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Re: [Phys-l] drop a metal cylinder through a solenoid



On 03/27/2012 01:36 PM, Carl Mungan wrote:
It's a lovely and very clear explanation!

:-)

Perhaps my only remaining question would be: How would you frame the
explanation in the lab frame instead?

That's a good question. Interesting and important. It is
always good to look at a problem more than one way.

This problem is about ten times harder and trickier in the lab
frame, but it is doable. I strongly recommend doing it in the
comoving frame first. We've already done that, so let's have
a crack at the lab-frame version.

In that frame we're maintaining
a charge separation across a finite gap d by essentially the Hall
effect.

I'm not convinced that "Hall effect" is exactly the right
description. Ordinarily the Hall effect is proportional to
current, and there's no actual current in the lab frame,
given that the bob is electrically neutral overall. Also,
the sign of the Hall effect is determined by the sign of
the mobile carriers, whereas the effect we are considering
is not. That is, you could make the bob out of a P-type
or N-type semiconductor and get the same voltage.

Can I think of that like a capacitor? If so, how do I get
away with using the vacuum permittivity?

As I see it, the key physics issue is to understand the mechanisms
that maintain equilibrium inside the bob ... and the key pedagogical
issue is to understand why that is not necessarily obvious.

This may help with both issues: This is something I learned in
school. I remember the exact moment. Dave Middlebrook had invited
Ralph Morrison to come in as guest lecturer. (Yes, *that* Ralph
Morrison.) He said
"When in doubt, go back to the Maxwell equations. They're
always right. All these other things like Ohm's law and
Kirchhoff's laws ... those are just approximations, and
they're not always right."

To me this is the essence of good teaching, good learning, and good
thinking: It is important to keep track of what's fundamental and
what's not ... or at least what's reliable and what's not.

Let me now add a corollary to Morrison's advice: When in doubt,
go back to the Lorentz force law. That's always right.

Note that the Maxwell equations are not very useful without the Lorentz
force law.

In equilibrium, the q E term must balance the q v × B term in the
Lorentz force law. In the lab frame, this produces a nonzero E
field inside the metal ... but q E is not the total force.

Can I think of that like a capacitor?

Yes. I didn't understand how that was possible until just now,
but the answer is yes.

If so, how do I get
away with using the vacuum permittivity?

That's an excellent question. It's a tricky problem, in the sense
that I can think of about ten ways of getting the wrong answer.

To get the right answer, let's apply Morrison's advice again.

Back in the bad old days, people would sometimes write the Maxwell
equations in terms of four fields: E, D, B, and H ... where D is
related to E by the permittivity of the material. However, as
emphasized by Feynman and others, that is not the fundamentally
correct way to do it. If you look at things microscopically, E
and B are the only fields ... and є0 is the *only* electromagnetic
constant that appears in the Maxwell equations:
http://www.av8n.com/physics/maxwell-ga.htm#eq-max-v

The macroscopic permittivity of the material can be explained by
charges moving around at the microscopic level ... in accordance
with the microscopic Maxwell equations that involve only є0 and
do not involve properties of the material. If you account for
*all* of the charges (including the internal charges) there is
no D, just E.

It is common knowledge that in a capacitor *at rest* in the
lab frame, a dielectric material changes the charge/voltage
relationship, because screening charges move around inside the
capacitor. If you look only at the charge on the plates, you
get more charge per unit voltage.

However, in the situation we are discussing, the field inside
the material is not screened. The Lorentz force law has one
term that tells the charges to move in a direction that would
result in screening ... but it has another term tells them just
the opposite.

That is seriously tricky, but interesting.

I'm now skipping part of the analysis. There is another part
that accounts for the energy of the charges. (Hint: This
involves the forces of constraint that hold the charges inside
the bob. Otherwise it would be impossible for the magnetic
field to influence the energy of the charges.)

======================

As I see it, there are two high-level take-home messages:

1) Physics is simple when analyzed locally.

Misner/Thorne/Wheeler is fond of emphasizing this point.

As a corollary, in 99% of all situations, you are better off
analyzing things in the instantaneously comoving frame. That's
certainly true for our pendulum.

2) It pays to keep track of what's fundamental and what's not.

I don't want to overstate that. I'm not saying that fundamental
is always good, or that approximations are always bad. Indeed,
conventional engineering approximations exist because they are
helpful in a wide range of situations.

What I am saying is that in a tricky situation, you need to be
able to switch gears. You need to switch off the seductive
simplifications and go back to basics.

In some sense, the pendulum problem is super-easy: If you set
it up the right way, you can solve it using nothing more than
high-school physics. It takes several steps, but each step
is easy. It's easy in the sense that walking along a narrow
parapet is easy ... so long as you stick to the righteous path.

On the other hand, the problem is hard in the sense that there
are lots of not-so-good ways to set it up. There are lots of
ways to fall off the parapet.

=====================

The "parapet walking" metaphor has pedagogical implications.

It's one thing to tell a student "don't fall off" but it's
another thing entirely to make constructive suggestions as
to /how/ to stay on the parapet, and how to recognize a
potential misstep before it's too late.

We have already seen one constructive suggestion: Work the
problem more than one way. If the alleged answer in the comoving
frame differs from the alleged answer in the lab frame, it means
you need to re-do the problem, probably several times, until you
find the bug(s).

Another bit of constructive advice is to do little sanity checks
along the way.
-- For example, you know the effective capacitance in the lab
frame cannot depend on the conventional dielectric constant
of the material, because a metal has /infinite/ dielectric
constant, and we know that the bob is not going to develop
infinite charge per unit voltage.
-- Also, in my notes in recent days, you saw I checked the
scaling several times. Not only did I check the dimensions
(which is a simple way of checking /some/ of the scaling)
but I made several additional "dimensionless scaling"
checks: Scaling with respect to the frequency ω, scaling
with respect to the length L, scaling with respect to the
resistivity ρ, et cetera.
-- You can also check your methods by finding some simple
analogous system and practicing on that. In this case,
you can practice on a lumped RC circuit. (The hard part
is figuring out what's analogous and what's not.)

There's probably a lot more to this story ... stuff that we
should be able to explain to students but can't, because we
cannot quite put our finger on it. Physics isn't so much
different from riding a bike, in the sense that it's easier
to do it than to explain it. A lot of it is subconscious.