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

[John Denker <jsd@av8n.com>]

On 04/01/2014 09:58 AM, I wrote:
> As a general principle, it is better to emphasize
> stuff that is correct, rather than making a fuss over 
> stuff that is incorrect.  In that spirit, here's how 
> I understand the sign that appears in Lenz's law:
> It is what it is, because of certain positive and 
> negative signs that appear in the Maxwell equations 
> and in Ohm's law.
>
> If you want a mnemonic, think of a perfect conductor,
> which excludes and/or traps magnetic flux entirely.
> (This is not to be confused with a /super/ conductor
> which actively expels flux, via the Meissner effect.)
> A less-than-perfect conductor excludes and/or traps
> flux temporarily.  Also, the existence of the term
> "skin depth" tells you the sign of the effect.

Here's an more detailed derivation, based directly on 
the fundamental laws:

Start with a transformer.  For simplicity, assume a 
1:1 turns ratio.  Imagine the two coils are loosely
coupled, so this is a very imperfect transformer,
i.e. the mutual inductance is not as large as the
self-inductance. 

We drive the primary with a current, with a sinusoidal
waveform.  Let the secondary be open-circuited at this 
stage of the game.

There will be some voltage across the primary, which
we can calculate based on the primary current and the
self-inductance.  So far so good.

Now .... short-circuit the secondary.  This means that
the secondary voltage is now smaller than it was in
the open-circuit case.  This in turn tells us that 
there is less flux_dot threading the secondary.  This 
means that the induced current in the secondary is 
oriented /opposite/ to whatever flux the primary is 
trying to push through the secondary.

That's the basic outline of the argument.  Let's 
now look into some of the details.  We have used 
one of the Maxwell equations directly, in the form 
   flux_dot = voltage

We have also implicitly used the /stability/ property
of Ohm's law.  This came in when we assumed that a
short circuit resulted in decreased voltage.

Proving stability for Ohm's law is more than I have
time to deal with at the moment.  Suffice it to say, 
for now, that it has got nothing to do with conservation
of energy.  It has more to do with entropy.

A backwards sign in Ohm's law V = IR would lead to 
negative dissipation in the power law P = I^2 R.
This does *not* violate conservation of energy, in
the same way that positive dissipation does not.

Also note that in carefully-engineered systems you can
have a negative resistance at certain frequencies:
  https://www.google.com/search?q=%22negative+resistance+reflection+amplifier%22

In any case:  If you remember nothing else from this
discussion, always keep in mind that
  equilibrium ≠ stability ≠ conservation of energy

Also
  energy ≠ entropy