Re: [Phys-L] Lenz's law

[John Denker <jsd@av8n.com>]

On 4/25/24 2:12 PM, Richard Tarara via Phys-l wrote:

> Some on this list, maybe all on this list, will say that asking WHY?
> takes you into metaphysics (philosophy) not physics.

That is 100% true as stated.

For that reason, the wise teacher will deflect questions about ``why
does this happen'' and replace them with a series of better questions:
 -- How do we know this must happen?
 -- How do we remember it?
 -- How is it related to other stuff we know?

In the present case, Lenz's law is tantamount to saying that there is
no minus sign in Ohm's law. In other words, if you push on a charge,
it diffuses in the direction of the force, not in the opposite
direction. By way of mnemonic, note that if the reverse were true, it
would violate the second law of thermodynamics, since a resistor would
produce useful energy out of nothing, rather than dissipating energy.

Lenz's law can be seen as a mathematical consequence of the Maxwell
equations and Ohm's law. It is useful as a mnemonic, to remind you of
the oppositeness inherent in the equation for the magnetic field of a
long straight wire.

If you are a masochist, you could work out the mathematics in terms of
cross products. You need one cross product to find the direction of
the induced current, and another to find the field produced by said
current. You have to be fastidious about the order of the factors,
because cross products are anitcommutative.

If you're tempted to do that, I suggest using a computer-algebra
system to check your work, to guard against dropped minus signs.

YMMV, but for me I find it better to use geometric algebra. Bivectors
and all that. I can teach somebody about bivectors and use them to
explain Lenz's law in less time than it would take to explain anything
in terms of cross products. And they would be vastly more able to
remember it. Bivectors are easier to visualize.

BTW the same goes for gyroscopic precession: I can teach somebody
bivectors from scratch, and use that to explain precession, more
easily than I could explain anything in terms of cross products, even
if they already (supposedly) know about cross products. And retention
is much better.

So here is how I explain Lenz's law:
 -- There is no minus sign in this Maxwell equation:
        Voltage equals flux dot
 -- There is no minus sign in Ohm's law.
 -- There is oppositeness inherent in the formula for the magnetic
  field of a long straight wire. In terms of bivectors it is
        F = (1/r) ∧ I
  where the factors must be in that order. That means that the edge
  of the bivector nearest the wire is oriented opposite to the wire.

This is worked out in detail, with diagrams, here:
  https://www.av8n.com/physics/straight-wire.htm#sec-lenz