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*From*: John Denker <jsd@av8n.com>*Date*: Wed, 5 May 2021 11:05:18 -0700

On 5/5/21 4:00 AM, Carl Mungan wrote:

2. I always thought of permeability being to magnets what

permittivity is to dielectrics. But now instead permeability is being

made analogous to conductivity. Does permittivity likewise map onto

some kind of analog of conductivity?

All of the above. AFAICT there is not any deep issue here.

There are lots of different analogies. The existence of

one mapping does not rule out another mapping, or really

affect it in any way.

There are lots of situations where you can have series

and parallel combinations, which invite analogies to

Ohm's law. For example, the topic of /security/ involves

series and parallel combinations. Increasing the thickness

of your front door might add security (in series) but not

if the side door is standing open (in parallel).

We can really drive home this point with a physics example.

It's obvious that an electrical LC oscillator is deeply

analogous to a mechanical mass-on-a-spring oscillator. But

the question arises: what is "the" mapping:

a) L <=> mass

C <=> spring

q <=> position

b) C <=> mass

L <=> spring

φ <=> position

The answer is that either one works perfectly well. In the

language of classical mechanics, the two mappings differ by

a contact transformation.

Note that the Lagrangian is the same in either case (within

an unimportant minus sign). So the variational principle

δL = 0

is exactly the same in either case. Pick a variable (e.g. q

or φ) and turn the crank. Euler-Lagrange blah blah.

q and φ are the most elegant choices, but other choices for

the "position" variable are allowed also.

I really recommend doing this calculation. I know a lot of

physicists who vividly remember doing this the first time.

It's one of those "Wow, physics actually works" moments.

It's a rite of passage.

And by the way, q and φ are dynamically conjugate. This comes

in handy if you are labeling the axes in phase space ... or

writing down the Heisenberg uncertainty principle in electrical

units.

While we're on the topic of "the same equations have the same

solutions" and mechanical analogies: I know a lot of physicists

who have worked out the physics of a piano string using the

methods of /field theory/. Sum over modes and all that. This

another of those "rite of passage" moments.

**References**:**[Phys-L] change in inductance with iron core***From:*Carl Mungan <mungan@usna.edu>

**[Phys-L] magnetic circuits ... was: change in inductance with iron core***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] magnetic circuits ... was: change in inductance with iron core***From:*Carl Mungan <mungan@usna.edu>

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