bob. try these two links and let’s discuss
https://secure292.sgcpanel.com:2096/cpsess0101557516/webmail/Crystal/index.html?login=1&post_login=87960434097590
https://secure292.sgcpanel.com:2096/cpsess2278565679/webmail/Crystal/index.html?mailclient=sqmail
(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.
11 thoughts on “Displacement Current within Good Conductors: Gone MIA but why? ”
I did a *very* rough calculation to find the ratio of the magnitudes of Poynting V for the conduction current divided by he Poynting V for the E&M wave traveling *outside* the wire. If my math is right the ratio is enormous because of the enormous magnetic field inside. The ratio appears to be approximately equal to the ratio of magnetic field magnitude to electric field magnitude.
Hayt et al couldn’t completely ignore energy density outside a good conductor even though it was relatively negligible. They chose to not ignore it on logic alone. Where else would the fast moving E&M *be*?
I think they were fine not caring much about the exact magnitude of the energy density due to displacement current. They could get away with discarding energy density from displacement current *inside*, because even if that contributed twice the amount of total energy as that from the outside it is still negligible relative to conduction current total energy.
You may want to do this same calculation to see what you get.
One other point. I’m speculating here, but Jackson had mentioned that, using the Drude model, he could assume dielectric constant “near unity” for non-conducting properties of the good conductor. He calculated this by considering the ultra high frequency case which is where the conductor acts like an insulator. So a question …
If dielectric constant of the non-conducting wire is about 1, but it is something like 1.5 for surrounding insulation, does this explain a bit more how all that energy inside went missing? One needs to account for *all* dipoles no matter where they are when we calculate “things” related to displacement current both inside and out. If the overall dielectric constant is mostly due to the insulation, then wouldn’t that explain how models based on outer displacement current alone might still be fairly accurate?
Jackson 2nd ed. Ch. 8 section 1. He is clear about the existence of the displacement current inside the conductor and he states clearly he is neglecting it. A proper treatment of displacement will involve starting with eq (8.4) and keeping the displacement term.
I fixed a mistake in the original post. this is what I wrote ….
Third, when Jackson treats energy propagation in a good conductor for lower wavelengths he also ignores terms related to displacement current.
this is what I meant …
Third, when Jackson treats energy propagation in a good conductor for lower frequencies he also ignores terms related to displacement current.
Am currently research TEM, TE and TM modes in two cases:
1. coax cable with insulation between.
2. parallel cylindrical cables with insulation between.
We know TEM is dominant mode for these structures, but are TEM modes sustained in the conductors themselves? How are they dependent upon skin depth. Same questions for TE and TM modes. For any of these in conductors, what is the speed of propagation in axial direction?
First thing to note: both TM and TE modes have cutoff frequencies for many cases. I’ll assume the above two are included here, so … a strong hint that at 60 hz neither of these modes can exist.
Since all I want to know is whether an E&M wave can propagate *within* a good conductor at a good % of the speed of light, I will proceed to consider only the following case for now which concentrates on TEM mode only …
A modified coax cable, where both the inner and outer conductors are hollow cylinders with thickness smaller than the skin depth. The inner conductor is filled with the same dielectric as everywhere else in the coax cable. We now have a case where neither of the conductors satisfy the “singular waveguide” case (i.e. the case where TEM mode is not allowed).
For many treatments of E&M propagation within a conductor, the displacement current is often ignored. This ignored component is responsible for significant energy transfer. It is an E&M wave propagating at about the speed of light. That term most definitely carries energy, and it is driving the free carriers. Please see J.D. Jackson section 7.5 – he models a good conductor as dielectric with free charges.
As the force driving the carriers it is losing energy in a good conductor. Charges with mass are accelerating. Energy must come from the electric field corresponding to the displacement current. After this interaction the driving electric field becomes very *weak*, because energy must be conserved.
That is a very key point! The displacement current is very weak in regions where it is driving the conduction current. Now look at Jackson and see his footnote about the very small but normally (perpendicular to the boundary) directed E field. And how it is discarded!
If had a good chart showing how insulated cables change characteristics with frequency, I might be able to show experimental proof of displacement current inside the conductors as well as its strength. In addition to velocity factor, are there other characteristics of cables *attributed to the insulation* that might also be frequency dependent?
While I still need to take time to figure out TEM propagation for various 2-conductor geometries, I’m now curious about what happens in *thick* wire at high frequencies, depending on geometry. Assume that regardless of the geometry (i.e. coax or parallel-line) the conductors are 4 times thicker than the skin depth, a good amount of displacement current near the center of the conductors will not be driving current in this region. That means it won’t lose much energy as the driving force of the accelerating charges in this region. It should therefore be strong here and more responsible for overall velocity of propagation than it is for low frequency cases (large skin depth). In other words, in this case the dielectric constant due to the conducting medium should now be more responsible for the velocity factor (for instance) than before.
One thing that could be immensely helpful here: are there charts showing transmission characteristics of coax cables that have nothing but air as the insulation? I guess those wouldn’t be transmission lines but rather antennae (otherwise arcing would occur for larger voltages). Still, if there is a body of knowledge of wire pair configurations where there is no insulation (just air), then we now have a case of conductor dielectric constant higher than that of the medium surrounding it. This could be an interesting case to consider.
We know that field energy just inside the surface of a current carrying conductor is by far mostly magnetic. Where did all this magnetic energy come from? “Free” charges are accelerating due to the electric field from displacement current. Energy is lost from that field while it drives charge acceleration: work is being done by the E field here. The accelerating charges create the extremely strong magnetic field component that is most responsible for the E&M energy within the conductor. The magnetic field is changing which creates another component of electric field. The net result is out of phase E&M fields.
Meanwhile, energy is lost non-conservatively mostly due to thermal transfer to the lattice. Much of the energy from the E field is being absorbed thermally. If it were not (imagine a good conductor of low density at a very low temperature), all processes are the same but less E field energy is lost to the lattice and can therefore maintain substantial carrier acceleration further from the surface and into the conductor. Absorption and the resultant attenuation of conduction current still exists but less so in this case. Now the processes appear as more reactive than lossy relative to cases of higher temps and denser conductors.
A note regarding some confusion resulting from derivations for *thin* skin depth conditions: Jackson considers only this case for his treatment of “Fields at the Surface of and within a Conductor”[2]. He begins by mentioning this treatment is only part of a successive approximation scheme, but he doesn’t complete the rest of the scheme in the sense that he doesn’t appear to complete the approximation by including terms which effectively model cases where skin depth is large compared to a smaller depth of interest. I believe that results in lack of clarity later.
Just after his perfect conductor treatment at the beginning of this section, and just prior to solving for continuity of tangential H field across the boundary, he states that the case in the “transitional region” is complicated. He is clear that for the transitional region treatment in the case of finite conductivity (i.e. a non-perfect conductor) things are complicated, partially because “there cannot actually be a surface layer of current”. He continues to address this case by mentioning that his treatment will involve a successive approximation scheme. What follows is a series of assumptions, again starting with perfect conductor behavior right at the surface and then a statement that “we make use of the fact that the spatial variation of the fields normal to the surface is much more rapid than variations parallel to the surface.” He then proceeds to match tangential magnetic field across the boundary. Everything afterwards is related to spatial variations of fields.
Jackson is proceeding as if fields on either side of the boundary have achieved quasi steady state conditions. His treatment is restricted to curls and gradients and other derivatives with respect to spatial dimensions. His only hint at time dependence for this part of the approximation is related to two things:
1. Frequency and wave number indicative of plane wave propagation with sinusoidal variations relative to time and distance.
2. The phase relationship between the E&M fields.
He has left out the derivation of magnetic field strength dependence on time. I believe he has left it out because he is approaching this case as the perfect conductor case. In that case it takes *ZERO* time to establish the resultant E&M field strengths and phase relationships right at the surface. In that case there is no reason to discuss how the fields were established.
I believe this is why Jackson ignores displacement current entirely in this treatment. For his treatment right at the surface he mentions that the non-perfect conductor case is the same as the perfect conductor case. He is assuming that field strengths are instantaneously established by instantaneously moving carriers.
Jackson 2nd Ed. Section 8.1
He calculates power flow just outside the conductor and matches it to rate of dissipation of energy inside due to ohmic losses. But there is no discussion of how the power from these fields outside are contributing to the carrier energy that is transferred to the lattice inside. Although Jackson *appears* to imply the carriers are being driven by the energy flow from outside, he makes no attempt at explaining how that might be the case.
The kinetic energy of the carriers within the conductor (aside from their random thermal motion) is due mostly to the E field of the displacement current *inside* the conductor which is doing work on them. The displacement current *inside* is losing energy by accelerating the carriers, and it will therefore have a weaker E field than that outside. There is a *net* flow across the boundary because displacement E inside has now been depleted. To be thorough I’ll need to match boundary conditions for E. Hopefully I can get to that soon.