Re: [Phys-L] Lenz's law and conservation of energy

[John Denker <jsd@av8n.com>]

On 04/03/2014 08:14 AM, Philip Keller wrote:
> A u-shaped circuit is closed by a bar that can slide across the
> rails. There is a magnetic field directed down into the plane of the
> rails.  I apply a constant force to drag the bar to the right.

OK.

> There are a number of ways to predict the direction of the resulting
> current.  One of them is to say that the increase in the enclosed
> flux due to the increased area of the loop must be opposed by the
> outward field caused by the resulting counter-clockwise current.

That sounds fine to me.

> Is that not an example of Lenz's law?

Clearly several key elements of the physics are the same here, 
the same as in classical instances of Lenz's law.  In this case 
we have a changing loop, and in the other case we have a changing 
field.  If you want to label these the same, that's OK with me.
If somebody else wants to emphasize the differences, that's also 
OK with me.  I can't get very excited about terminology at this 
level.  De gustibus non disputandum.

> And if the current were to flow in the direction opposite to that
> predicted by Lenz's law, would I not get a current that would help me
> to drag the bar?  Couldn't I then let go of the bar and let that
> induced current continue to accelerate the bar for me, thus producing
> free energy?

I am still skeptical of this kind of energy argument.  It strikes
me as a Kiplingesque just-so story.

The problem is, if you teach students to make this argument, they
will apply it to other situations where it gets the wrong answer.

As an obvious and important example of what I'm saying, suppose
we set up two masses fairly near each other in otherwise-empty
space.  I drag one mass toward the other at a steady rate.  So
far, this is very closely analogous to the wire-on-rails example.
So far, so good.

☠  Now we make the argument that if there were a force in the same 
☠  direction as the imposed motion, it would allow me to get energy
☠  "for free".  The system would be unstable.  This obviously cannot
☠  happen, so ............

The indented paragraph is completely wrong.  In reality there *is*
a force in the direction of motion.  In reality there *are* some
systems that are unstable.  Gravitational instability *does*
release energy.

Maybe it's obvious to you that the wire-on-rails system is stable,
but AFAICT you haven't proved this, much less explained it.

At the very least I strongly suggest this be labeled a /stability/
argument not a "conservation of energy" argument.  That's because
unstable systems conserve energy just like stable systems do.  In
particular, almost any distribution of dust in interstellar space
is unstable against the effects of gravity, unstable against 
condensation into stars and galaxies ... yet it conserves energy 
just fine.

If you want to teach your students how to make stability arguments,
that's 100% fine with me, but please let's do it right.  Start by
explaining that equilibrium ≠ stability ≠ damping ≠ conservation.
This stuff gets short shrift (or none at all) on the typical high-
school syllabus, but it is at least as interesting and at least as 
useful as a lot of stuff that does get emphasized.

On the other hand ..... you do not need the energy argument to
fully explain the wire-on-rails situation.  It suffices to know
that the resistivity is positive in Ohm's law.  In an ordinary
resistive chunk of metal, charges move so as to null out the 
electric field.

   This applies to electric charges in an electric field, but
   *not* to dust particles in a gravitational field.  The latter
   move so as to augment the field!  This is how you know that 
   there cannot possibly be any deep conservation of energy 
   principle involved here.  Energy arguments apply equally to
   electromagnetism and gravitation.

Again:  You do not need the energy argument to explain the wire-
on-rails situation.  My advice:
  a) Stay away from energy conservation arguments in this situation.
  b) If you want to make a stability argument, that's fine, but 
   only if you have already laid the proper groundwork.
  c) If you want to introduce the ideas of equilibrium, stability,
   damping, spontaneity, et cetera, Lenz's law is not the best
   introductory situation.  There are lots of simpler and clearer
   examples.  I generally start with weights taped to a bicycle 
   wheel:
      http://www.av8n.com/how/htm/equilib.html


On 04/03/2014 10:23 AM, Jeffrey Schnick wrote:

> > I think the arguments that have been put forth so far are:
> > 1) This is instability, not a violation of the conservation of energy.
> > 2) This would be a violation of the second law of thermodynamics, not a violation of conservation of energy.
> > 3) If this were a violation of conservation of energy, there would be no such thing as paramagnetism.
> > 4) Flux only penetrates superconducting materials to a certain skin depth, hence this would not be a conservation of energy violation.
> > 5) Lenz's law comes from Maxwell's equations, not the other way round.
> > 6) Lenz's law has nothing to do with conservation of energy.
> >
> > I don't buy any of these arguments.

Well, I don't buy any of the counterarguments.

Most importantly, the aforementioned list is incomplete.

7) If you teach students to make this argument, they will get 
 the right answer applying it to electrostatics ... but they 
 will get diametrically wrong answers applying the same argument
 to gravitation and various other systems.  Therefore there 
 cannot possibly be any deep principles involved here.  Unstable
 systems exist in nature.  Really they do.  Lots of them.  They 
 conserve energy.

> The fact that it would lead to instabilities does not mean that it
> would not lead to a violation of conservation of energy.

We agree that one can imagine a system that violates the first
law at the same time that it violates the second law, but who
cares?  The fact remains that there are plenty of unstable systems
in nature, systems that are unstable even though they violate
*neither* the first law nor the second law.
  -- Interstellar dust is an important example.
  -- If you want an example you can exhibit in class, start
   with a wooden, plastic, or non-magnetic metal tray, and
   lay down some small bar magnets on it, one by one, spacing
   them out to keep the peak local density as small as possible.
   Before long it will go unstable.

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

I am not making this stuff up.  There is a serious formal theory
that deals with dynamical systems, including the stability thereof.

If you know enough theory, there is a simple (albeit not particularly
elementary) proof that the electron distribution in a passive chunk 
of resistive material is stable.  You can write down a Lyapunov 
function for it.
  http://en.wikipedia.org/wiki/Lyapunov_function

Ohm's law describes a system that is /overdamped/ and 100% dissipative,
i.e. not described by the second law of motion.  It's Aristotelian 
dynamics: charges at rest remain at rest, and charges in motion tend 
to come to rest.  Another example of Aristotelian dynamics occurs 
in fluids when the Reynolds number is very low, e.g. for bacteria
"swimming" in water.  Humans swim using kinetic effects that absolutely 
do not work for bacteria.

My point remains that you cannot /assume/ that physical systems are
stable.  You have to prove it on a case-by-case basis.  Conservation
of energy doesn't prove it, because all systems conserve energy, yet
not all systems are stable.

If you have a Lyapunov function, that is a nifty way to prove
stability.  In an overdamped system, system energy is likely to 
be a Lyapunov function ... but in other systems it definitely
won't be.

> Argument 4 has not been fleshed out enough for me to be able to comment on it.

Nobody is relying on argument 4.  You've misstated it anyway.  The
skin effect phenomenon applies to ordinary non-super conductors 
such as copper and iron.  If you know about skin depth it serves 
as a reminder for the sign of Lenz's law.  It's not a derivation,
just a mnemonic.  OTOH if you don't know this, there's no harm 
done either way.

BTW you don't need to wait for every possible argument to be spelled
out in detail via email.  Hint:
  https://www.google.com/search?q=%22skin+depth%22
  https://en.wikipedia.org/wiki/Skin_effect