Displacement Current within Good Conductors: Gone MIA but why?

bob. try these two links and let’s discuss



(the following is from an email to Constant314 who is editor of the Wikipedia entry for “Speed of Electricity”)

I believe we’ll end up having to jump at least three hurdles before all is said and done …

1.) We really need to study these books in detail and that means more time is needed for me (at least) because I have other things to do. I know you do as well.

2.) We need to come to terms with the reasons for the different approaches, at least in the cases Hayt vs. Jackson

  • I’ve glanced at Hayt and displacement current is ignored entirely 1/3 of the way down p359. This does not make it zero. He means tiny in relation to conduction. I don’t see discussion on this point afterwards. I think this omission leads Hayt into making incorrect statements at top of p360. I believe I know why he did and and how it’s gone for so long without correction. I explain this in the second to last paragraph at bottom of this email.
  • Jackson has a similar treatment and statements about *most* of the energy propagation normal to surface. He then includes a footnote about the ignored displacement term. I’ve attached a relevant page with footnote at bottom
  • Hayt makes an important point at top of p358 about the “loss tangent”. He moves on here to invoke equations and constants that I need to guess at. I need the relevant pages to know that I am making good assumptions
  • Both Jackson and Hayt are being (using your word) “terse” with respect to clarity in relation to the importance that skin depth plays in their arguments. For instance, at this point on Monday morning I ran out of time and quite can’t figure out if Jackson’s treatment later in the book involves *only* thin skin depth relative to cross sectional dimensions of the wire. I need to walk away and come back to it. If he is treating only a case of small skin depth relative to wire radius then I really need to get my hands dirty to apply his equations to the opposite case. It is that opposite case which is relevant for electricity flowing in most wires of interest to most people reading this wiki entry. Yes that’s my opinion and I hope you agree. Case in point, even if we are talking high frequencies, thin wire is often chosen (but not always!) such that its radius is small in relation to skin depth. It think it will be necessary to make this distinction in the wiki section eventually.

3.) We need to agree on what I think will end up being *the* crucial point: The topic is and always has been the “Speed of electromagnetic waves in a good conductor”. That should never have been interpreted as meaning “Speed of electromagnetic waves in a good conductor only for the component of largest magnitude of energy transmission”. It is an enormous distinction and I believe this distinction is what resulted in (my opinion) careless statements at top of Hayt p360.

Maybe #3 should be #1 on the list, because I think we need to start here first. The section is entitled “Speed of electromagnetic waves in a good conductor”. What follows should be an explanation starting with which E&M waves can and do propagate within a conductor. For my purposes as a physics guy, I want to know what the speed is of E&M field propagation in the axial direction. I want to know that first and foremost, because I want to know primarily how an electric field *within the interior* can fluctuate and propagate *within the interior* so quickly from one end of a wire to another. As it is E&M propagation it does not need to carry much energy to be able to sustain itself without attenuating to zero at the other end of the wire. Dipole motions everywhere they exist are sustaining it here (NOT at the exact center however for unpolarized waves. there shouldn’t be any problem posed by that point). Its Poynting vector does not need to be significant in relation to the normal flow. Rather, it needs to carry phase information and yes, some amount of field energy, from one location along the wire to another quickly. Is that interior E&M wave in phase with axially propagating waves elsewhere? I want to know this primarily. That would mean that for a 60 hz signal, fields in the interior must involve waves of very long wavelength moving at a speed close to that of light.

Can a conducting medium sustain such a wave?

If I’m reading Hayt correctly he is saying “No”. That is incorrect. Please read on …

Early on in our discussion this past weekend I mentioned my surprise about the magnitude of the magnetic field. Jackson makes it clear that almost all the energy within the interior of the conducting medium is magnetic. It should be very clear now to both of us that this large H is responsible for the most significant component of *energy flow*. Solve for the Poynting vector inside a conductor and you see it is *almost* exactly normal to the surface. That is a statement about just how large the magnitude of energy flow is relative to the axial direction. This is not proof that axial energy flow can be discarded! Rather, what it implies is mathematical proof that by far *most* of the energy flow is normal to the surface. By far most, but not all. Fine. I think even Hayt is fine with this point.

Please look at my Jackson page I’ve attached. At bottom is a footnote clarifying that inside the conductor there’s a small electric field *normal* to the surface. Note the imaginary symbol i here. This is a fluctuating electric field in the direction normal to the surface. It is small as expected. But how fast is its phase information propagating within the wire? We know it’s frequency, but what is its wavelength?

Except for the following two paragraphs I need to stop here for three important reasons. First, I/we really need to know if Jackson is considering a case of small skin depth relative to wire radius. Second, although terse, Jackson doesn’t hesitate to remind that we are making approximations about the direction of the fields right at the surface. We really need to know to what degree other components of the fields are being ignored for this treatment. Third, when Jackson treats energy propagation in a good conductor for lower frequencies he also ignores terms related to displacement current. However he does not say these terms are zero. I want to find out why he didn’t choose to come back to this point. Maybe he did and I missed it. Maybe that’s the entire point of the footnote attached.

Finally two points to be leave with for now …

I believe Hayt and EEs simply just DO NOT CARE about the tiny energy flow in the axial direction. You made a significant point when you wrote “There is a well developed theory that connects the field values in the dielectric with the field values in the conductor and it accounts for the transmission lines frequency dependence, time domain behavior, delay, dispersion, conductor loss, power drawn, power delivered, and skin effect.” I’ve been thinking about this, and I think it all makes sense now. These are the things EEs and perhaps 99% of others are going to care about most. Hayt’s treatment *does* get energy flow right. It *does* correctly assume the axial flow of energy is negligible relative to the normal flow. His incorrect statements at the top of p360 simply do not matter at all with regard to how well all these things mentioned are calculated (I mean the things within the quotes above). It is absolutely true that energy transmission due to axial displacement current can be *almost* entirely ignored when figuring those things out. So who cares?! Well I do, because I don’t want to know about relative energy transfer. I want to know about the various speeds and directions of E&M waves within a conductor. With regard to the wiki section in question, I do not care about the relative magnitudes of the respective E&M energy flows.

The last point to be made here is that Jackson mentions something that is a big hint and should guide me when I get time to jump back into all this. He mentions that H and E are out of phase inside a good conductor with a phase angle and wave number magnitude that do *not* depend on penetration distance into the conductor. This screams to me that somehow the fields related to conduction current inside the conductor are varying in the axial direction “in step” with those varying outside. I am back to questioning how the *total* current inside can maintain a constant phase relationship with the voltage. How can this be accounted for if a 60 hz signal is penetrating normal to the surface at 3.2 m/s? How can there be no lag outer to inner along the axial direction without the existence of an axial component propagating *inside* the conductor at near light speed? What else can maintain such an E&M profile that keeps “in step” with the outside? The questions are mainly for myself, because it’s quite possible Jackson makes assumptions that I missed here. Maybe he means the profile to be just within a very tiny distance into the conductor from the outer surface. So more time is needed for me to better grasp this.

More on this later this week, but not sure exactly when. I really do need to get to other things. In the meantime, more than likely I will passing this email along to others who know Jackson. I’m considering hitting up a physics prof or two or three to get their “feeling” on all this.

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Displacement Current within Good Conductors: Gone MIA but why?