Talk:Skin effect/Archive 1

Archive 1

Archive 2

Lamb and 1883

According to Paul Nahin's biography of Oliver Heaviside, Horace Lamb published a paper on the skin effect in January 1883 in spherical conductors, and Oliver Heaviside generalized that in 1885. --Wtshymanski 23:23, 7 Jun 2005 (UTC)

What is a spherical conductor please? I think you would need a whole load of balls to make any useful conductor.--Light current 05:43, 9 September 2005 (UTC)

The spherical conductor that Lamb was interested in was the Earth's core. [1] --Heron 11:46, 9 September 2005 (UTC)

Ah ha! Or just one big ball! --Light current 13:39, 9 September 2005 (UTC)

Think of a magnetically-"induced" current and you'll see how one might use a sphere as a simple model to understand the effect.

Atlant 13:10, 9 September 2005 (UTC)

Large power transformers

Litz wire will be used in large power transformers. If this means in 50/60 Hz distribution transformers, this is the first I've heard of it!! Are any pictures available? --Light current 05:40, 9 September 2005 (UTC)

Just been looking in my copy of Higher Electrical Engineering (Shappard, Moreton, and Spence) pub Pitman 1970 ISBN 0 273 40063 0 (a standard work for first/second year undergrads in Britain). In the section on power transformer construction I would like to quote a short extract from the section on windings. The coils are made of varnished cotton or paper covered wire or strip and are circular in shape to prevent high mechanical stress.... If ever there was a place to mention Litz wire this was it. But they dont. Thats because its not used. So can we please delete this erroneous statement. --Light current 17:55, 9 September 2005 (UTC)

THanks Atlant for pointing me to those pictures. I was trying to see if there was any indication of special windig wire (Litz) being used. But there is not enough detail in the photos to see.--Light current 19:04, 9 September 2005 (UTC)

I think of "Litz wire" in terms of RF transformers, but having seen large power trnaformers being built I can assure you that the heavy current windings are in multiple parallel strands, rather like Litz wire but much larger in cross-sectional area. --Wtshymanski 20:05, 9 September 2005 (UTC)

In that case, I'll revert my deletion of that line from the article. It would be nice if you could find a reference, though, that proves that these multiple strands are there to mitigate the skin effect, and not for some mechanical reason (e.g. to reduce bending stresses). --Heron 21:57, 9 September 2005 (UTC)

The term used is "transposed" conductors. The transformers are wound with stips of copper, bundled two stips wide and some number tall. The thickness of the strip is chosen to give less than 1/2 skin depth at power line frequency to manage skin / proximity losses. The width is chosen to be 1/2 of the width of the disired bundle width. The stips are stacked up as tall as needed to give the total cross section needed to handle the current. One space for conductor is left open so that the stips can be "rolled" and maintain the overal retangular bundle shape. The change of position is called a "transposition". The tranpositions are spaced such that each strip will be in each possible position in the bundle some intiger number of times. Like litz, the individual strips are insulated with a lesser amount of insulation and more insulation is applied around the bundle. Same idea as litz wire but the use of retangular wire and retangular bundles results in less wasted space in the winding.

Look at the first picture on this page: http://www.copper.org/innovations/1999/09/transformer_innovations.html

They also mention copper foil windings, also a method of mitigating skin loss.

Lightning teaser

Since we know that current travels down the outside of conductors at high frequency, why is it that lightning conductors are solid of rectangular X-section and not just made from hollow copper pipe? (much cheaper) --Light current 19:09, 9 September 2005 (UTC)

Serious lightning conductors usually aren't made of solid copper; instead, they are braided from a number of strands of relatively fine-gauge (17 gauge'ish?) copper to form an overall conductor that's maybe 3/4 inch in diameter with a lot of included air-space.[2]

It's just the copper grounding wire we buy at Radio Shack that is solid. I think it's function would probably be better-described as "electrostatic discharge" where it helps to keep the antenna or what-have-you at ground potential and not allow the build-up of a charge that might attract a lightning strike. I'm sure that this solid style wire would demonstrate a pretty high impedance in the face of real lightning strikes.

Atlant 00:17, 10 September 2005 (UTC)

Well, over here, al lightning conductors are made of solid copper sometimes in an insulating sheath of pvc or something, pinned to the walls of tall municipal buildings and churches etc. --Light current 07:04, 10 September 2005 (UTC)

Any way, the question remains, why are lightning conductors not made from hollow copper piping. (apart from the fact it may be more difficult to bend.)--Light current 18:40, 10 September 2005 (UTC)

You can make a down conductor from a hollow copper tube, e.g. a water pipe, but do not try to make it the few tenths of a mmm, which is the area where a lightning current flows, as this will make it evaporate immediately if a lightning strikes. The massive conductor is much more robust both thermally and mechanically and even if a waterpipe can easily do the job, it is more tricky to bend. NSV 10-10-2005, 1250UTC

Solid or stranded conductors are used primarily for mechanical strength and thermal mass. When conducting very large peak currents, a hollow conductor may actually collapse due to "magnetic pinch" effects. The higher the di/dt, the greater the effect - robust lightning conductors designed to withstand high current positive lightning strikes must be physically strong. Bert 20:03, 10 October 2005 (UTC)

Think Fourier analysis: lightning has high frequency radio components because of the sharp edges (ie rapid rise and fall times) of the pulses. The pulses themselves are typically 5 coulombs at 40 kA, meaning about 0.13 ms wide and therefore 8 kHz fundamental frequency. Skin depth can be calculated from this. StuFifeScotland 19:05, 28 October 2006 (UTC)

Silver plating

What happens to the skin resistance when the silver tarnishes?--Light current 22:00, 10 September 2005 (UTC)

I did some simple experiments on this some years ago. I took lengths of copper waveguide and measured the microwave attenuation (using a vector network analyser) before and after they were internally weathered or chromate conversion coated. Remarkably, neither had any significant effect! I concluded that either: 1) the surface 'corrosion' layer had the same high conductivity as the clean metal (very unlikely!); 2) the layer was such a good insulator that the current effectively moved deeper into the clean metal; or 3) the corrosion layer is so much thinner than the skin depth (which was less than a micron, ie < 0.001 mm) that only a small percentage of the current is flowing in it. StuFifeScotland 19:05, 28 October 2006 (UTC)

Change of Phase With Depth

A little known fact is that the phase of the current changes with depth, as well as the amplitude. This is touched on at the skin depth page, though in very mathematically language. It is quite intriguing to realise that, in the middle of the conductor, the current may actually be flowing the opposite way to that on the surface! StuFifeScotland 19:05, 28 October 2006 (UTC)

Small note about minor edit summary

In my edit summary, I mentioned a person the nearest google to 'opoku kofi'. Having done a little research - as I should have done in the first place - it now appears that 80.87.70.4 has recently been adding little bits of text to (amongst other things) engineering-related articles. My edit summary thus contained irrelevant information.
Just clarifying. --Shirt58 10:27, 6 February 2007 (UTC)

Effect for Dummies

the second ext link is an oversimplification. they correctly mention that the sd approximation only works when s << r. secondly, skin effect matters at 60 hz because the conductors used are huge. yet the article acts like this effect is only at RF. it *is* audible at audio frequencies, although the other issues with such a hypothetical cable might swamp it out.66.217.164.195 03:43, 14 August 2007 (UTC)

Skin effect and Faraday cages

While reading for this article, I discovered that some of the effects that are attributed in Wikipedia to Faraday shielding (see Faraday cage article) are in fact due to the skin effect. To put it briefly, Faraday shielding is due to the behaviour of electrostatic charges trying to maximize their distances from each other, while the skin effect is due to a magnetic field, created by a changing current, acting on that current. On the other hand, perhaps someone really smart will tell me that both effects are just different aspects of some deeper principle. I'll wait for comments before wading in to the Faraday cage article. -- Heron

The only area I can see where there may be confusion is in the Faraday cage article where it says that the cage has a dual purpose - the additional one being RF screening. However, I dont think this screening effect arises from skin effect but by the conducting walls efficiently reflecting the EM radiation (ie keeping out or keeping it in.--Light current 19:34, 11 September 2005 (UTC)

There are other areas of confusion in that article. Under "Real-world Faraday cages", the TEMPEST shield, cordless phone and microwave oven paragraphs all describe an RF-reflecting shield, not a Faraday cage in the electrostatic sense. I'm just not sure whether this means that they are not Faraday cages, or that they are Faraday cages but are used in a way that Faraday could not have envisaged. --Heron 20:03, 11 September 2005 (UTC)

Well, I think if they have 6 conducting walls, they are Faraday cages, but they are also RF screened rooms. THere is, IMHO, no practical difference. So I would go with your latter statement ;-) --Light current 20:09, 11 September 2005 (UTC)

In serious applications, i'd guess the bars and plating are grounded. there is TEMPEST as applied to computer gear, then there is shielding used in walls and windows. the power lines are filtered as well.66.19.70.85 00:55, 17 August 2007 (UTC)

Litz Translation

Presently the German Litzendraht is translated as "braided wire", but I'm not sure that's correct, as litz is typically twisted rather than braided. Ccrrccrr 03:36, 17 January 2007 (UTC)

the wiki article on litz wire says it's braided. it's been my experience that it is not twisted, but i don't work with it much. i'll ask an RF engineer if i run into one.66.19.70.85 00:57, 17 August 2007 (UTC)

Litz wire is both twisted and braided. The structure is complex.Trojancowboy (talk) 02:26, 3 June 2009 (UTC)

There's about 6 different sorts of litz wire in common use. Some types (intended mostly for the lower frequencies, say ~1 khz) are simply twisted. Higher frequencies use more complex patterns.- (User) Wolfkeeper (Talk) 03:47, 3 June 2009 (UTC)

If you want to be helpfull, get permission from this Litz wire manufacurer to use some of their photos and data. I have used many of their products over the years.Trojancowboy (talk) 15:35, 3 June 2009 (UTC)

Welding section

"Iron rods work well for dc welding but it is impossible to use them at frequencies much higher than 60 Hz."
This doesn't make sense to me. Shall we change it to read:
"Iron rods work well for dc welding but it is impossible to use them for AC welding (if there is such a thing?) at frequencies much higher than 60 Hz."--TFJamMan (talk) 11:41, 12 August 2009 (UTC)

I forget why, but AC is used when welding aluminum.- (User) Wolfkeeper (Talk) 11:49, 12 August 2009 (UTC)

Okay but then you wouldn't use iron rods for welding aluminium. Unless we know what this person was trying to say I suggest we delete this ambiguity until it can be resolved in this discussion page.--TFJamMan (talk) 11:27, 26 August 2009 (UTC)

Table removed from article

rough draft -- please double-check these numbers, then remove this notice

material	skin depth at 60 Hz	skin depth at frequency f
(general)		$d={\sqrt {\rho }}/{\sqrt {\pi f\mu }}$
Copper	8 mm	$d={65~\mathrm {mm} }/{\sqrt {f/\mathrm {Hz} }}$
Transformer iron	0.1 mm	$d={0.80~\mathrm {mm} }/{\sqrt {f/\mathrm {Hz} }}$
Aluminum	10 mm	$d={82~\mathrm {mm} }/{\sqrt {f/\mathrm {Hz} }}$
Nickel	3 mm	$d={23.5~\mathrm {mm} }/{\sqrt {f/\mathrm {Hz} }}$
Steel	???	???
Water	???	???

the correct values for copper are in Engineering Electromagnetics by Hayt. Steel is not an element, and transformer iron/steel is laminated layers. also the tables are pretty much meaningless because the effect is not well-formed at 60 hz in many applications. all things being equal, the skin depth will be greater in materials with low conductivity.66.19.70.85 00:50, 17 August 2007 (UTC)

Please no put no units in formulas, this is wrong and makes it unclear! Better to use: Copper || $d={65~}/{\sqrt {f}}$ with d in mm and f in Hz. —Preceding unsigned comment added by 57.67.164.37 (talk) 14:25, 1 September 2009 (UTC)

Merge proposal

Right now, skin effect is more about conduction of ac current whereas skin depth is more generally about electromagnetic waves. I think that separation is useful, as the two articles serve different audiences, even though it's fundamentally the same phenomenon. I think it would be more useful to have lots of cross linking between the two, rather than to merge them. So I oppose the proposal to merge. Ccrrccrr (talk) 22:59, 1 April 2009 (UTC)

I agree with the proposal to merge. Skin depth is the cause of skin effect and should not be a separate article. Skin effect is important for short wave propagation in the ionosphere, underwater communication with low frequency radio waves, and ground waves for radio frequency propagation. In these cases skin depth is on the order of hundreds of kilometers, hundreds of meters and tens of meters, respectively. Trojancowboy (talk) 02:35, 3 June 2009 (UTC)

• Agree with merge. I think Ccrrccrr's concerns are valid, though: there seems to be two distinct audiences here (perhaps engineers and physicists?), so we need to be sure not to lose too much info. Specifically, the material effect section should include the table in the skin depth article. I disagree with the notion that skin depth causes skin effect; rather, skin depth is a measure of skin effect which is useful for certain calculations. Both the skin effect and the skin depth are simply results of the laws of electromagnetism. --W0lfie (talk) 17:52, 30 June 2009 (UTC)

• Oppose The merge-to article does not cover the material, so I oppose this.- (User) Wolfkeeper (Talk) 20:24, 30 September 2009 (UTC)

skin effect

Does using 2-3 thin wires instead of 1 thick wire in winding of transformer has any effect on temp rise in windings (due to skin effect) Thanks ```` —Preceding unsigned comment added by Sachin111 (talk • contribs) 09:19, 1 June 2008 (UTC)

Yes it helps and is essential for high frequency transformers.Trojancowboy (talk) 02:37, 3 June 2009 (UTC)

The table containing skin depth for copper depending on frequency seem to be incorrect, at least the formula in the introduction secition (and the same one on http://www.rfcafe.com/references/electrical/skin-depth.htm) gives different results. —Preceding unsigned comment added by 217.83.179.133 (talk) 16:57, 21 November 2009 (UTC)

Flat plate

In the example of the thick flat plate, it doesn't make sense that the current would only flow through one skin depth. I would think that it would travel along the outside, sort of like rectangular tubing, as in the wire example. An explanation would be nice.

Current flows at all depths in the conductor, but the magnitude of the current density (amps per square mm, for example) decreases exponentially as the depth increases. --Wtshymanski (talk) 20:23, 28 April 2010 (UTC)

Could someone provide a reference?

Could someone provide a reference for the differential equation describing the current J? —Preceding unsigned comment added by 64.26.167.106 (talk) 22:04, 30 April 2008 (UTC)

See Engineering Electromagnetics, by Hayt, chapter 11, section 5, page 362 in the 5'th edition. By the way, sometimes a subscript is used to indicate partial derivatives, but in this case, the equation is just an ordinary equation and not a differential equation. Constant314 (talk) 23:41, 21 May 2010 (UTC)

Formula error??

The formula R=roh/d*(l/pi*(D-d)) with d=sqrt(2*roh/(w*my)) has an problem

for f=0; the therm of d gets big (infinity) for f=1Hz the therm is even quite big! ==> d>D

So the R gets negative!!!! Why that?

For my measurements I did it would fit good if the formula would be: R=roh/d*(l/pi*(d-D)) ==> but then we get some problem for the big frequencies!

Could you please give a reference for the formule? Or the logical approach?

thx —Preceding unsigned comment added by Bananajoe84 (talk • contribs) 17:24, 6 December 2009 (UTC)

The formula only applies when D is much larger than d, so R never goes negative. And yes, d and D get infinitely large at 0 Hz. But we never actually get to zero Hz. Constant314 (talk) 18:16, 26 May 2010 (UTC)

skin effect

I want to know the relation between inductance and skin effect —Preceding unsigned comment added by 116.68.197.34 (talk) 10:03, 10 March 2010 (UTC)

The only effect I can think of is that skin effect will increase the resistance of your inductor, lowering the Q factor (i.e. the inductor becomes more lossy). Of course the distribution of current in a cross-section of the inductor winding will be effected by the other nearby windings - but this isn't really skin effect, it's a proximity effect. GyroMagician (talk) 11:53, 11 April 2010 (UTC)

See Engineering Electromagnetics, by Hayt, chapter 12, section 2 (in the 4'th and 5'th editions). Hayt analyzes a coaxial transmission line and shows that as frequency increases inductance decreases due to skin effect. In the particular example there was no proximity effect. The basic argument is that skin effect denies the volume of the wire as a place that the magnetic field can store energy. Less volume means less stored energy which means less inductance. Constant314 (talk) 22:21, 21 May 2010 (UTC)

Interesting, thanks for that. I guess I've always assumed PEC, so that the magnetic field is entirely stored in the dielectric between the conductors. I'll need to think about that for a while... GyroMagician (talk) 10:53, 25 May 2010 (UTC)

What does PEC mean? Constant314 (talk) 12:23, 27 May 2010 (UTC)

Ah, sorry, perfect electric conductor (=> no field inside the conductor, skin depth=0). I've been spending too much time simulating stuff recently! GyroMagician (talk) 20:39, 27 May 2010 (UTC)

Revise formula

I believe there is something wrong in the AC resistance formula, specifically the one which is supposed to be more precise:

$R={{\rho \over \delta }\left({L \over {\pi (D-\delta )}}\right)}$

Assuming rising from very low frequencies R gets very low but negative values, then up to a certain frequency at which $\delta$ tends to D. In this case, R tends to $\infty$ and any further increase in frequency (i.e: higher frequencies) will make sense. It should be revised more carefully, or there must be a frequency limit above which the formula will be valid.--Email4mobile (talk) 10:18, 10 September 2010 (UTC)

The text above the formula imposes the condition that D is large compared to δ. Consequently, the approximation has a valid range. I changed the initial equal sign to an approximation. Glrx (talk) 15:26, 10 September 2010 (UTC)

Hollow-center conductors

"Hollow-center conductors such as pipe can be used to take advantage of skin effect."

I see there has been quite a bit of editing on that. It looks to be like everybody has the right idea but cannot agree on the exact words.

Saying that hollow conductors take advantage of the skin effects seems to suggest there is an advantage to having a hollow conductor. Manufacturing a hollow conductor of the same cross section as a solid conductor is more expensive. It is more expensive to transport, handle and install. It has a larger cross section so it probably gets high wind loading. All in all, in general it is a disadvantage to have hollow conductors. But, since skin effect is operative for large diameter high current power lines, the copper in the middle just isn't being used. Copper is expensive stuff, so in spite of the disadvantages of hollow conductors, it is still worth it to use hollow conductors. Having knowledge of the skin effect allows you to optimize your overall cost. Then once you have a hollow conductor, you can take advantage of that and run some steel down through the middle for strength. Though, I think copper wound onto a steel cable is more common than actual hollow conductors.

And the "Hollow-center conductors such as pipe" suggests that pipe is commonly used for carrying electricity but it is not; pipe is commonly used for carrying water.

So, my suggestion is this:

"Taking advantage of knowledge of skin effect, pipe-like hollow conductors are sometimes used to minimize the overall cost of power transmission." Constant314 (talk) 22:54, 26 November 2010 (UTC)

It buries the information. Here's something more direct with a concrete example. "When the skin effect is significant, most of the conduction occurs near the surface of a conductor and virtually none near the center. Engineers will sometimes economize on copper by using hollow conductors such as pipe. The copper litz wire used in the 60 kHz WWVB transmitter has a nonmetalic core." Glrx (talk) 01:03, 27 November 2010 (UTC)

But they don't really use pipe; they use an engineered conductor. But maybe in their industry they call it "pipe".

By the way, I've heard that Litz wire with a non-conducting center conductor referred to as ersatz Litz Wire.Constant314 (talk) 03:12, 27 November 2010 (UTC)

The hollow conductor is lighter. That's the advantage. Hollow conductors are used all the time in communications satellites, where light weight is critical. The hollow conductors (which carry RF frequencies) are called pipe or tube or hollow conductor or hollow transmission line. Binksternet (talk) 09:36, 27 November 2010 (UTC)

A hollow conductor is not lighter than a solid conductor of the same cross section which would work just as well if there were no skin effect. Skin effect does not confer any advantage; skin effect is almost always a disadvantage, but knowledge of skin effect allows you to mitagate the disadvantage. Regarding satellites, are you referring to wave guides? —Preceding unsigned comment added by Constant314 (talk • contribs) 14:40, 27 November 2010 (UTC)

A hollow conductor is lighter for very high frequency conduction, since the center of a solid core conductor is not used by high frequencies due to skin effect. Of course, you are right in terms of DC or low frequencies and no skin effect. Binksternet (talk) 17:56, 27 November 2010 (UTC)

I think we are all on the same page with this. My complaint is semantic. We don't take advantage of skin effect to use a hollow conductor; rather we take advantage of the ability to manufacture a hollow conductor to mitigate skin effect. In fact, even if there were no skin effect we could still use a hollow conductor if there were an advantage to do so. Constant314 (talk) 19:19, 27 November 2010 (UTC)

I don't know the industry terms for hollow conductor. Copper tubing has been used for HF transmitting coils. It's also been used in VHF resonators. Price and convenience are probably issues there -- it's easier to get copper tubing rather than large diameter solid wire. Copper tubing is also used in some induction heaters -- but that may be more so they can run coolant through the coils rather than saving copper. Some resonant HF transmitting loop antenna designs use copper pipe straight from the hardware store. I have no problem with dropping the words pipe or pipe-like until there is a WP:RS, but pipe does seem appropriate. The litz wire non-conducting core is mentioned in a NTIS/WWVB publication. I have not heard the modifier ersatz; there are types specified by some manufacturers; see litz types. Glrx (talk) 16:40, 27 November 2010 (UTC)

When I think of "pipe" I think of something steel and galvanized and optomized for strength and corosion resistance instead of conductivity. I'd be happy on the pipe issue if you just said "conductors such as copper pipe " instead of plain old "pipe".Constant314 (talk) 18:55, 27 November 2010 (UTC)

How about "When the conductor radius is large with respect to the skin depth, the inner portion of the conductor carries little current. It is possible to save copper and weight by using hollow conductors. In some cases off the shelf copper tubing and copper pipes can be used. Instead of a hollow conductor, the center may have a strength member that is a non-conductor or poor conductor." It mentions pipe. It lists the advantages of using hollow conductors. It avoids the semantically troublesome construct of "taking advantage of skin effect".Constant314 (talk) 14:43, 28 November 2010 (UTC)

That works for me. Binksternet (talk) 16:26, 28 November 2010 (UTC)

I've been looking in the Elecrical Engineers Handbook, 1969 by Fink. For buses in power stations it mentions bars, tubes, rods, angles, channels, hollow square tubing and structrural shapes, but no pipes. It does say that copper and aluminum tubes are available in "pipe sizes". For power distribution there is no mention of hollow conductors but there is copper over steel core and aluminum over a steel core. Interestingly, it notes that for conductors reinforced by steel that the resistivity increases "somewhat" with current density suggesting some mild non-liearity. Maybe a hysterisys loop in the steel?Constant314 (talk) 01:25, 1 December 2010 (UTC)

I wish I'd saved a set of prints for the substation at my former employer; the bill of material for the 13.8 kV bus into the building identified it as "Schedule 40 copper pipe". The 11th ed of Fink and Beatty calls out different pipe sizes. And they really are hollow tubes -sometimes they even put a length of ACSR into them to damp vibration. --Wtshymanski (talk) 03:37, 1 December 2010 (UTC)

Just curious, what is copper pipe used for if not used as a conductor?Constant314 (talk) 14:10, 1 December 2010 (UTC)

Well, in junior high shops, we used to cut little pieces to make rings. I once built an antenna out of copper pipe; no, wait, that was a conductor. It's too soft to make good furniture legs, and too heavy to use as ski poles. Maybe you could ask someone in a red vest at the plumbing store why they keep so much of it around. --Wtshymanski (talk) 15:00, 1 December 2010 (UTC)

Atomic bonds

As the fine page says, "At 60 Hz in copper, skin depth is about 8.5 mm". This should immediately indicate that we don't need to talk about atomic bonds. It's a field effect that can simply be described using classical electromagnetics. Skin effect is clearly seen in regular copper wire. We don't need to talk about exotic materials and nanoscale structures, especially in the lede. And we certainly don't need 7 references at the end of a sentence. Hence, I removed them. GyroMagician (talk) 15:02, 5 December 2010 (UTC)

Can you explain to the reader how this "Skin effect is due to eddy currents induced by the alternating current." process occurs, and by what mechanism?.Francis E Williams (talk) 15:18, 5 December 2010 (UTC)

All the following explanations are based on classical electromagnetics. #1 has the easiest math. #2 and #3 have intuative appeal. #3 I think needs to consider the mutual coupling between the shells.

1. The expression for skin efffect comes directly from the solution of the wave equations. The high frequency electromagnetic wave simply attenuates rapidly as it penetrates a good conductor. As the E field vanishes then so does th curret density. See almost any of the following Griffiths, Harrington, Hayt, Kraus, Jackson and Marshall.

2. The magnetic field circulating a filament of current within the conductor is such that it opposes current that is deeper and reinforces the current that is shallower. See Johnson.

3. The circular wire can be viewed as a set of concentric shells. As the diameter of the shells decrease, the inductance increases. Thus current on the inside is opposed by greater inductance than current along the outside. See Skilling and Terman

Constant314 (talk) 15:48, 5 December 2010 (UTC)

I see that you are mainly focussing on accepted theoretical knowledge that is currently and historically available. I can see that the frustrations of working with Wiki to allow simpler understanding of a subject are many.Francis E Williams (talk) 17:41, 5 December 2010 (UTC)

I share your frustration. Although one can find a rigorous development of skin effect from the solution of the wave equation in dozen college E&M textbooks that range from introductory to advanced, it is hard to find any that put it into a narative that offers an explanation suitable to the non-specialist.Constant314 (talk) 21:54, 5 December 2010 (UTC)

Personally, I like no.1 - that a conductor damps the E-field. By that argument, for a perfect conductors, all current flows on the surface (skin effect in superconductors anyone?). One of the things I enjoy about electromagnetics is that most of the field hasn't really changed since Maxwell and friends wrote down their fundamental description. Sure, quantum mechanics has developed since then, and our basic understanding of matter has changed a bit, but most macro-scale electromagnetic phenomena can still be described using a set of laws derived in the 1800s. And we're still finding new ways to use them.

In the spirit of keeping wikipedia accessible to the non-specialist, I don't see any reason to invoke anything more complicated than classical theory. New research sometimes corrects misconceptions, but more often it explains some extreme cases, while providing the same result over the 'normal' range (quantum for small, relativistic for fast, etc). Classical models are almost always more intuitive, and easier to describe to the novice. There is no such thing as steady state (steady-state is not DC), but it is very useful for describing a system, and is used daily by working scientists because it makes their problem solvable. Science is often abut making the 'right' assumptions or simplifications. As a very smart man once said "Make everything as simple as possible, but not simpler." If you don't need to consider the switch-on event, don't include it - typical antenna transmission, for example.

Returning to the content of this page, what exactly is not comprehensible? I'm really asking - sometimes, having made the leap of understanding oneself, it can be hard to see what others do not understand. Guidance here is always useful, but it has to be specific. What's missing? GyroMagician (talk) 20:18, 5 December 2010 (UTC)

I think what is missing is a good write up of number 1.Constant314 (talk) 20:28, 5 December 2010 (UTC)

Francis, in defense of the wave equation explanation, it is rigorous within the domain of classical electromagnetic theory. If you accept Maxwell's equations as true then the skin effect will arise out of the math. The wave equation is particularly simple. It is second order, linear and not time varying. Not every person with an undergraduate degree or higher in math and physics major could derive the result, but most of them (I hope) could verify the math if shown the steps in sufficient detail. Further, the wave equation explanation correctly predicts that the skin depth is inversely proportional to the ""square root'' of frequency. Any other explanation posted to the article ought to do as well. That doesn't mean that you have to derive the result, but the result ought to be derived rigorously somewhere in a citable public source.Constant314 (talk) 14:45, 10 December 2010 (UTC) Constant314 (talk) 18:34, 11 December 2010 (UTC)Constant314 (talk) 18:35, 11 December 2010 (UTC)

Resistance of a slab

The resistance of a flat slab (much thicker than δ) to alternating current is exactly equal to the resistance of a plate of thickness δ to direct current.

Should this not point out that the current on only one side of the slab has been considered? In other words, a ribbon conductor would actually have half of the stated resistance. Pointing that out detracts from the main point, but the literal statement in the current version appears to be off by a factor of 2. No? Interferometrist (talk) 21:32, 6 February 2011 (UTC)

I went ahead and rewrote it without reference to a "slab," thus removing this ambiguity. Interferometrist (talk) 23:29, 7 February 2011 (UTC)

Skin dept in poor conductors

I checked Stratton, Jackson, Harrington and Hayt. They all made the qualifying statement that the formula for skin depth applied to good conductors, which suggests that the formula would be different for poor conductors. But none of them had a formula for poor conductors. However, some of them had a general formula that might be algebraically manipulated into the suggested form. I'll look at that this weekend.Constant314 (talk) 15:50, 6 January 2011 (UTC)

Thanks for checking the refs. Glrx (talk) 16:56, 6 January 2011 (UTC)

If I combine the formula from Harrington page 50 assuming no dielectric loss and no magnetic loss, with the formula for the square root of a complex number in cartesian coordinates found here Square roots of negative and complex numbers with a little manipulation, I get the result posted by Nadovich. Of course it would be better to get a good reference; I may have made a mistake. Constant314 (talk) 00:12, 7 January 2011 (UTC)

I have Jordan's book and can confirm that the formula for skin depth of a general (good, bad, in between) conducter is there in section 5.06 on page 130Constant314 (talk) 16:05, 10 January 2011 (UTC)

I have found the equation for poor conductors to be in error. The graphic of the equation is the problem. Using the reference, Advanced Engineering Electromagnetics, Constantine A. Balanis, ISBN 0-471-62194-3. The equation in chapter 4, page 149, eqn 4-34 is depth = i/alpha = 1/[w * sqrt(ue)*{(1/2)*[sqrt(1+(1/wep)^2) -1]}^(1/2). or simply all of the curly bracket needs to be put in the denominator. —Preceding unsigned comment added by 144.188.24.25 (talk) 17:32, 31 January 2011 (UTC)

Here's an unevaluated link about poor conductors. http://farside.ph.utexas.edu/teaching/em/lectures/node102.html Glrx (talk) 18:18, 31 January 2011 (UTC)

here is the formula $\delta =\left({1 \over \omega }\right)\left\lbrace \left({{\mu \epsilon } \over 2}\right)\left[\left(1+\left({1 \over {\rho \omega \epsilon }}\right)^{2}\right)^{1/2}-1\right]\right\rbrace ^{-1/2}$

The Curly braces are raised to the power of -1/2 which effectively puts them in the denominator.Constant314 (talk) 19:33, 31 January 2011 (UTC)

I'm pretty sure that this formula is .... essentially wrong. Certainly misleading! The problem is that epsilon in a conductor (even a rather poor conductor) at normal frequencies, is NEGATIVE. So this perhaps could give a correct result for dielectrics with a small conductivity (where you'd hardly ever talk about "skin effect" in the first place!) but in normal situations (conductors!) you have a negative number under the radical. I can see how this formula MATHEMATICALLY reduces to the normal forumla for a good conductor (under the assumption epsilon>0) but that's beside the point (and relies on a false assumption).

On a very related note, the conclusion that "In a very poor conductor, with .... the equation simplifies to ..... In a very poor conductor skin depth is independent of frequency." is also wrong on two accounts. First, you'd again need such a ridiculously small conductance (in which case indeed epsilon > 0) and/or extremely high frequency (where again, epsilon > 0) that this isn't useful. Secondly in a conductor (in the usual sense of the word) at normal frequencies (< 1 THz) epsilon is not only <0 but a function of frequency. Indeed if you think about it, " skin depth is independent of frequency." doesn't even make sense from a dimensional analysis.

Now, I can see a way of rearranging this formula (would that be "original research"?) such that epsilon is placed inside the inner radical such that it can go negative and still get the normal formula (in which epsilon cancels out), which would be better. But I feel it is misleading to include a formula with epsilon in it, even if that cancels out in ordinary cases, UNLESS you at least point out that it is normally negative in conductors AND that it is frequency dependent.

I'd simply suggest to take these both out, unless someone is really interested in talking about the "skin depth" of what would normally be called an insulator!! I'll wait for someone who has been working on the article to do that: revert this to version 16:07, 5 January 2011 Glrx Interferometrist (talk) 16:08, 4 February 2011 (UTC)

I'll add one qualification to what I just wrote:

Indeed if you think about it, " skin depth is independent of frequency." doesn't even make sense from a dimensional analysis.

Well, if you DO want to talk about a very poor conductor such that the conductance is so poor that epsilon isn't affected by the conductivity AND one whose dielectric constant is not frequency dependent, perhaps not too different from epsilon0 even at the sufficiently high frequencies where the condition for the approximation is met, then you can indeed come up with a distance according to dimensional analysis. Namely the size of a cube of the material whose resistance from face to face is equal to (for instance) eta0, the impedance of free space, or more generally eta = eta0 / sqrt(relative dielectric constant). Indeed the factor sqrt(eps/mu) in that equation is 1/eta. So this makes a little sense, but still I don't think its very useful. With a resistivity of 100 ohm-meters (really poor conductor!) the approximation becomes valid above ~ 100MHz, and then if the dielectric constant is of order 1 I get a skin depth of 6 meters (back of envelope accuracy), similar to the wavelength. Is that useful at all? (I doubt it!) Interferometrist (talk) —Preceding undated comment added 16:48, 4 February 2011 (UTC).

Sorry, but one more qualification to the last qualification:

one whose dielectric constant is not frequency dependent, perhaps not too different from epsilon0 even at the sufficiently high frequencies where the condition for the approximation is met

Actually that's not usually the case, at least with anything you'd use to conduct a current. Analyzed microscopically, a material can be a poor conductor 1) because its electrons collide very frequently with phonons; or 2) because there are very few charged particles (electrons) per cubic meter providing the conduction. In the first case, epsilon<<0 and what I said doesn't apply. In the second case, yes, epsilon might be >0 and comparable to epsilon0; this might apply to a plasma, for instance (I haven't worked out the numbers). So this formula is even less useful considering that. Interferometrist (talk) 17:08, 4 February 2011 (UTC)

And FYI, a normal conductor has a epsilon > 0 at around f > 10^15 Hz ;-) Interferometrist (talk) 17:11, 4 February 2011 (UTC)

Do you happen to have a reference that discusses negative epsilon? Constant314 (talk) 22:46, 4 February 2011 (UTC)

Try these if they are useful [3],[4],[5].Francis E Williams (talk) 00:14, 5 February 2011 (UTC)

Thanks for looking it up. I guess I should have asked for about negative epsilon in ordinary conductors. By the way, that third example looks like "anomolus dispersion" where apparently wierd things happen near a resonance. Hayt says this about epsilon in metals "It is customary to take $\epsilon =\epsilon _{0}\,$ for metallic conductors", which is not an absolute statement that it is so.Constant314 (talk) 00:58, 5 February 2011 (UTC)

Well those references are interesting in a different context, but I was simply talking about the definition of epsilon: ratio of D to E. With bound charges (dielectric) BELOW the resonance frequency, the electric susceptibility chi is positive, but negative above resonance (also generally with an imaginary part, but ignore that for now). The chi from all electrons add, along with the permitivity of free space, to get epsilon. Where epsilon is a negative real number (thus ignore the imaginary part if it's small) the refractive index n is purely imaginary and the material is a perfect reflector, or more generally when the real part is negative then n has a large imaginary part and the material is essentially opaque. This is true in between the resonant frequency and the point where epsilon becomes positive again (a bit away from resonance where it dies off).

In the case of metals, there is no restoring force on the free electrons, so you can call the "resonant frequency" equal to 0 and you can use the same math to get similar results: a negative epsilon (and no transmission but high reflection). This is true UNTIL you get to the plasma frequency at which point epsilon goes positive, n becomes real (with a small or not so small imaginary part which attenuates transmission). Now the metal is transparent. But this is only at a frequency MUCH MUCH higher than we're talking about. At ordinary (radio, even IR) frequencies epsilon is negative. This is covered in Feynman vol. 2 towards the end (sorry I don't have it in front of me) in the discussion on refractive index and metals.

Hayt says this about epsilon in metals "It is customary to take $\epsilon =\epsilon _{0}\,$ for metallic conductors"

I can only imagine that he's saying that if you IGNORE the free electrons of the metal, then there is no electric permittivity beyond that of free space. But those electrons drive the net epsilon way negative, as I've detailed. Or do we have a problem with definitions here?? Interferometrist (talk) 12:00, 5 February 2011 (UTC)

I was going to point to a graph of the electric susceptibility due to a lorentzian, a really familiar plot, but had a hell of a time finding it online! But here's the closest I could find in a quick search:

http://ee.stanford.edu/~solgaard/5gain.pdf

Well, look at the 4th pane. The bottom curve is for the real part of the electric susceptibility, but it's upside down! The top curve is the imaginary part. Take the case of a very high Q (sharp) resonance: the top curve approaches a spike and as you advance in frequency the bottom one (negated) goes from 0 to infinity, then from negative infinity to 0. That's what I'm talking about. For a metal, the "resonance" is at zero frequency so you only have the higher frequency tail of this curve. (The graph shown is upside down because they're discussing lasers and media with population inversions. The top graph isn't plotted upside down because it is for GAIN rather than attenuation. But the shapes are the same as we're talking about, given that understanding.) Interferometrist (talk) 12:31, 5 February 2011 (UTC)

I have Feynman. Perhaps you can remember the subject where this discussed of maybe the subject of the chapter.

Regarding Hayt, the free electrons are accounted for by a positive conductivity term. Jackson says using a conductivity or and imaginary component of epsilon are equivalent.

Regarding the Stanford link, that is about lasing material, not about an ordinary conductors. This also looks like anomalous dispersion.

see the following for a discussion of anomalous dispersion. Basically it says that near a sharp resonance phase as a function of frequency has a very strong slope.

• Jackson, John David (1999), Classical Electrodynamics (3rd ed.), John-Wiley, ISBN 047130932X page 310.
• Stratton, Julius Adams (1941), Electromagnetic Theory, McGraw-Hill page 324

Jackson does say that in laser application the imaginary part of epsilon goes negative (indicating gain) near certain resonances. He also gives graph of epsilon for a material with molecular resonances and shows that the real part of epsilon varies above and below unity. It has negative slope in the vicinity of a resonance, but it never goes negative.

So, do you have reference about ordinary conductors and metals having negative epsilon at DC?Constant314 (talk) 15:00, 5 February 2011 (UTC)

Hi, I've been looking at all of this closely and finally think I have resolved the ambiguities! First, let me correct myself (sort of) about epsilon being negative for metals at low frequencies. I should have said that epsilon is MAINLY imaginary but with a smaller negative real part. At some "low" (but actually very high!) frequencies below the plasma frequency (thus optical frequencies) the negative real part is of greater magnitude than the imaginary part. However at lower frequencies the real part goes ~1/f whereas the imaginary part goes at 1/f^2 and dominates it when you get down to "normal" electronic frequencies. So I wanted to clarify that. However it doesn't solve the problem at hand: for such an epsilon (very large magnitude and mainly imaginary) certainly doesn't apply to or even make sense in the equation in the article -- more on that in a minute!

As far as the actual behavior of what I call epsilon, I did find a good reference which is less confusing than the last one I pointed you to (having to do with lasers):

http://www.ece.ucsb.edu/courses/ECE228/228A_W11Blumenthal/Lecture3-228a.pdf

Page 12 has the formula which is absolutely correct (also in Feynman) for the squared index of refraction n^2 = epsilon/epsilon0 due to a single atomic resonance only (unfortunately the LHS should be modified according to the Claussius-Mossotti relationship by multiplying it by 3/(n+2) but forget that for now). This function (minus 1 to get the susceptibility alone) is plotted on page 17 and THIS is what I had meant to show you last time. [I found the one for the laser where you have a population inversion and is negative of this -- that's the only difference -- but forget that now, we'll just use this more appropriate one]. The case for metals can be obtained by setting omega0=0, so you only have the right hand part of the graph where Re(chi)<0. In case this graph isn't clear (it IS poorly labelled!) the curve that peaks at omega0 is the imaginary part of chi (leading to absorbtion) and the S-shaped curve is the real part of chi. Also, I believe the graph is mislabelled: "anomolous dispersion" just means the portion in between -1/2 and +1/2 on the graph, where the slope of Re{chi} is negative. At >1/2 the dispersion is "normal" again. But that's not important.

Now you wrote something important but which also surprises me when I consider the consequences:

>Regarding Hayt, the free electrons are accounted for by a positive conductivity term. Jackson says using a conductivity or and imaginary component of epsilon are equivalent.

Yes, I can see this now: there are TWO definitions of epsilon! And thus 2 definitions of D!!! I had never heard that before (and still don't like thinking this way) but the choice depends on what you call a "free" charge which defines D (div D = rho_f). The 2 forms are also mentioned in the WP article on permittivity where Hayt's definition is first given (applying to dielectrics, as in the normal discussion of the dielectric constant), but then in the section titled "Lossy medium" the "complex permittivity" is defined which INCLUDES the effect of currents/charges in metals (and a definition of D where these are NOT included in rho_f) -- this is the definition of epsilon which I have always been talking about, and is clearly NOT the definition of Hayt which is also the one which applies to the general equation for the skin effect in this article.

There is an unfortunate mistake here, though: the permitivity article implies that in cases of media without conduction epsilon is NOT complex, thus purely real, but that clearly isn't the case and you will get incorrect results if you ignore the imaginary part around absorbtion resonances. But that article IS right in saying that the imaginary part has to do with LOSSY media. However not all loss (and thus imaginary part of epsilon) is due to conduction currents: there is also loss on the microscopic scale (leading to the imaginary part of the curve I pointed out) involved in the polarization. And the article does point that out further down, Unfortunately it incorrectly claims this causes non-linear effects, a misunderstanding (do I now have to edit that article too?!).

So we can now talk about 2 kinds of epsilon:

"Real epsilon" due only to charges bound within an atom and NOT to charges in a metal (free electrons). div D includes space charges due to free electrons in metals but not polarization inside atoms. Unfortunately "Real epsilon" is not strictly a real number except in lossless dielectrics.

"Complex epsilon" due to all aspects of a material (polarization AND free electrons). Now div D only includes externally applied charges, NOT space charges due to the divergence of conduction inside a metal.

The second form, "complex epsilon" is better for solving the wave equation etc. without taking conduction into account separately. And is more general because it combines in the same number the effect of loss due to conduction or within atoms (especially in practical cases where you can observe the loss but can't easily pinpoint its origin!). It is what I have been using as epsilon (also in the other discussion) but clearly isn't the one that goes into the general formula shown on this page (in which case I'd agree that you could say epsilon=epsion0 for most metals, although this DOES ignore some polarizability due to non-conduction electrons, probably becoming significant only at high optical frequencies).

Taking all of this into account, I will now rewrite the section and invite you to view the results, and either agree or revert it if you must -- we can discuss it further and tweak the wording in either case. But in particular, the discussion about "poor conductors" is a bit misleading since this is only in relation to a specific frequency: at sufficiently low frequencies a certain "poor conductor" is a good one and follows the top approximation, not the bottom one.

To answer a couple of other things you brought up:

>I have Feynman. Perhaps you can remember the subject where this discussed

My copy is at a different office. When I'm next there I will look it up and get back to you. But I'm pretty sure that its the same formula as on page 12 of the pdf I mentioned.

>Jackson ..... gives graph of epsilon for a material with molecular resonances and shows that the real part of epsilon varies above and below unity.

alright, this is the same as the plot on page 17.......

>It has negative slope in the vicinity of a resonance, but it never goes negative.

Perhaps not on a particular graph in a particular case. But it certainly can: it depends on whether chi goes low enough to make epsilon = 1 + (sum of chi's) get below 0. When it does (as it will just above a sufficiently strong and narrow resonance) then epsilon < 0 and there is no transmission through the material (even without considering absorbtion). This particularly applies to metals: you make omega0 zero (thus treating all electrons as "bound" with a restoring force that has reached zero!). Only above the plasma frequency (in the UV) is chi > -1 at which point the metal becomes transparent (but still attenuating due to gamma). A "perfect" metal where gamma=0 has a purely imaginary epsilon below the plasma frequency and thus is a perfect reflector (no loss!). Above the plasma frequency it is partially reflective and the rest transmits (again, no loss). And its conductivity is infinite; its skin depth is zero.

>So, do you have reference about ordinary conductors and metals having negative epsilon at DC?

Well as I admitted, the proper result at low frequencies isn't that complex epsilon is actually negative, but that it is mainly imaginary with a smaller negative real part. As you go down in frequency the real part becomes less signficant. As you APPROACH zero frequency, the imaginary part goes to infinity. Again, that is for "complex epsilon" not the one Hayt is using.Interferometrist (talk) 15:35, 6 February 2011 (UTC)

I take it that you no longer think the following equation is incorrect:

$\delta =\left({1 \over \omega }\right)\left\lbrace \left({{\mu \epsilon } \over 2}\right)\left[\left(1+\left({1 \over {\rho \omega \epsilon }}\right)^{2}\right)^{1/2}-1\right]\right\rbrace ^{-1/2}$Constant314 (talk) 23:33, 6 February 2011 (UTC)

Right, I no longer have my original objection, given the different possible meanings of epsilon (as I had mentioned) and thus ambiguities about the definitions of D. I will (soon) write something else about this on a talk page in order to see if there is a consensus on terminology (well, the answer appears to be NO). If you wish (as I do) to have the wave equation derived from the 2 maxwell's equations for curl E and curl B not having any source terms (explicit currents, charges) away from actual sources, and you insist that the refractive index n^2 = epsilon and D = epsilon E, then you are FORCED to use my version of complex epsilon. But that clearly isn't the one that would properly go in the formula but rather Hayt says that currents in metals are to be handled apart from epsilon, in which case this formula is possible. Though even in that case, epsilon can still be complex which doesn't make exact sense in this formula (you'd get a complex skin depth: what's that mean?) unless Hayt EVEN wants to treat loss in a dielectric as due to conduction currents which doesn't make any sense (but I believe an "equivalent conductivity" could be defined at any one frequency which would make the math work, but isn't really what's going on). So this is my dilemma. As I said, I will post more later on this subject.

Also, you had wanted the reference in Feynman: his equation for epsilon/epsilon0 (=n^2) is given in section 32-6, eq 32.38 (in my edition, at least). That is the same equation as in (for instance) page 12 of the lecture notes I linked to above, if you set omega0=0. But now I notice that Feynman avoids calling this quantity epsilon or defining D=epsilon E (see what he writes around eq 32.18). So I guess one way of getting around these confusions is to not talk about epsilon at all! But if you do define it as espsilon0 * n^2, then it clearly isn't what Hayt means in that equation, as you can appreciate. Interferometrist (talk) 16:35, 7 February 2011 (UTC)

Every source I checked, including Feynman, Jackson, Hayt and others use the greek letter sigma as the conductivity of a conductor. Complex permittivity is used for dielectric loss. The formula

$n^{2}={\frac {\epsilon }{\epsilon _{0}}}\,$

appears to be reserved for good dielectrics and is not applied to conductors.Constant314 (talk) 17:36, 7 February 2011 (UTC)

That's an interesting observation about how various people refer to these quantities, but you didn't think through the complications and ambiguities. For instance, a material can well have conductivity AND a (perhaps complex) permittivity due to polarizability of its atoms. How do you write n^2 then? One formulation is shown in the WP article on permittivity#Lossy medium where the complex permittivity that I use epsilon-hat is defined as a real epsilon minus j sigma / omega. But that means that you ascribe loss in a dielectric to "conduction currents" given by sigma even though no such currents exist in reality. But the math works.

My inclination is to SIMPLY use epsilon-hat which takes ALL dielectric AND conduction effects into account in one complex number. Then you can solve for the wave equations away from driving sources without separately solving for additional "source" terms (charge or currents) caused by the wave itself. You can still say D= epsilon E and n^2 = epsilon/epsilon0. The opposite approach is also valid (but less useful): just talk about the "microscopic" physical laws using E and B (don't even talk about H, D, P, M, or epsilon) but then every material included in the problem must be explicitly taken into account in terms of the charges and currents induced in it by the wave. If you take a middle approach, where some things are and some things are NOT included in epsilon, then you are always stuck trying to legislate which aspects are assigned to polarization and which to conductivity. The equation in the WP article assigning the real and imaginary parts of n^2 to epsilon' and sigma is also a concise notation but artificial since sigma no longer means conduction by free electrons. Even people who don't use complex epsilon as used in that equation, usually DO allow epsilon due to dielectrics to be complex (so you can talk about resonances) and properly identify n^2 with that epsilon (where no other conduction takes place). So by going away from complex epsilon you are stuck with an epsilon that is STILL complex, or you have to talk about conduction which doesn't exist. And if you do talk about a complex epsilon plus conduction, then two different people could measure the optical properties of the same substance and disagree about how to divide up where the loss occurs, so they don't agree on those numbers. What's more, at high frequencies not real small compared to the plasma frequency, you also have the effect of the inertia of the conduction electrons which is not described by sigma (unless you let that become complex too!) so you would have to ascribe it to a "polarization" (actually negative polarization) which doesn't exist in reality. Do you include that in epsilon then? Or complex sigma?

Do you see the problem? Interferometrist (talk) 18:48, 7 February 2011 (UTC)

As far as "thinking it through", I'm just reporting what the sources say. 100% of the sources I checked distinguish between displacement current and conduction current. I looked in the following:

• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1964), The Feynman Lectures on Physics Volume 2, Addison-Wesley, ISBN 020102117XP {{citation}}: Check |isbn= value: invalid character (help) p=32-10
• Griffiths, David (1989), Introduction to Electrodynamics (2nd ed.), Prentice-Hall, ISBN 013481374X p=369
• Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill p=6
• Hayt, William (1981), Engineering Electromagnetics (4th ed.), McGraw-Hill, ISBN 0070273952 p=400.
• Jackson, John David (1999), Classical Electrodynamics (3rd ed.), John-Wiley, ISBN 047130932X p=312
• Jordan, Edward; Balmain, Keith G. (1968), Electromagnetic Waves and Radiating Systems (2nd ed.), Prentice-Hall p=120
• Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, ISBN 0070354235 p=445
• Marshall, Stanley V. (1987), Electromagnetic Concepts & Applications (1st ed.), Prentice-Hall, ISBN 0132490048 p=114
• Purcell, Edward M. (1963), Electricity and magnetism (1st ed.), McGraw-Hill, ISBN 070048592 {{citation}}: Check |isbn= value: length (help) p=115.
• Ramo, Simon; Whinnery, John R.; van Duzer, Theodore (1965), Fields and Waves in Communication Electronics, John Wiley p=235
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory (4th ed.), Addison-Wesley, ISBN 0201526247 p=167
• Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 993013846 {{citation}}: Check |isbn= value: length (help) p=401
• Stratton, Julius Adams (1941), Electromagnetic Theory, McGraw-Hill p=14.

Although, Jackson agrees that conductivity could be handled as a complex permittivity, except at DC.

Now I will give you my opinion why. Many practical situations can be modeled with a constant, frequency independent conductivity and a constant frequency independent permittivity. Modeling it with only a complex permittivity would require a strongly frequency dependent permittivity. And it covers the DC case.

Also, this article is about skin effect. I think we have wandered too far from the subject of the article. The plasma frequency stuff is interesting, but I think belongs on some other article. On this article, I think we can tacitly presume that the frequency is well below the plasma frequency.Constant314 (talk) 04:17, 8 February 2011 (UTC)

Hi, I don't have time to look at this in detail now, but THANK YOU for checking so many references. If there is a consensus on terminology, then I certainly don't want to challenge it (nor have I in anything in this article). But I wonder if that is exactly the case. Do they all have the same definition of D, epsilon, and curl H, and which charges or currents are "free" or considered part of P? Is epsilon constrained to be real (otherwise it's value is ambiguous as its imaginary part can be traded off with sigma). I DO think there is a problem here, but it has no reprecussions regarding this article.

But one thing: "skin effect" = "penetration depth" of an electromagnetic wave and applies at ALL frequencies >0. For instance, light penetrates aluminum about 10nm = the skin depth at 500THz. Largely due to conduction electrons but no one can actually measure the source of that (or needs to!); we can only measure its complex refractive index and thus complex epsilon (as defined in that other WP article) which is sufficient. But look, I wasn't saying not to express things in terms of conductivity when that is the main thing physically occurring! Only that its effect can be lumped into the complex epsilon which is especially useful in solving the wave equation in a medium (you can then ignore all "free" charges and currents), and again that has nothing to do with this article or (as I came to realize) that general formula for skin depth which uses "epsilon" (but in which it cancels out in practical cases). (BTW, I haven't checked that formula out in detail, but my formula for the skin depth is simple: one radian of wavelength lambda/2pi divided by the imaginary part of n = (epsilon/epsilon0)^2 using the formulation of complex epsilon). So we don't disagree about anything affecting the article, but I am still trying to (and thank you for looking these up) understand what different people mean by these terms which may well cause confusion (as may be seen on the Poynting vector talk page for instance). Interferometrist (talk) 12:59, 8 February 2011 (UTC)

I should have mentioned this: the formula I just gave assumes NO magnetic permeability (mu=mu0). As you might have gathered my main work is in optics :-) but it should be correct using n= (c^2 * espsilonhat * mu)^2 I believe. Sometime when I'm bored I'll compare that with the "big formula" to find out what definition of epsilon and sigma in that formula makes these equivalent. Interferometrist 13:19, 8 February 2011 (UTC)

And of course I meant sqrt(c^2 * espsilonhat * mu) :-) Interferometrist 13:25, 8 February 2011 (UTC)

I finally went ahead and calculated the skin depth in the very general case based on the very simple formula I told you in words, but completely in terms of real numbers which made it come out very messy!! See what I am going to write below. Also, where did the "big formula" originally come from? I now understand the extent to which it is valid (but I didn't get to figuring out what modifications to sigma and epsilon one would have to make in order for such equivalent quantities to be inserted into that equation in order to obtain the general equation). In the process I noticed that it could be written somewhat nicer with the radical in the numerator. I won't change the original form given in a reference, but I will for the rewritten form I added (since it WAS my own manipulation of the expression, and I trust that doing so isn't considered original research). Check for my new comments on this page. Interferometrist (talk) 20:00, 8 February 2011 (UTC)

Never mind my stupid question: I can see the references right there in the footnotes! Interferometrist (talk) 20:04, 8 February 2011 (UTC)

If you want to add the high frequency near optical information, perhaps you should add a new section called extremely high frequency or something like that.Constant314 (talk) 14:20, 9 February 2011 (UTC)

See below Interferometrist (talk) 14:38, 9 February 2011 (UTC)

Induction cookers....

"A practical consequence is seen by users of induction cookers, where some types of stainless steel cookware are unusable because they are not ferromagnetic."

Really?? I thought all steels including stainless steel are ferromagnetic! (A magnet sticks to it!) Or is mu lower for certain types? So you're saying that the induction cooker relies on a smaller skin depth so that the alternating magnetic field gets fully absorbed by the pan? That would mean aluminum pans are disallowed, is that right? I've been interested in buying one but they're still a bit pricey AND have serious restrictions on what sort of pans you can use (but I hadn't looked at the details).

What you added is great, but why don't you add a few words and spell it out a little more clearly for the reader: that pans which are not ferromagnetic have too large a skin depth. Interferometrist (talk) 16:51, 7 February 2011 (UTC)

Could prattle on a bit longer, I suppose. It's very entertaining running around with a magnet seeing what sticks and what doesn't. Check the stainless steel article; austenetic stainless steel (non-ferromagnetic) can be transformed to martensic steel by working it. You can actually go around the inside of a kitchen sink and see parts where the magnet sticks, and parts where it doesn't.

Aluminum pans don't work because the skin depth is so deep, and the resistivity is so low, that the effective "surface resistance" per unit area is too low. The pot acts like a shorted turn of a transformer - the induction cooker can only induce so many amperes in that shorted turn, and if there's not enough resistance in the shorted turn, not enough heat gets produced. Foil gets hot, though - thinner than a skin depth, so there's enough surface resistance. I've melted aluminum foil on my cooktop, see the induction cooker article.

Yes, induction tops are pricey, but I like the cleanliness and relative safety; I'll never pay for it out of energy savings, but it is nice to hit a button and see the boiling stop like turning off the jets in a hot tub. As for pots, we had to give away two aluminum pots - and we really need a second frying pan, but we were able to use most of the "fleet". --Wtshymanski (talk) 03:37, 8 February 2011 (UTC)

Now you've got me interested and I'm going to be checking my pots and pans with a magnet! If I have one or two that qualify I'm going to buy a single element induction hotplate for about 50 euros (and not just to use for unwise experiments, like I used to do with microwave ovens ;-) Interferometrist (talk) 13:04, 8 February 2011 (UTC)

Alas, all of the pans in my house aren't attracted by a magnet at all! Much to my amazement. I also noticed the same -- unexpected! -- phenomenon testing it on my metal kitchen sink as you described. Anyway, I have a coin-sized but very strong magnet I'm going to take around with me for testing things. I was about to slip it into my wallet .... but then saw my bank card and pulled my hand right back.... Interferometrist (talk) 15:10, 9 February 2011 (UTC)

My favorite kind of Wikipedia moment is when I learn something; the stainless steel article pointed out the austenitic structure is changed by cold-working and that one deep drawn stainless part (like a sink!) could have varying permeability, and I, too, was delighted by running a magnet around the bowl of my sink. Places that sell pots and pans often have fridge magnets for sale, too. Some European pans are apparently marked "Induction" to show their compatibility. That will let you know that the skin depth in the pan is shallow enough to make it useable <bringing this discussion back on article topic>. --Wtshymanski (talk) 15:21, 9 February 2011 (UTC)

Discharge tube circuits

[The skin effect] can also be of importance in the design of discharge tube circuits.[1]

[1] "The Formation and Implosion of a Current Sheet in a Gas Discharge containing Reversed Magnetic Fields". Proceedings of the Physical Society Volume 79, Number 1 Issue 1 January 1962. Retrieved 6 February 2011.

With all due respect to the person who added this reference after I tagged it, I really really think that this one application where the skin effect is important doesn't belong in the WP article. I hadn't known anything about discharge tubes, but unless there are a lot more references (and ones more recent than 1962!), and ones by engineers in fields other than atomic weapons (!), I'd tend to conclude that the importance of the skin effect in this regard could hardly be mentioned as a third item (following RF, then power mains) in the introduction. I'm removing the sentence. Thanks anyway for finding a reference in response to my skepticism, but in this case it swayed me in the opposite direction. Interferometrist (talk) 21:22, 9 February 2011 (UTC)

Other corrections applying at higher frequencies

Constant314: If you want to add the high frequency near optical information, perhaps you should add a new section called extremely high frequency or something like that.:

That's not a bad suggestion but is slightly misstated. As you can surely appreciate, the fundamental laws of physics don't change when you go up in frequency (ignoring quantum-only effects like pair production) so "high frequency" is all relative. You saw this with the "poor conductor" correction which would more correctly be called a (different) high frequency correction, that cutoff frequency being extremely high (I mention 10^18 Hz) for metals, but had to give the example of silicon where it would happen in the megahertz range. There are two other critical frequencies where even the big formula fails. First when you get near/above 1/tau_c where tau_c is the mean time between electrons' collisions. Second is when you approach the plasma frequency. I generated the math to handle these situations (yes, even in terms of epsilon', epsilon" and sigma, but you need to specify one more thing such as the carrier density N, tau_c or omega_p). But that's "original research" so I can't write it into the article unless someone can find such a reference. I'm holding back on editing for that reason.

But the frequencies I'm talking about are only extremely high in the case of metals (where you have a large N and omega_p). In the case of the plasma in the ionosphere (whose characteristics change greatly and are also affected by the magnetic fields.....) in one example, N=30e9 /m^2, sigma=.1S, tau_c=.1msec, and it thus acts like a metal (low loss) but everything is shifted to much lower (radio) frequencies. Above 1Khz the "high frequency" regime is entered with a skin depth ~40m. Then above 1MHz the plasma frequency is approached and the skin depth increases. At 1.6MHz it becomes essentially transparent, delta=18km. (if my calculations are correct).

The easier thing for me to do would be just to state the agreed-upon depth of penetration of an EM field into an attenuating medium: lambda/2pi / Im(n) and NOT give a detailed explanation of how n can be predicted from all of the possible effects (dielectric, atomic resonances, conductivity with N electrons/m^3 and tau_c, and of course mu). Would that be good? Along with examples?

Any opinions out there (there are supposed to be 59 people watching this page!) in addition to Constant314 I would welcome :-) Interferometrist (talk) 14:59, 9 February 2011 (UTC)

Hello from an occasional watcher of this page. Coming back after many months, I am pleased to see a bunch of new, good content. However, I think that many people reading this article are interested in practical problems of using hi-conductivity metal conductors at frequencies between say 1 Hz and 100 GHz-- or more likely 50 Hz to 5 GHz. So I would support having the "high-frequency" corrections in a section near the bottom, with the stuff at the top only having a pointer to that later section.

I think a similar approach to the new stuff about inductance versus frequency would be appropriate. I would like to see the examples section before the inductance section.Ccrrccrr (talk) 14:38, 19 February 2011 (UTC)

I thought the table of line parameters in the inductance section led naturally to the examples section, but I changed it as you suggested.Constant314 (talk) 14:46, 19 February 2011 (UTC)

Basically I agree with you: it would be sufficient in discussing the skin effect to talk about the effect of current flow in a metal at normal frequencies. But there was a desire to include a fancier formula that used to be entitled "high frequency correction" but where (as usual!) "high" is relative, as I pointed out with silicon where this happens at a few megahertz. If you do that, then there are a few other corrections to be noted too which can apply to plasmas, for instance, at radio frequencies. But I would not mind getting rid of all but the basic formula (since that's what people are probably interested in) and refering the reader to the article on penetration depth in relation to electromagnetic radiation including optical frequencies and low conductivity. Of course if you take that approach, the section on the inductance of a wire is also of no practical concern to anyone, but I'll stop repeating myself..... Interferometrist (talk) 19:26, 20 February 2011 (UTC)

Article section "Effect on Inductance"

I don't know who wrote this, but it's not too terribly coherent. Perhaps interesting (I didn't know that the inductance is reduced by the skin effect) but isn't complete or even readable! Does someone want to fix it? Or trash it?Interferometrist (talk) 21:21, 6 February 2011 (UTC)

Do you have any suggestions? Is there some part in particular that is difficult to read?Constant314 (talk) 15:39, 7 February 2011 (UTC)

I assume you aren't being serious! I finally removed it altogether, especially realizing what a small effect was being discussed and then only applying to the self-inductance of a single wire (not a coil). Interferometrist (talk) 23:28, 7 February 2011 (UTC)

Alright, I see you restored that section, and added a nice diagram and an outline explanation of the phenomenon. I still think that this effect is quite obscure and of almost no practical consequence and shouldn't be in the WP article, but for now I will just edit it to put its (un)importance in perspective.

*If you look at the history of this talk page you will see that there was a specific request for effect on inductance. At least one person wanted to know.

*You will also see a comment from GyroMagician from which I infer that at least one other person was able to read and comprehend the information.

*If you look at the telephone cable data you will see that there is at least a 20% change in inductance, much of it right in the DSL band. Both velocity and impedance are frequency dependent. The effect is not insignificant. Constant314 (talk) 23:34, 20 February 2011 (UTC)

It also leads to the following paradox. Can someone out there resolve this for me? If the skin effect reduces the self inductance per meter of the inner conductor of coax, and the capacitance per meter doesn't change, then why does the coax's characteristic impedance (sqrt(L/C)) and speed of propagation (1/sqrt(LC)) not change? You don't even have to think about the skin effect for now: what about the difference between coax using a solid inner conductor (at low frequencies, say) and one with a tubular inner conductor (equivalent of the skin effect) which has a smaller inductance per unit length according to this. Assume the wire's mu=1 and no dielectric so epsilon=1. Now everyone knows that this propagates signals at exactly the speed of light. BUT, if the inductance were to change then that wouldn't be true at all frequencies! What's wrong here? Interferometrist (talk) 18:50, 20 February 2011 (UTC)

I presume that for coax that the impedance and velocity parameters are specified for frequencies where the skin depth is already well developed.Constant314 (talk) 23:34, 20 February 2011 (UTC)

Actually I think I've resolved the paradox. The normal explanation of coax cable ASSUMES that the conduction is at the surface (which is why the outer diameter of the outer conductor doesn't enter into the equation, for instance). Or equivalently, it ignores the magnetic field inside the conductor. The TEM wave is strictly in between the two conductors, in free space where indeed the wave propagates at c (with no dielectric). BUT if you DO have conduction in the interior of either conductor, THEN the EM wave is no longer propagating strictly in a vacuum but is partly contained inside the conductor where of course the propagation velocity is far different (insofar as you can talk about "propagation" in a reflecting medium) and slows down the wave. And the characteristic impedance of the coax must also be a bit higher at low frequencies. The insignificance of this effect is illustrated by the fact that no one (am I wrong?) quotes the characteristic impedance as a function of frequency (except insofar as leakage conductance has an effect at low frequencies, a different issue) or talks about coax as being dispersive. Interferometrist (talk) 19:50, 20 February 2011 (UTC)

The characteristic impedance of coax is very high at low frequency once the frequency is low enough that resistance dominates the inductance.

regular old RG58 is dispersive. Look at this picture from the Time domain reflectometer article

TDR trace of a transmission line with an open termination.

you can see that the reflected pulse is spread out with respect to the stimulating pulse.Constant314 (talk) 23:34, 20 February 2011 (UTC)

Also, I didn't change it out of respect (again, I think the whole section should be removed for a WP article) but if we are going to have a table concerning telephone cable why bother including the irrelevant columns? Just the L per kilofoot (and why don't you change it to meters?!) is what's relevant. The referral to extensive tables and the formula (with symbols that aren't defined!) seem particularly out of place (why not put that in a WP article on telephone cable?). The point in regards to the skin effect (the subject of the article!) has already been made. Interferometrist (talk) 19:14, 20 February 2011 (UTC)

The table is there and in those units because that is the way I found the information in the reference. The raw data is virtually indisputable, where as my interpretation of it might be considered WP:OR.Constant314 (talk) 23:34, 20 February 2011 (UTC)

Propagation speed does change with frequency in a coax. Group delay issues can dictate against using baseband signaling. IIRC, group delay was an issue for 10 Mb/s Ethernet on coaxial cable: RG-58 coax segments could only be 1/3 the length of RG-8/U segments; skin depth issues are proportional to conductor diameter and the two coax types are roughly a 1:3 ratio. The very high speed serial data formats (e.g., USB 3.0, SATA 6) use active equalizers.

I have a dim memory of some coaxial cables trying to match increased skin loss with another effect, but I don't recall what the other effect was.

To reduce skin effect issues, engineers will go to larger diameter cables, but there's a limit. At microwave frequencies, the cable diameter must be small to avoid other propagation modes.

From another perspective, skin loss is proportional to sqrt(f) but dielectic losses are proportional to (f)[6]; at some point, the dielectric losses in ordinary coax will dominate the skin effect; tan delta of 0.00001 is trouble but hard to achieve.

Consequently, practical issues may weigh against covering secondary skin effect issues here. Right now I'm neutral.

Glrx (talk) 23:21, 20 February 2011 (UTC)

Well thank you both for your answers. Although I'm a little confused about RG-58 vs. RG8, since it would seem that if the issue is "baseband" phase fidelity (dispersion) then a smaller conductor would actually be better from the standpoint of the skin effect affecting the inductance per unit length (and thus phase velocity) since there would be less change when the skin effect kicks in (but I haven't thought it through. And it would appear that a larger b/a would give less weight to the magnetic field inside the conductor, but that raises the characteristic impedance and not be acceptable). On the other hand I recognize that at a radio frequency the attenuation of RG-8 is less so that for broadband RF there would be better fidelity with RG-8, however that's a different issue. Or aren't you talking just about this issue? :-) And I recognize that the characteristic impedance at low frequencies is very different (governed by the series resistance and shunt conductivity, the latter being small) but at those frequencies (where the cable length << a wavelength) the importance of the characteristic impedance is reduced.

Regarding RG58 vs RG8, you may find this surprising, but it is in Hayt and Grover, the inductance attributed to the field inside the center conductor is independent of the radius. A larger wire has more volume but the current density is less and the intensity of the magnetic field is less and it balances out to make the inductance of the center conductor independent of radius. Constant314 (talk) 01:57, 21 February 2011 (UTC)

But thank you for the interesting facts about coax transmission and equalization problems! I guess I'm wondering (as I guess you were) whether the issue of the skin effect IN RELATION TO the change in inductance (which you call a "secondary" effect) is so important compared to other low frequency effects. And of course the major role of the skin effect is at high frequencies causing more resistive loss (the main attenuating factor, no?) but the article has already talked about that. I didn't mean to make a big deal about this one section of the article, but I honestly think it more belongs in the article about coax and transmission lines. Some WP articles are way too long with a lot more detail that you'd expect in an encyclopedia, but thankfully this isn't one of them. I was worried that with this discussion about the shift in the inductance of a wire, it would be going in that direction. I was also worried that it would be confusing to an average person with a knowledge of electronics who sees "inductance" in terms of a lumped component consisting of a coil, after all. I included a sentence to make that clear. The increased resistance of wire at high frequencies may be of some practical importance to such a person, but thinking about the distributed inductance inside a coax isn't; you just read the spec sheet to learn how it behaves. Interferometrist (talk) 00:41, 21 February 2011 (UTC)

I used to design TDR equipment for the phone company. It had several pulse widths. We had to explain over and over that the long pulses were slower than the short pulses and that there wasn't anything wrong with the instrument. The frequency dependence of propagation velocity is important to these people. They may not understand it as a frequency depedent velocity, but it impacts measurements that they make every day. I'm going to take a wild guess that there are 100,000 TDR's of all types in the hands of telephone technicians worldwide. So there are 100,000 people or so who's daily activity is directly effected the propagation velocity of telephone cables.Constant314 (talk) 01:34, 21 February 2011 (UTC)

I agree with you that the skin effect on inductance is almost nil with regard to lumped inductors. In fact, the only situation that I am aware of where the change of inductance is important is in transmission lines.Constant314 (talk) 17:27, 22 February 2011 (UTC)

New graph by Zureks

Thank you for the improved graph of the skin depth vs. frequency which includes a few different types of steel, showing the differences in their skin depth (presumably due to differences in mu) which I find interesting (and which may be related to the conflicting values for the skin depth of iron formerly cited in the article which surely related to different crystalline forms). Maybe, though, it would help to improve the caption as well so that someone (like me!) could appreciate it better.

Did you generate this yourself, or copy it? In either case I think the curve labeled Mn-Zn needs to be explained. What does that mean, exactly? Zinc is a good conductor and while manganese isn't such a great conductor, its skin depth would still appear to be smaller than what the graph implies. Can you explain to me, and then (in the caption) to the readers as well? Thanks, Interferometrist (talk) 22:45, 24 February 2011 (UTC)

All the information about how the graph was generated is on the image description page. But answering your specific questions:

1. I generated the graph myself using the equation for "good conductors" and material data from sources given on the file page.
2. Mn-Zn - magnetically soft ferrite. Neither Mn nor Zn are good magnetically, but the ferrite is quite good soft ferromagnet, used commonly for high frequency applications. The permeability is moderate, but because it is really an oxide not pure metal the conductivity is very low so the skin depth is in different league as compared to regular metals.
3. Steel. Yes, the difference in skin depth is related mostly to the differences in μ. The stainless steel (410) is ferromagnetic, but not very good magnetically so the depth value is larger. Fe-Si denotes grain oriented electrical steel which is much better magnetically. Fe-Ni is the high-nickel alloy, superior magnetically hence lowest value of skin depth.

Regards --Zureks (talk) 08:57, 11 July 2011 (UTC)

Into and out of the screen

There have been several recent good faiths edits by User:Tiscando to explain the color scheme in a cross section of the wire. To me, those edits added confusion, so I reverted them. Many physics books have diagrams that describe currents flowing into or out of the page. The text has been altered again. I would like some support in reverting (or better explaining). I would add some text that explains the view is a cross section, and the current flow is perpendicular to the screen. I would keep the into and out of screen statements. Glrx (talk) 15:59, 21 July 2011 (UTC)

maybe into and out of the figure instead of the screen.Constant314 (talk) 17:38, 21 July 2011 (UTC)

Hence the first of my edits, "into the (computer) screen" was confusing to me, so thanks for disambiguating it to "diagram"; I think that is the best term to use. Tiscando (talk) 13:05, 6 August 2011 (UTC)

Temperature analogy

A periodic temperature as a boundary condition will induce a temperature “skin depth” analogous to the electromagnetic skin depth, varying as (alpha/omega)^(1/2), where alpha is the thermal diffusivity, and omega is the angular frequency of the surface temperature. This “skin depth” occurs during the heating of the earth’s crust by the sun, and in some industrial processes. (See, for example, “Analysis of Heat and Mass Transfer” by E.R.G. Eckert and R.M. Drake, McGraw-Hill, 1972, p.208-214.) Is there a place for this either here or somewhere else? Psalm 119:105 (talk) 12:57, 1 October 2011 (UTC)

I think the analogy is a bit too complex to explain skin depth well; in the climate in my region, we're familiar with the idea that water piping, etc. must be buried below the "frost line" but Wikipedia analogies have to speak to more people than those living in a (nearly) sub-Arctic climate. And just how familiar is the reader going to be with the variation of temperature with depth, or the notion of "thermal diffusivity"? Some of those deep South African mines get really hot, so this is counter to the temperature gradient driven by air temperature. Could be referenced and discussed in anything relating to soil temperatures, but I think the fundamental physics are far enough apart that the analogy isn't really a help. --Wtshymanski (talk) 13:12, 1 October 2011 (UTC)

Kft?

What exactly is the unit 'Kft' that is used in the table at the very bottom of the page? CodeCat (talk) 18:33, 20 December 2011 (UTC)

Thousand feet. --Wtshymanski (talk) 18:53, 20 December 2011 (UTC)

Please put some "Whys" instead of "Whats"

I'm trying to understand various things to do with EM waves, yet all I can see on wikipedia is a bunch of "Whats", decribing what it is in words, then what it is formulae. But I want to know why!!!! Everything on this page is useful if you wanted to calculate the effect, or if you never heard of it and wanted to know what it is, but I've read all that, I know WHAT it is, I want to know why it is. Please, it's driving me nuts!!! -OOPSIE- 10:56, 27 February 2007 (UTC)

Agreed! From what (little) I understand, it's due to electromagnetic inductance and / or eddy currents or some such, but I'd love for there to be a nice explanation in the article. Nickwithers 03:44, 2 March 2007 (UTC)

I'll do my best. Meanwhile, take a look at the latest Eddy Currents article and the Discussion page of Skin Depth. I'll work on the Skin Effect and Skin Depth articles when time allows. At the moment, I agree that they are good engineers' articles, but the physics is rather cloudy. StuFifeScotland 15:08, 27 March 2007 (UTC)

+1 for some explanation. It's a good thing that we have forumlas for this effect, but what properties of matter cause this? Why the AC waves like to propagate on the outer region of a conductor? Which force makes them act like this? PAStheLoD (talk) 11:27, 14 July 2008 (UTC)

Here is a hand waving explanation. The only way to have current deep inside a conductor is to have some electric field in there to create and sustain a current. The only way to get electric field into a conductor and sustain it is too propagate it into the conductor from the outside. The ability of conductors to resist penetration by high frequency fields gets greater as the frequency increases. But then why that and why does it take the form that it does? I don’t have an intuitive explanation for that. Constant314 (talk) 12:37, 27 May 2010 (UTC)

I think i am very wrong... "Heating due to Skin effect is caused by the conductor having a poor magnetic permeability. A material with a poor magnetic permeability takes longer for the current to move into the material so when an alternating current is moving slowly (low frequency) there is time for it to utilise a deeper amount of the material, but when it is moving quicker (higher frequency), there is less time for it to seep into the material. When there is a small amount of surface area for the electrons to pass through, the density of the electrons build up causing friction between the electrons as they move." any punts at this? Sirnails (talk) 15:52, 22 May 2012 (UTC)

Howard Johnson says "A large value of magnetic permeability shrinks the skin depth to an incredible degree" see http://www.signalintegrity.com/Pubs/edn/steelplated.htm Constant314 (talk) 02:38, 23 May 2012 (UTC)

Bundled wires

The reason for bundling wires in HV lines is definitly NOT the skin effect, but the prrevntion of corona losses. By creating a bundle, the electric field strength on the surface is brought below the critical value of 17kV/cm (resp. 43 kV/inch ), which is the startpoint for corona effects, which create even bigger losses than the skin effect and which have surface corrosion as effect ! — Preceding unsigned comment added by 188.250.162.187 (talk) 20:18, 23 July 2012 (UTC)

Clarification of the resistance formula

I spreadsheeted the formula given (the one below) and found that there are issues not described in the text. Thinking there was an error in the formula i created my own formula but it has the same issues. I found that if the skin radius is greater than the radius of the conductor, the output will not be negative, instead, if you give 1mm greater than the conductor, the answer will be the same as 1mm smaller than the conductor.

The formula i chose to use: ρL/(π(D-δ)δ)

The formula i created: ρL/(π(r²-(r-δ)²)) (note: requires wire radius not diametre)

Should the skin area be greater, this formula is to be used: ρL/(πr²)

If the conductor is not round and the skin area is greater, use: ρL/A

For convenience, here are some values for permeabilies and resistivity in Ω·m. Relative Permeability figures from Nathan Ida's "Engineering Electromagnetics"

material	resistivity (Ω·m)	R permeability (μ)
gold	2.44×10^-8	0.999998
copper	1.68×10^-8	0.999994
lead	2.2×10^-7	0.999983
silver	1.59×10^-8	0.99998
aluminum	2.65×10^-8	1.000021
iron	see below	see below
graphite	(3to60)×10^-5	0.999956
tungsten	5.6×10^-8	1.0000019
mercury	98×10^-8	0.999968

Iron permeability figures are 7000, 6000 and 100 for 4%Si, 100%Fe and 0.9%C.

Iron resistivity figures are 4.72×10^−9, 9.6×10^-6 and 1.7×10^-5 for 3%Si, 100%Fe and 0.5%C

Stainless resistance: 304 is 6.897E-07 316 is 7.496E-07 According to someone with a username of Ron07663 on a forum, he used 316 and 305SS and after deep drawing, the 316 steel was "1.15 right out of the tool and 1.0003 after solution anneal. The 305 parts measured 1.003 right out of the tool." on the same forum it is said that it depends on the amount of deformation; "Annealed 304 or 316 SS should be on the order of 1.0 to 1.02." (this could be heresay).

Charlieb000 (talk) 21:41, 6 April 2014 (UTC)

Question about the back EMF at the center of the conductor

Sorry if the question is trivial, but why the back EMF is strongest at the center of the conductor? Saung Tadashi (talk) 14:35, 27 December 2014 (UTC)

Here we discuss the article, not the content. It is also not the place to ask questions about the subject—see wp:talk page guidelines. Someone might be able to help at our wp:reference desk/Science. Good luck! - DVdm (talk) 14:50, 27 December 2014 (UTC)

I thought the answer to this would be trivial. However, I just looked through 18 books on E&M. Of the 14 that derive skin effect, not one of them developed it from the inductive coupling of successive cylindrical shells of current. Every one of them derived the skin effect from the attenuation constant of a plane wave penetrating a conductor. I think that it would be legitimate to ask for a citation regarding the entire explanation.Constant314 (talk) 16:01, 27 December 2014 (UTC)

I added a citation establishing that the counter emf is greatest at the center. Constant314 (talk) 17:41, 28 December 2014 (UTC)

Promoting the simpler form of the equation for skin depth in the general case

The general formula for skin depth is shown in two forms. The first form is

$\delta =\left({1 \over \omega }\right)\left\lbrace \left({{\mu \epsilon } \over 2}\right)\left[\left(1+\left({1 \over {\rho \omega \epsilon }}\right)^{2}\right)^{1/2}-1\right]\right\rbrace ^{-1/2}$

and the second form is

$\delta ={\sqrt {{2\rho } \over {\omega \mu }}}\;\;{\sqrt {{\sqrt {1+\left({\rho \omega \epsilon }\right)^{2}}}+\rho \omega \epsilon }}$

The first is manipulated into the second, by multiplying the right hand side of the first form by unity in the form

$\left\lbrace \left[\left(1+\left({1 \over {\rho \omega \epsilon }}\right)^{2}\right)^{1/2}+1\right]\right\rbrace ^{-1/2}\left\lbrace \left[\left(1+\left({1 \over {\rho \omega \epsilon }}\right)^{2}\right)^{1/2}+1\right]\right\rbrace ^{+1/2}$

and then applying ordinary rules of algebraic simplification.

The simplified form was first added by Interferometrist in Feb 2011. No one has challenged it and I have verified it to be an exact algebraically equivalent formula. The simplified form is better for several reasons: it is simpler, the low frequency approximation is obvious, the high frequency limit is simple to compute and the simple form is numerically stable in that it gracefully converges to the high frequency limit whereas numerical evaluation of the first form suffers numerically from finite precision arithmetic effects. Since the simplified form is exactly the same, I suggest that we suppress the complicated form and replace it with the simplified form along with a note that the form in the reference is algebraically equivalent. Constant314 (talk) 19:41, 15 August 2015 (UTC)

Yes, thanks, you understood exactly my intention. I'd say go ahead and edit it as you suggest. I remember having added that equation and explanation as to the low and high frequency asymptotic forms directly revealed by it, but that must have been quite long ago so I'm not up to speed on the matter and would rather not edit it myself. But I figured at the time (and have seen examples many times elsewhere on WP) that algebraic manipulation of a formula is not OR, and is thus appropriate if it aids clarification for the reader. Interferometrist (talk) 17:15, 6 September 2015 (UTC)

Caption problem on second image

Caption on second image: "The 3-wire bundles in this power transmission installation act as a single conductor. A single wire using the same amount of metal per kilometer would have higher losses due to the skin effect."

First off, the caption starts by assuming that it's obvious in the picture what "3-wire bundles" refers to, which it is not.

Supposing one assumes that what appear, in the photo, to be individual cables are actually 3-wire bundles, presumably the 3-wire arrangement is chosen precisely because it does _not_ act as a single conductor, contrary to the first sentence, and in agreement with the second.

So I'm not sure what the caption author was trying to say, but it's not coming through. Gwideman (talk) 21:25, 1 October 2017 (UTC)

Internal Inductance or self inductance

I have been looking at a number of transmission line and PCB design reference books. Every one of them uses the term internal inductance for the inductance associated with the magnetic field that is internal to the wire. I presume that the self inductance of an isolated wire would be associated with both the internal and external magnetic field. The formula the weeks gives

$\mathbf {Z} _{int}={\frac {k\rho }{2\pi R}}{\frac {J_{0}(kR)}{J_{1}(kR)}}$.

only includes the internal inductance. You are right that it is the impedance per unit length. Constant314 (talk) 17:56, 29 January 2018 (UTC)

It's just that in most cases the "internal inductance" is small compared to the actual inductance. I don't see it as a separate inductance but as part of the inductance that is LOST (a negative number) due to areas where the magnetic field due to the wire's current is reduced due to the part of the current that is not flowing inside a radius r (compared to a uniform current profile when there is no skin effect). In other words, it is more of a correction (<0) DUE to the skin effect which is why it's even mentioned on this page. And it is only important for the inductance calculated for a transmission line where the magnetic field is localized due to the opposite current in the return conductor, so that the portion of the field inside the conductor itself becomes significant. For an isolated wire of any length this effect is tiny and I'll bet it couldn't even be measured.

At least that's my understanding, but I don't have the reference books handy which is why I was asking someone who does to check that formula, and see if it even applies to a single wire. Interferometrist (talk) 20:01, 29 January 2018 (UTC)

Also note that if the correction (dependent on the skin depth dependent on frequency) were very large even for coax, then the characteristic impedance of a coax would depend on frequency. You and I have never seen that mentioned in coax specs (have you?). Interferometrist (talk) 20:09, 29 January 2018 (UTC)

I was also going to say that this formula I find extremely opaque especially after realizing k is complex. Bessel functions are difficult enough and I believe this is the first time I've seen them with a complex argument. I wish there were a way to write this relationship that is more useful for understanding. Interferometrist (talk) 20:21, 29 January 2018 (UTC)

I believe that you are correct that for an isolated wire the internal inductance is relatively unimportant. In transmission lines it can be important. In plain old telephone service, the internal inductance is significant at voice frequencies. There is a sort of backwards reason that internal inductance is important at higher frequencies. If you notice the case of round wire, once the skin effect is well developed, the reactance of the internal inductance asymptotically approaches the resistance. Many approximations for the high frequency resistance of conductors of other shapes (especially PCB conductors) use this fact (it is not entirely true, but often good enough and many sources just assume that it is true), because it is easier to compute the internal inductance than it is to compute the resistance.

As for the formula for round wire, I believe that I faithfully reproduced the formula from Weeks, but it doesn’t hurt to get more eyes on the source. Weeks, by the way, is cited by Hayt. Constant314 (talk) 20:31, 29 January 2018 (UTC)

Thanks. I went ahead and figured out the Bessel functions for |kR|<<1 even for a complex quantity. In the case where the skin depth >> R (thus no skin effect), the formula gives ZERO imaginary part, thus NO internal inductance. Does that make any sense though?? Or did I make a mistake? I was just substituting in the expressions for k and delta in the previous section. Interferometrist (talk) 20:50, 29 January 2018 (UTC)

Yes, at DC the imaginary part of the internal impedance goes to zero because it is frequency (zero) times internal inductance (not zero). Constant314 (talk) 20:59, 29 January 2018 (UTC)

Well I see what you're saying but I didn't exactly say DC, just a large skin depth compared to R which could be due to low conductivity or a very small R where the internal inductance is supposed to reach a constant. Maybe I need to take the approximation one order further and then divide by frequency to get the inductance from the reactance as you say. Interferometrist (talk) 21:10, 29 January 2018 (UTC)

As frequency goes to zero, Weeks's formula has a phase that goes to +45 degrees, indicating an inductive part.Constant314 (talk) 01:38, 31 January 2018 (UTC)

Yes, I have it all sorted out now! (Though I think you're confused about the 45 degrees, it sounds like you're thinking of the HIGH frequency limit). I went through all the algebra using two terms for each Bessel function, and got exactly the right answer, though it still surprises me! I should have listened closer to what you were saying since this is obviously something you're very familiar with. And in particular let me apologize for foolishly asking why the characteristic impedance of cable doesn't decrease with frequency both because it clearly does if I had just looked at the telephone cable figures on the same page, and moreover because I now see that we have actually had this discussion on this talk page before! Which I would have known if my memory weren't so awful. Thanks for all your help! Interferometrist (talk) 01:54, 31 January 2018 (UTC)