# Voltage on a conductor due to skin depth

As we know due to skin depth effect the Alternating Current flows at a depth from the conductor surface (which means there is no voltage on the surface of the conductor, it is only at a depth from the surface).
Then how is it possible for voltage to get appeared across some other conductor which is connected to the sourcing wire (when wrapped around to tap power).

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As we know due to skin depth effect the Alternating Current flows at a depth from the conductor surface (which means there is no voltage on the surface of the conductor, it is only at a depth from the surface).
You misunderstood the skin effect. The current density is largest at the surface.

The current density J at depth x is

$$J_s e^{-\frac {x} {d}}$$

where d is the skin depth, and $J_s$ the current density at the surface.

davenn
Gold Member
2019 Award
and to add to what Willem2 has said, as the frequency increases the skin effect becomes more pronounced
That is ... the skin depth decreases as freq increases.
This is why it is common to find conductors in the UHF and up into microwave frequencies silver or gold plated to maximise conduction qualities of that skin zone. At microwave frequencies, the copper wire becomes nothing more that a support surface for the silver plating.

cheers
Dave

PS ... just a bit of good advice to you ....
Its really wise NOT to start a sentence with " As we all know.... "
unless you are really sure of your facts

Baluncore
2019 Award
Rahu. Welcome to PF.

It takes time for a surface current flow to soak into a conductor because it sees it's own reflection.

Higher frequencies have less time to soak in before they reverse, so they penetrate less.

davenn
Gold Member
2019 Award
Rahu. Welcome to PF.

It takes time for a surface current flow to soak into a conductor because it sees it's own reflection.

Higher frequencies have less time to soak in before they reverse, so they penetrate less.
not really a good description , Baluncore.
doesnt have anything to do with time taken for anything to "soak in" anywhere

try (from wiki) ....
An alternating current in a conductor produces an alternating magnetic field in and around the conductor. When the intensity of current in a conductor changes, the magnetic field also changes. The change in the magnetic field, in turn, creates an electric field which opposes the change in current intensity. This opposing electric field is called “counter-electromotive force” (back EMF). The back EMF is strongest at the center of the conductor, and forces the conducting electrons to the outside of the conductor.

cheers
Dave

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Baluncore
2019 Award
davenn said:
The back EMF is strongest at the center of the conductor,
I think I disagree with your model.
An EM wave will propagate on the surface of a conductor at close to the speed of light, 300 Mm/s.
An EM wave will propagate through a solid conductor as a wave.
The velocity through solid copper at 60 Hz is only 3.2 m/s.
The refractive index of copper in air at 60 Hz is the ratio 3x108 / 3.2 = 95 million.

What is happening now on the surface of a conductor has no immediate effect at the centre, the electric currents and magnetic fields won't get to the centre until quite a bit later, often many cycles later.

That is why I find the descriptive term “soak” so applicable to understanding skin effect.

berkeman
Mentor
There is some confusion here, so I'm closing this thread temporarily until I can sort out the technical details tomorrow.

berkeman
Mentor
Okay, I will re-open this thread now.

Baluncore -- your explanations are not accurate on this subject (you are usually correct, but this time there seems to be some misunderstanding).

The explanation at wikipedia is good:

https://en.wikipedia.org/wiki/Skin_depth
wikipedia said:
Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor, and decreases with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor. The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller

Baluncore
2019 Award
Ah' such is the dominant paradigm. I apologise for my deviance.

nsaspook
Okay, I will re-open this thread now.

Baluncore -- your explanations are not accurate on this subject (you are usually correct, but this time there seems to be some misunderstanding).
I can see his point, not all discussions of 'skin depth' are simply about electrical conductors and AC currents (like ground-penetrating radar). A better term might have been 'diffusion' instead of 'soaking' when discussing the propagation of EM waves in conductive and/or dielectric media.

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Baluncore
2019 Award
nsaspook, thanks for the understanding. But students new to EM are too delicate to be thrown out of wikipedia and into the jungle of the real world. Wikipedia has twice ruled the day here.

berkeman
Mentor
nsaspook, thanks for the understanding. But students new to EM are too delicate to be thrown out of wikipedia and into the jungle of the real world. Wikipedia has twice ruled the day here.
No, sorry Baluncore. On this technical issue you have the wrong understanding of how skin depth works. There is no justifying what you posted, and no disrespecting the technical explanation at wikipedia or in our textbooks.

meBigGuy
Gold Member
The wikipedia article is pretty simplistic.

A few quotes from http://fermi.la.asu.edu/w9cf/skin/skin.html (BTW the animation illustrates Baluncore's model quite nicely)

"Except for the various factors of 2i and 2$\pi$ , the skin depth is essentially the wave length in the metal. The main thing to notice are that the free space wave length is enormously larger than the skin depth."

"Notice that the skin depth being so small dominates Eq. 10, so for conductors many skin depths thick, you will always get this exponential decay as you move away from the surface. In particular you will still get this behavior even for surfaces with large radii of curvature once you are many skin depths from the surface. "

"Plots of the magnetic field and the current density are shown in figure 1. It is amusing to see that the current is not all flowing in the same direction in the conductor. From the figure, we see that there is a wave traveling and decaying into the conductor. When the current density is a maximum at the surface, the current deeper than about 1.5 skin depths is flowing in the opposite direction. The magnetic and electric fields have the same behavior. "

nsaspook
I don't see how anyone one here is 'wrong', as usual there is more involved.

The wikipedia article is pretty simplistic.
A few quotes from http://fermi.la.asu.edu/w9cf/skin/skin.html
This is the important thing to remember about skin effect.
The skin effect is the description given to the phenomenon where electromagnetic fields (and therefore the current) decay rapidly with depth inside a good conductor.
From the from wiki also.
An alternating current may also be induced in a conductor due to an alternating magnetic field according to the law of induction. An electromagnetic wave impinging on a conductor will therefore generally produce such a current; this explains the reflection of electromagnetic waves from metals.
Skin effect is not just something that happens on circular wire conductors isolated in space, it's also part of a whole system of energy transmission that uses the wiring (good conductors of many shapes) to constrain EM fields near them by reflection at a media boundary (air/vacuum <-> good conductor)

Saying the "back EMF is strongest at the center of the conductor, and forces the conducting electrons to the outside of the conductor" assumes (very simplistically) a circular solid conductor but we also see separate skin effects on the inside and outside surfaces of the shield on a coax conductor and a stripline conductor with a dielectric on one side and air on the other.

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sophiecentaur
Gold Member
My old copy of Panosky and Philips "Classical E and M" just introduces the skin effect as the consequential reduction of the E field at depth in a conductor - a result of the boundary conditions on an incident wave. This basically applies to a wave travelling along a wire just as to a wave incident on a plane reflector. I have read the term 'evanescent mode' applied to the wave beneath the surface too.
Personal pictures / models for what's "going on" can often be misleading when presented to other people although they can work perfectly well for an individual. Maxwell's mathematical model is pretty satisfactory for most purposes, I think.

meBigGuy
Gold Member
If you are going to criticize so called "personal pictures" please have the competence to describe exactly how they are misleading. I doubt you actually understand the picture or read the article I linked to. It is the result of "Maxwell's model". It illustrates the fields resulting from the effects that cause the skin effect. I challenge you to find fault with it or illustrate how it is "misleading".

sophiecentaur
Gold Member
If you are going to criticize so called "personal pictures" please have the competence to describe exactly how they are misleading. I doubt you actually understand the picture or read the article I linked to. It is the result of "Maxwell's model". It illustrates the fields resulting from the effects that cause the skin effect. I challenge you to find fault with it or illustrate how it is "misleading".
I am afraid that you are far too confrontational in your posts - not just to me and not just on this thread. It spoils a thread when personal remarks creep in. That article is, as one would expect, more or less word for word what Panofski and Philips say (without the animation) and I am not even sure what words of yours you think I was criticising.

I have an issue with words like "soaking", "back emf" and "diffusion", each of which have other connotations and can distract from message from the direct mathematical treatment. No problem when trying to make it 'easier' to approach but when trying to deal with a problem in detail, surely Maxwell is the best way to communicate and derive the relationships.

If you want to reply, please try avoid the personal angle. It really isn't necessary. Participating on PF is supposed to be a pleasant experience.

nsaspook
I have an issue with words like "soaking", "back emf" and "diffusion", each of which have other connotations and can distract from message from the direct mathematical treatment.
In my defense "diffusion" as in diffusion equation has a specific meaning when used in relationship to EM energy movement in a conductor but maybe it is something that's not commonly used in teaching students about skin effect and energy coupling unless you need to understand EM shielding.

Electromagnetic Diffusion

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sophiecentaur
Gold Member
I agree. Diffusion is a more applicable term than the other two terms. But does the idea go further than the straightforward field calculation that Maxwell yields in the skin depth case? I don't know enough about the diffusion approach and can't see the whole article.

meBigGuy
Gold Member
@sophie
"I am not even sure what words of yours you think I was criticising."

BTW, I was unaware that:
1. Skin depth is ~one wavelength deep in the conducting medium
2. The complicated field relationships (with respect to direction and magnitude)
3. That the center of the conductor is many wavelengths from the surface (when it is many skin-depths thick, which isn't uncommon).

If you look at the wikipedia picture (of internal rotating eddy currents), and the picture in my link, you see two views of the same phenomena.

I hadn't heard of the "soak/diffusion" model before this thread. I think the EM diffusion nsaspook refers to is a broader subject than the "soaking" that Baluncore referred to (but they are obviously referring to the same physical phemonena).

davenn
Gold Member
2019 Award
BTW, I was unaware that:
1. Skin depth is ~one wavelength deep in the conducting medium
where did you get that from ?

if that were the case, then skin effect would not be the issue that it is on microwave frequencies
as the wavelength would be substantially larger than most of the conductors used

take 10 GHz for example wavelength is 3cm. Not too many 3cm thick conductors used at that freq ... mainly thin microstrip lines but skin effect is an issue

from the link in post #18 ....

so you can see that skin depth is tiny compared to wavelength of freq in question

cheers
Dave

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meBigGuy
Gold Member
The point is that the wavelength *IN THE MEDIUM* is 1 skin depth.

So, the wavelength of 60Hz IN COPPER is ~8mm.

From http://fermi.la.asu.edu/w9cf/skin/skin.html :

"Except for the various factors of 2i and 2$\pi$ , the skin depth is essentially the wave length in the metal. The main thing to notice are that the free space wave length is enormously larger than the skin depth. "

This ties in to Baluncore's assertion that the center of the copper wire can be multiple wavelengths delayed from the surface. If you look at the fields picture at the beginning of the article you see how the fields (and therefore the currents) change direction as you descend into the wire. If you look at the wikipedia eddy current loops picture, you see the complex currents associated with this from a 3D perspective.

Baluncore
2019 Award
The skin depth is the wavelength, but at the phase velocity in the conductive medium. That is why the concept of refractive index is so important.

The exponentially attenuated layers of alternating currents at deeper levels, (initiated by previous half waves of the sinusoidal surface current), tend to cancel and so sum to zero.

sophiecentaur
Gold Member
The skin depth is the wavelength, but at the phase velocity in the conductive medium. That is why the concept of refractive index is so important.

The exponentially attenuated layers of alternating currents at deeper levels, (initiated by previous half waves of the sinusoidal surface current), tend to cancel and so sum to zero.
Whilst this is strictly correct (and very interesting, too), it isn't the best thing to hit students with until they understand the real significance of what it really says. The readily available parameter for a metal tends to be σ and not the phase velocity so, imo, it is best to use the conventional expression for skin depth. (Fewer balls to keep in the air at once.)

sophiecentaur