Re: AC electricity

[Leigh Palmer <palmer@SFU.CA>]

At 6:39 PM -0500 1/18/01, David Bowman wrote:
> Regarding Ludwik's and leigh's comments:
>
> Leigh is not quite correct when he says: "In a simple resistor carrying a
> current the Poynting flux which accounts for the Joule heating effect
> comes in radially from infinity."

I'm taking an analytically simple geometry here, an infinite, straight
wire of resistance per unit length r, carrying a DC current i*. You may
approach this situation as nearly as you wish in practice, say with wire
wrapped around the Earth's equator, or even more nearly with gedanken
materials. I'm also taking the DC case, so of course the part about
predicting the future is nonsense, but the reification of the energy
flow is the most condemnable nonsense; that's my point.

David is correct in saying the flux does not extend to infinity in the
case of a finite circuit. One could imagine a superconducting hoop
carrying a persistent current I. After a long time the hoop is made to
go normal and has a constant resistance R. The persistent current will
decay exponentially in time, and Joule heat in the amount 1/2 LI^2 will
be dissipated, where L is the self-inductance of the hoop. If one
integrates the Poynting flux over a closed surface surrounding the
entire loop over the time taken for the process to go to completion,
that integral will be found to depend upon the geometry of the surface,
and it will diminish to zero as the surface goes to infinite distance
in all directions. (The integral represents the total electromagnetic
field energy that was external to the chosen surface at the time the
loop went normal.)

All of the claims made in the preceding paragraph reflect my confidence
in a computational method which has worked for ~100 years. Not a single
integral was actually carried out to verify those claims.

Leigh

* If the details matter, then by symmetry the Poynting vector is purely
radial. Explicitly, it is the cross product of a longitudinal electric
field vector and an azimuthal magnetic field pseudovector, resulting in
a purely radial Poynting vector. If one integrates the Poynting vector
over a concentric cylindrical surface one finds an inward flux per unit
length which is exactly equal to the expected Joule heating power per
unit length. The fact that this mathematical process yields the correct
result using a model which suggests that the energy flows inward from
infinity (and that such a process will do so in many, many other system
analyses) does not lend any substance to this picture. One should not
suggest that such a calculation can be used to explain Joule heating.