs per second, and the charge on electron B is some number of coulombs.) Then we can calculate the force on A as a result of this minimal current, and the motion of A that results.
But we could also do the calculation another way ,by forgetting about the wire, and just saying that electron B is travelling through space, and exerts an electrostatic force on A, according to Coulomb's law. We could calculate the motion of A that results from this electrostatic force.
We ought to get the same answer both ways. But do we?
Suppose we have a beam of light that is travelling along the x axis, and the electric field is perpendicular to the x axis, say in the y direction. We learned in freshman physics how to calculate the vector quantity that represents the intensity of the electric field at every point on the x axis; that is, at every point of the form (x, 0, 0). But what is the electric field at (x, 1, 0)? How does the electric field vary throughout space? Presumably a beam of light of wavelength λ has a minimum diameter on the order of λ, but how how does the electric field vary as you move away from the core? Can you take two such minimum-diameter beams and overlap them partially?

I did ask #6 to the physics instructor, who is a full professor with a specialization in high energy theory; she did not know the answer.

[ Addendum 20090204: I eventually remembered that Noether's theorem has something to say about the necessity of the energy concept. ]

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