Whether the speed of alternative/direct current is faster than light?

[Mr.GaGa]

Whether the speed of alternative/direct current is faster than light?

What's the speed of alternative and direct current in metals? c, less or greater? It is impossible to be c because metal is not vacuum.

 

[space guy1]

according to what I know, the charges move anywhere from 66-99% the speed of light. the electrons themselves only move on the range of millimeters per hour.

 

[ghwellsjr]

according to what I know, the charges move anywhere from 66-99% the speed of light. the electrons themselves only move on the range of millimeters per hour.

The charges are the electrons and move rather slowly but not that slowly, I don't think. The voltage signals travel down pairs of wires at about two-thirds of c although the charges travel down one wire and back on the other one.

 

[Drakkith]

Also, in an AC circuit, each electron only moves a few millimeters one way before being forced back the other way when the direction of current changes. A DC circuit would be different as the direction of current never switches.

 

[pervect]

The actual speed with which electrons move in metals is very slow. Google for "drift velocity".

http://en.wikipedia.org/w/index.php?title=Drift_velocity&oldid=444660015" has a calculation of this for an example of 3 amperes through a 1mm copper wire, getting .00028 meters/second.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html" also has some calculations for drift velocity.


Now, you might not be interested in the actual (slow) speed with which the electrons move, but rather the speed at which the signal propagates.

The speed of signal through a propagation through a wire is hard to define, the wire distorts the signal a and it depends on the environment and the frequency.

A high bandwidth "wire", like coaxial cable, doesn't distort the signal and the speed is easier to define and measure. However, the speed through a cable depends mostly on the insulator, and not very much on the metal. The usual coax cable transmits signals about 75% of c, you get close to 'c' if you use an air dielectric (i.e. air insulator).

See Wiki again, http://en.wikipedia.org/w/index.php?title=Wave_propagation_speed&oldid=444855003.

 

[Q-reeus]

It's amazing how the old Drude model of conductivity in metals continues to be perpetuated, even at eg Wiki and Hyperphysics as per Pervect's links in #5. The Wikipedia article there starts off right in mentioning that 'free' electrons rattle around at the Fermi velocity, but then uses a calculation going back to the Drude model concept. Fact is Sommerfeld around 1927 figured out that picture of an extremely small mean drift velocity of at most mm's/sec contributed by all the 'free' electrons is wrong. Actually only a tiny fraction, those unpaired and close to the Fermi surface, are free to conduct and move with a mean 'drift' velocity close to the Fermi velocity, typically around 10⁶ m/sec! A brief article comparing the Drude and Sommerfeld values: http://tau.nanophys.kth.se/cmp/hall/node1.html

 

[ghwellsjr]

A million meters a second! I find that really hard to believe.

 

[Drakkith]

A million meters a second! I find that really hard to believe.

Why's that? It's not that fast. Light itself travels at 299,792,458 m/s.

 

[Mr.GaGa]

according to what I know, the charges move anywhere from 66-99% the speed of light. the electrons themselves only move on the range of millimeters per hour.

Why the value is 66-99% c ? Would you like to give some references to show the result?

 

[Drakkith]

Why the value is 66-99% c ? Would you like to give some references to show the result?

Per here: http://en.wikipedia.org/wiki/Electric_current#Drift_speed

Any accelerating electric charge, and therefore any changing electric current, gives rise to an electromagnetic wave that propagates at very high speed outside the surface of the conductor. This speed is usually a significant fraction of the speed of light, as can be deduced from Maxwell's Equations, and is therefore many times faster than the drift velocity of the electrons. For example, in AC power lines, the waves of electromagnetic energy propagate through the space between the wires, moving from a source to a distant load, even though the electrons in the wires only move back and forth over a tiny distance.

That is referring to the propagation of the force I believe, not the charges (electrons) themselves.

 

[chaszz]

A few questions.

1. Is it correct to say that in a current the movement, or drift, of the electrons themselves is small, but the movement of the charge is much faster? That is what I gather from the preceding posts.

2. Is the movement of the charge the same thing as the propagation of the electromagnetic waves? If so, is the charge somewhere on the electromagnetic spectrum, and if so, where on the spectrum? Or is the charge itself the 'electro' component of 'electromagnetic', and the magnetic component is caused by it (the charge)?

3. If the direction of the current in AC is continually reversing, how does the current do any work? It would seem to me to cancel itself out.

 

[sunroof]

A few questions.

1. Is it correct to say that in a current the movement, or drift, of the electrons themselves is small, but the movement of the charge is much faster? That is what I gather from the preceding posts.

2. Is the movement of the charge the same thing as the propagation of the electromagnetic waves? If so, is the charge somewhere on the electromagnetic spectrum, and if so, where on the spectrum? Or is the charge itself the 'electro' component of 'electromagnetic', and the magnetic component is caused by it (the charge)?

3. If the direction of the current in AC is continually reversing, how does the current do any work? It would seem to me to cancel itself out.

The speed of charge(electron) has nothing to do with that of the electromagnetic field. It is much slower than the latter. (Wiki: http://en.wikipedia.org/wiki/Drift_velocity )

 

[Drakkith]

A few questions.

1. Is it correct to say that in a current the movement, or drift, of the electrons themselves is small, but the movement of the charge is much faster? That is what I gather from the preceding posts.

Replace charge with EM field or force and you are correct.

2. Is the movement of the charge the same thing as the propagation of the electromagnetic waves? If so, is the charge somewhere on the electromagnetic spectrum, and if so, where on the spectrum? Or is the charge itself the 'electro' component of 'electromagnetic', and the magnetic component is caused by it (the charge)?

No, a Charge is a property of matter (particles) that causes it to experience a force. An Electron is an Electrically Charged particle. Hence we call it an electric charge: http://en.wikipedia.org/wiki/Electric_charge

3. If the direction of the current in AC is continually reversing, how does the current do any work? It would seem to me to cancel itself out.

I think that if the frequency was so high as to reverse before the EMF (electromotive force) can propagate through the circuit, then it would interfere I believe. However the frequency of AC circuits is low enough to give the EMF plenty of time to propagate throughout the circuit. Remember, it's speed is a high proportion of c, the speed of light.

 

[ghwellsjr]

An electron has a negative charge which means two things:

1) Surrounding the charge is a negative voltage that is inversely proportional to the distance from the charge. If you move this charge, this field of voltage adjusts itself at the speed of light to its new position.

2) A charged particle will experience a force along the direction of a spatial change in voltage field. So two charged particles separated by a distance will experience a force along a line between them because the voltage field is changing in that direction.

The same could be said for the positive charges on protons. For all practical purposes, the number of positive and negative charges are approximately equal so the voltage field in any area is generally quite low. However, if you can get an electron to vibrate, you can create a wave in the voltage field that propagates outward at the speed of light. When this voltage wave reaches another electron far removed from it, it will cause it to vibrate.

There, now you have the condensed theory of light.

If you can collect an excess of charge on a piece of metal which allows the free-flow of electrons, you can create a voltage on it. At any location on the metal, there will be enough charges in the surrounding metal to add up to the same voltage everywhere. Now if you take two such pieces of metal and put charges of opposite polarity on them, you will have a measureable voltage between them. You can then put a piece of wire between them, you will get a momentary measureable current flow in the wire.

There, now you have the condensed theory of electricity.

Now instead of a short piece of wire going directly between these two pieces of charged metal, you take a very long piece of wire going many miles distance and coupling back along a parallel path, before you connect both ends to the pieces of metal, there will be no voltage on the wire but as soon as you make both connections, all of a sudden the electrons in the two pieces of metal are transferred to the beginning of the two pieces of wire and electrons start to flow. These moving electrons create a changing voltage difference between the two wires that propagates down the wires at a substantial fraction of the speed of light and the disturbance in the charges along the wires propagate along with it but the charges are not moving very far.

There, now you have the condensed theory of a transmission line.

 

[Per Oni]

It's amazing how the old Drude model of conductivity in metals continues to be perpetuated, even at eg Wiki and Hyperphysics as per Pervect's links in #5. The Wikipedia article there starts off right in mentioning that 'free' electrons rattle around at the Fermi velocity, but then uses a calculation going back to the Drude model concept. Fact is Sommerfeld around 1927 figured out that picture of an extremely small mean drift velocity of at most mm's/sec contributed by all the 'free' electrons is wrong. Actually only a tiny fraction, those unpaired and close to the Fermi surface, are free to conduct and move with a mean 'drift' velocity close to the Fermi velocity, typically around 10⁶ m/sec! A brief article comparing the Drude and Sommerfeld values: http://tau.nanophys.kth.se/cmp/hall/node1.html

Chapter 1 The Drude Theory of Metals, bottom of page 16 “Solid State Physics; Ashcroft / Mermin”

But even in a current as large as 1amp/mm^2, v=j/ne is only of order 0.1 cm/sec.

Chapter 2 The Summerfield Theory of Metals, halfway page 51 same book:

There is thus a wide range of phenomena in which the behaviour of a metallic electron is well described by classical mechanics.

That book was written in 1975 and is generally accepted as a brilliant piece of work. However I do accept that the differences between Fermi speed, thermal velocity, drift speed, signal speed are never well explained anywhere.

 

[PhilDSP]

It's probably worth considering the 2 types of current: conduction current and displacement current. Conduction current is the movement of the charge while displacement current is the movement or transport of EM (light) energy which propagates at c.

You can calculate the displacement current using the Poynting theorem. The displacement current travels often at a 90 degree direction to the direction of travel of the charges. The displacement current is what creates EM waves.

Or rather the energy travels in a direction that is 90 degrees from the direction of the conduction current vector while the displacement current is in the opposite direction.

 

[harrylin]

It's amazing how the old Drude model of conductivity in metals continues to be perpetuated, even at eg Wiki and Hyperphysics as per Pervect's links in #5. The Wikipedia article there starts off right in mentioning that 'free' electrons rattle around at the Fermi velocity, but then uses a calculation going back to the Drude model concept. Fact is Sommerfeld around 1927 figured out that picture of an extremely small mean drift velocity of at most mm's/sec contributed by all the 'free' electrons is wrong. Actually only a tiny fraction, those unpaired and close to the Fermi surface, are free to conduct and move with a mean 'drift' velocity close to the Fermi velocity, typically around 10⁶ m/sec! A brief article comparing the Drude and Sommerfeld values: http://tau.nanophys.kth.se/cmp/hall/node1.html

Thank you - that little piece of information had not reached me!

 

[Mr.GaGa]

The speed of charge(electron) has nothing to do with that of the electromagnetic field. It is much slower than the latter. (Wiki: http://en.wikipedia.org/wiki/Drift_velocity )

I read http://en.wikipedia.org/wiki/Drift_velocity and the velocity of charge(electron) is very slow. Actually, it is the electromagnetic fiel in conductors to transimit energy and signal whose velocity is much faster than light in vacuum. In particular, it is infinity according to the reference (Energy transport faster than light in good conductors, arXiv:/1101.1840v2). Really?

 

[Q-reeus]

from Per Oni in #15:
"That book was written in 1975 and is generally accepted as a brilliant piece of work. However I do accept that the differences between Fermi speed, thermal velocity, drift speed, signal speed are never well explained anywhere."

I include a pdf version of an article (Conduction.pdf) that gives a nice summary of Drude and Sommerfeld models (unfortunately original online link no longer works). Note it makes the point that the net result gives the same overall figure for conductivity (but very different results for other parameters of interest). That's probably what Mermin meant by classical models working so well. https://www.physicsforums.com/attachment.php?attachmentid=38139&stc=1&d=1313832109

from harrylin in #17:
"Thank you - that little piece of information had not reached me!"
Your welcome Harald.

from Mr.GaGa in #18:
"I read http://en.wikipedia.org/wiki/Drift_velocity and the velocity of charge(electron) is very slow. Actually, it is the electromagnetic fiel in conductors to transimit energy and signal whose velocity is much faster than light in vacuum. In particular, it is infinity according to the reference (Energy transport faster than light in good conductors, arXiv:/1101.1840v2). Really?"

How that article got published in a peer-reviewed journal is beyond me. No way can be true. For a near monochromatic wave propogating in a hollow waveguide for instance, the phase velocity v_p (speed a crest moves at) exceeds c, but the signal velocity v_s (speed an actual signal moves at) is correspondingly less according to v_pv_s = c². When dispersion is high this relation gets a bit fuzzy and the peak value of a pulse can, within the pulse envelope and over a limited duration, move faster than c, but never can the pulse leading edge exceed c, no matter what. See eg http://en.wikipedia.org/wiki/Superluminal_motion [oops - better one at http://en.wikipedia.org/wiki/Faster-than-light]


Finally, wondering why I have received just one email notification on this thread since my last entry #6! And yes, my settings are still set for automatic 'instant' notification.

 

[Per Oni]

Actually only a tiny fraction, those unpaired and close to the Fermi surface, are free to conduct and move with a mean 'drift' velocity close to the Fermi velocity, typically around 106 m/sec!

This part I don’t agree with. When looking at the article you provided in #19 you can see in the picture of the Fermi spheres that the whole surface is displaced by a certain distance. This means that all conduction electrons have changed their speeds towards –E. Of course I accept that they are not all moving with the same velocity. But since they are all moving towards –E, the average speed is still given by v=j/ne, which is extremely slow.

 

[sunroof]

For a near monochromatic wave propogating in a hollow waveguide for instance, the phase velocity v_p (speed a crest moves at) exceeds c, but the signal velocity v_s (speed an actual signal moves at) is correspondingly less according to v_pv_s = c². When dispersion is high this relation gets a bit fuzzy and the peak value of a pulse can, within the pulse envelope and over a limited duration, move faster than c, but never can the pulse leading edge exceed c, no matter what. See eg http://en.wikipedia.org/wiki/Superluminal_motion [oops - better one at http://en.wikipedia.org/wiki/Faster-than-light]

Yes. A superluminal phase velocity v_p>c is corresponding to a signal velocity v_s less than c because v_pv_s = c². However, a subluminal phase velocity v_p<c might represent v_s>c. See http://en.wikipedia.org/wiki/Phase_velocity

 

[Q-reeus]

This part I don’t agree with. When looking at the article you provided in #19 you can see in the picture of the Fermi spheres that the whole surface is displaced by a certain distance. This means that all conduction electrons have changed their speeds towards –E. Of course I accept that they are not all moving with the same velocity. But since they are all moving towards –E, the average speed is still given by v=j/ne, which is extremely slow.

No, the article Conduction.pdf makes it clear it is only the small fraction of unpaired electrons that form a potential reservoir for conduction, and it is from this fraction only that the distorted Fermi surface has relevance. There are a number of contributing factors and the full details are way beyond my capabilities. However, typically at room temp a fraction of ~ 10^-3 are unpaired. At the same time, owing to the electronic wave nature, mfp is greater than the Drude picture suggests, by about an order of magnitude, but this is countered by the random speed ~ v_F ~ an order of magnitude greater than Drude value v₀, so that the mft is roughly equal in the Drude and Sommerfeld models. The picture so far gives almost no enhanced drift velocity per electron if one assumes only initially random emissions of electrons being accelerated for the mft - the Drude concept. This still leaves then about three order of magnitude deficit in conductivity to explain. The other main factor is that emission is suppressed in directions away from the applied -E (unpaired electrons 'sink' into the paired conduction band levels), and enhanced in the direction of -E. That is what I believe is implied in the linked article. The actual level of vectoring I'm not sure of, but the author writes "j = nqv_F" for the current, suggesting only a small fraction of that ~ 10^-3 of unpaired electrons appreciably carries the current. Then there is the question of scattering being elastic or inelastic and to what degree correlated etc. For a full treatment things get quite complicated and Brillouin zones, Bloch modes and all that quantum physics stuff comes into play, which is taking this right away from SR & GR.

 

[Q-reeus]

Yes. A superluminal phase velocity v_p>c is corresponding to a signal velocity v_s less than c because v_pv_s = c². However, a subluminal phase velocity v_p<c might represent v_s>c. See http://en.wikipedia.org/wiki/Phase_velocity

No doubt that is referring to the part under sub-heading 'Slow phase velocity and superluminality'. Another example where Wikipedia cannot always be trusted. Conceivably the author of that piece was the same one who wrote the arXiv.org article linked to in #18. For an EM wave to propagate freely through a metal, it must have a frequency well above the plasma frequency f_p, which for eg copper is in the ultraviolet range. Even then, it won't get far beyond the micron range typically, as damping is pretty high. At mains frequency, permittivity is almost purely imaginary, and severe attenuation and reflection occurs - skin depth for copper at 50 Hz is slightly under 1 cm. Interestingly the speed of such a severely attenuated wave is much less than the speed of sound in the metal - so much for superliminal transmission! Try http://farside.ph.utexas.edu/teaching/315/Waves/node49.html for some finer details. As discussed in #5, transmission velocity is owing to coupling by the fields, not motion of the charges, and can only approach c, being considerably less if dielectric insulation is present. This is all standard transmission-line theory stuff.

 

[Per Oni]

@ Q-reeus
So far you admit that you are at odds with both articles in Wiki and Hyperphysics. To support your view you quote one ambiguous article which shows Fermi sphere pictures explaining one thing and a text explaining a different thing. Now just like you this whole theory is far beyond my capabilities, therefore we are both dependent on what the profs in physics tell us. Let’s take Kittel’s “Introduction to Solid State Physics” page 142 and look at figure 10. There, the whole Fermi sphere is (again) moved by a certain distance. But more important look at equation 42: j=nqv. where v is (as he calls it) the “incremental velocity” and n is the number of electrons per unit volume. This a different formula from the one your article suggests: j=n(Fermi) q V(Fermi).

As an aside: in my opinion your article mixes up the number of scattered electrons with the number of conduction electrons.

And yes I also don’t know why this thread is posted here.

 

[Q-reeus]

...To support your view you quote one ambiguous article which shows Fermi sphere pictures explaining one thing and a text explaining a different thing.

The ambiguity is with your interpretation of what is being said there, not the article itself. As per last entry, vast bulk of 'free' electrons are velocity paired and make no net contribution. Displacement of Fermi surface therefore can only be relevant for unpaired fraction. The author makes this plain.

...Let’s take Kittel’s “Introduction to Solid State Physics” page 142 and look at figure 10. There, the whole Fermi sphere is (again) moved by a certain distance. But more important look at equation 42: j=nqv. where v is (as he calls it) the “incremental velocity” and n is the number of electrons per unit volume. This a different formula from the one your article suggests: j=n(Fermi) q V(Fermi)...

Don't have Kittel but there is practically no chance any real conflicting viewpoint exists between the two. Try looking at the context a bit closer - sure he's not just deriving an *equivalent* expression in terms of Drude model? The author in my article does the same, in the same equation line to that I quoted. Point being shown there is that the Sommerfeld and Drude models finish up with roughly the same overall result for electrical and thermal conduction (but far apart for e.g. electronic specific heat). My copy of Solid State Physics, 2nd Ed'n, - J.S.Blakemore, p185 backs up precisely the account given in my uploaded article - only a tiny fraction of the 'free' electrons carry the current.

As an aside: in my opinion your article mixes up the number of scattered electrons with the number of conduction electrons.

You may feel that, but I see no mix up. It is a brief but perfectly standard treatment - that fully qualified university lecturer is no idiot.

And yes I also don’t know why this thread is posted here.

Well only that the OP thought faster-than-light might be a realistic prospect. It has gotten sidetracked since, which is pretty typical! :zzz:

 

[Samshorn]

The article Conduction.pdf makes it clear it is only the small fraction of unpaired electrons that form a potential reservoir for conduction, and it is from this fraction only that the distorted Fermi surface has relevance. ...typically at room temp a fraction of ~ 10^-3 are unpaired. ...the author writes "j = nqv_F" for the current, suggesting only a small fraction of that ~ 10^-3 of unpaired electrons appreciably carries the current...

This is an interesting topic. I think it's fair to say that, at least until fairly recently, the standard explanation taught to physics undergrads (as in Halliday & Resnick 3rd edition for example) is that the drift velocity v_d of the "conduction electrons" in a copper wire is j/(ne) where j is the current density, e is the charge of an electron, and n is the number of "conduction electrons" per unit volume. The question is, how many "conduction electrons" are there in a unit volume of a conductor like copper?

According to Halliday & Resnick, there is 1 free electron per atom in copper, and they claim that each of these "free electrons" is a "conduction electron". On this basis they compute that the drift velocity for a copper wire of diameter 0.064 inches carrying current density 480 amps/cm^2 is only 0.036 cm/sec, so it takes 28 seconds for the electrons to drift 1 cm.

Just doing a quick survey of the web, I find that it is standard to take n = 8.4E28 conduction electrons per cubic meter for copper, which agrees with Halliday & Resnick based on their "1 electron per copper atom" assumption.

I gather you are saying that the number of "conduction electrons" (those that actually drift down the wire and represent the net transfer is charge) is actually three orders of magnitude smaller than 1 per copper atom. Am I understanding you correctly? If so, it's remarkable that this understanding hasn't infiltrated into the undergraduate cirriculum yet. After all, Sommerfeld was a long time ago.

Can anyone cite a recent reference that actually computes the drift speed for a copper wire?

 

[Q-reeus]

I gather you are saying that the number of "conduction electrons" (those that actually drift down the wire and represent the net transfer is charge) is actually three orders of magnitude smaller than 1 per copper atom. Am I understanding you correctly?

Well the picture that emerges is a sort of two-track contribution. A small part is contributed by the bulk of that ~ 10-3 fraction via a Drude type mechanism, while the bulk of the current is contributed by a tiny fraction of that fraction - those emitted via Fermi surface shift with a strong directional bias toward the applied E. That fraction would then be roughly the factor 1/2 v_d/v_F smaller than the 'free' electron density you quoted, where v_d is the Drude drift velocity. A very small figure! Incidentally, the 10^-3 unpaired fraction is ball-park at room temp, and drops dramatically at low temp in accordance with Fermi-Dirac stats.

If so, it's remarkable that this understanding hasn't infiltrated into the undergraduate cirriculum yet. After all, Sommerfeld was a long time ago.

Yes it is a bit strange, but anyone going on to specialize in solid-state physics no doubt gets the proper picture - of which Sommerfeld model is just an idealized starter. Owing to the rough correspondence, it seems education institutes and various online sources choose the simpler Drude model out of convenience, or inertia maybe. Reasoning seems tro be it sort of does no harm to non-specialists.

Can anyone cite a recent reference that actually computes the drift speed for a copper wire?

Can't right now think of any specific calcs available online, sorry.

 

[Naty1]

Very interesting thread...

elswhere I just posted two questions on electrical resistivity..over 100 have looked with no replies...and Q-reeus' reference answered why the classical approach is off by a factor of roughly 100...

The fundamental difference brought by quantum mechanics was that electrons obey the Fermi-Dirac statistics, which takes into account the Pauli exclusion principle. Sommerfeld applied the new statistics to metals with the result that the conduction electrons fill successive energy states in pairs of opposite spin up to the so-called Fermi level

Do any of you have an insight on this related question: Why does tungston with only about 3.3 times the resistivity of copper get so hot...as in light bulbs...Is it mostly a function of the filament being so thin? I am wondering if there is something different in the conduction band energies of resisters that causes so much heat to be generated...
Thanks...

 

[Per Oni]

My copy of Solid State Physics, 2nd Ed'n, - J.S.Blakemore, p185 backs up precisely the account given in my uploaded article - only a tiny fraction of the 'free' electrons carry the current.

If you look at the text just above equation 3-77 of your book it states:

…to give the electrical conductivity in terms only of the total electron density and of….

Where then subsequently n is used in that equation. This n is also used in equation 3-79. So all I can deduct from that book is that “the total electron density” must be used.

Now allow me to be very confused indeed.

 

[sunroof]

For an EM wave to propagate freely through a metal, it must have a frequency well above the plasma frequency f_p, which for eg copper is in the ultraviolet range. Even then, it won't get far beyond the micron range typically, as damping is pretty high. At mains frequency, permittivity is almost purely imaginary, and severe attenuation and reflection occurs - skin depth for copper at 50 Hz is slightly under 1 cm.

:uhh: But how to explain electric energy can be sent to users miles away by copper wires whose frequency is very low(60Hz or 50Hz) in power industry?

 

[ghwellsjr]

:uhh: But how to explain electric energy can be sent to users miles away by copper wires whose frequency is very low(60Hz or 50Hz) in power industry?

What has the frequency got to do with energy transmission? It can be done at DC, too.

 

[Q-reeus]

If you look at the text just above equation 3-77 of your book it states:
Where then subsequently n is used in that equation. This n is also used in equation 3-79. So all I can deduct from that book is that “the total electron density” must be used.

Now allow me to be very confused indeed.

Your continued confusion is in seeing the *Drude equivalent* expressions that Kittel, Blakemore above quoted, and Mehring in the uploaded article, use, while somehow missing the accompanying explanations that the *real* situation is quite different in detail, but divergent factors cancel out for overall result. For instance, did you read the part on p185 of Blakemore where he remarks "We can understand now that for a degenerate electron gas, the mean free time for only a small minority of the total electron population is significant. It has been remarked that Sommerfeld drew an important distinction between the large number of "free electrons" and the much smaller number of "conduction electrons"."? He goes on to explain some deficiencies in the Sommerfeld model, which for instance retained the Drude assumption of elastic scattering, whereas inelastic scattering applies. In many real metals the fermi surface is far from spherical, and density of states, effective mass, mfp, are all anisotropic quantities. But that all gets into details far from the OP's interest.

 

[Q-reeus]

...Do any of you have an insight on this related question: Why does tungston with only about 3.3 times the resistivity of copper get so hot...as in light bulbs...Is it mostly a function of the filament being so thin? I am wondering if there is something different in the conduction band energies of resisters that causes so much heat to be generated...
Thanks...

Naty1: Would say the sole reason tunsten is (still) used is owing to it's very high melting point and thus high retained mechanical strength at those elevated temperatures where emission of light is reasonably efficient. Push enough current through any resistance and it gets hot of course, but copper for instance would melt at dull red heat. A nice feature of the filament lamp is the current self-limiting owing to positive temperature coefficient of resistance. A less nice feature is the development of a hot spot owing to uneveness in the inevitable slow sublimation of the filament.

 

[Q-reeus]

Some intended edits for #27: factor 1/2 should have been 3/2. Also, at zero temperature there is no thermally excited 'intrinsic' unpaired electron population, and high conductivity there is entirely due to Fermi surface displacement by an applied E, coupled with a very long mft limited only by 'intrinsic' lattice defects. Thus it may be best to restate the two-track conduction process as superposition of a minor Drude type drift mechanism applying to the thermally generated population of unpaired electrons, and a major, essentially temperature independent vectored emission Fermi surface displacement mechanism at the heart of the Sommerfeld model.

My own interest in this area sprang from finding some very interesting results for certain current configurations owing to higher order relativistic effects - synchrotron radiation being an extreme example of relativistic harmonic generation. While for high power Magnetrons harmonic generation is considered objectionable as it robs power from the intended fundamental frequency output, in other situations it could be a plus. Even though for metallic conduction electrons moving at ~ 0.3% c, the very small first harmonic ~ 10^-5 times smaller than the fundamental, that could be useful in some configurations. Suspected though that in the stop-start scattering of room temperature regime, it might be entirely suppressed, and was never convinced that even in the ballistic regime of tiny dimension very pure metals at very low temperature, conduction electrons really behaved elecrodynamically identically to truly free electrons (such as in a cyclotron). Would love to know though.

 

[Q-reeus]

:uhh: But how to explain electric energy can be sent to users miles away by copper wires whose frequency is very low(60Hz or 50Hz) in power industry?

Adding a bit to what ghwellsjr said in #31, found an online simulation that can help to get a feel for how 'throwing the switch' at one end of a DC line is initially a series of waves going back and forth - at c or less: http://users.ece.gatech.edu/~wrscott/applet_bounce/Reflect1.html