Electron speed

[lionelsoni]

I was well into my electrical-engineering career years ago when I first learned that electricity travels far slower than I would ever have imagined.  I was as surprised as anyone, but I did not doubt that it was a fact.  Ever since, I have wanted to calculate for myself the speed of electrons in a wire, particularly for the AC case, but never got around to it--until now!  I am posting the result here for the "electric train" fans who may find this sort of thing interesting.

When we flip on a light switch, the light comes on so quickly that we may imagine the electrons in the circuit traveling between the switch and the light in a fraction of a second--but in fact the electrons hardly move at all!  For example, a DC current of 1 ampere takes nearly half a minute to move the tiny distance of 1 millimeter in 14-AWG copper wire.  In an AC circuit, the electrons travel even less, merely vibrating in place over a distance of less than a thousandth of a millimeter, and never make any forward progress.

It seems like the electrons are moving fast, because they are already in place throughout the wires before the switch closes the circuit.  They do not need to travel from the switch to the light; instead, they simply push each other through the wire, and the electrons at one end move immediately when the electrical voltage pushes on the electrons at the other end.

A bicycle chain works the same way.  When the pedals begin to turn the chainwheel at the front, there is no delay for the sprocket at the back to begin turning and no wait for the chain to travel the foot or so from one point tothe other.  The chain is already in place and begins to move all at once, no matter how fast or slow the rider begins to pedal. 

Let us call the speed of any particular electron past any particular place on the wire, S.  In one second, a cylindrical volume of electrons with length S and with a cross-sectional area a passes the point of interest on the wire.

So the rate of the volume of electrons flowing in the wire is

  S * a

This has the units of cubic meters per second.  I will assume that the wire is 14-AWG, which has an area a of 2.081e-6 m*m.  We can convert the units to electrons per second by multiplying by the conduction-electron density in the wire, which I assume is made of copper:

  S * a * rho

where rho = 8.491e28 /(m*m*m).  Then we can convert electrons per second to coulombs per second by multiplying by the charge q of a single electron, which is -1.602e-19 C:

  S * a * rho * q

Coulombs per second is amperes, so we now have current as a function of speed:

  I = S * a * rho * q

Solving for S:

  S = I / a / rho / q = I * -35.33 um/C  

Where u substitutes for the lower-case Greek letter mu.  The minus sign reflects the negative charge of the electron and tells us that the electrons are moving backwards, in the direction opposite to whatever direction we assumed for S.

For a DC circuit, the current I is constant, so the speed S is also constant at -35.33 um/s for a current of 1 ampere.

In an AC circuit, the current varies sinusoidally as a function of time, and alternately in opposite directions.  Let the instantaneous current be

  I(t) = Irms * sqr(2) * sin(120 * pi * t)

where Irms is the root-mean-square current, that is, the square-root of the average of the square of the instantaneous current.  The distance D that the electrons move is the definite integral of their speed over a complete positive half-sinewave, like the interval from t = 0 to s/120:

  D = integral(S * dt, 0, s/120)

    = Irms * sqr(2) / a / rho / q * integral(sin(120 * pi * t) * dt, 0, s/120)

    = Irms * sqr(2) / a / rho / q / 60 / pi = -265 nm

[hielsie]

Certainly explains why my trains start real slow and then go real fast. Hielsie

[alank]

You know Bob, one of the reasons I stay tuned into this forum is to hear you talk about electricity.   Yes we share an interest in the trains, but listening to you and your explanations reminds me of being back in my college classes.   I think of the electrtons and used to claim I could see them and therefore could control their flow, but surely I couldn't come up with the explanation you gave here.    You remind me of my buddy Bill who we gave the label of the Big E along time ago.  Good for you that you have this ability and remain this sharp

When I first read your explanantion I thought of a couple of things.   I first thought of when you flip the switch.   Ever since my father gave me a lantern battery, a bulb, a piece of wire and showed me how to light the bulb, I thought electricity was fast.    It is instantanous when you flip the switch. 

I was also thinking of this summer as I stood at the face of Niagara Falls.   As you walk to the falls from the parking lot on Goat Island you see the water of Niagara River flowing to the falls.   You see the water and hear a flow.   At the face of the falls you see the water going over the falls in a rush, and some droplets of water seperate from the stream.   You think of them as electrons.   You see the current of the water both at the top and bottom where it enters the pool. Then it starts the journey to Lake Ontario.    

You can see the power of the water as it rushes over the falls, and also down below especially as the Maid of the Mist steers into the current.  Down river at Lewiston though, there is still a current, but the river seems calm.   So yes I can see your explanation. 

I had the good fortune of seeing one of my roomates this summer.   We met in Rochester and visited the college.  50 years, where did that time go. 

Thanks for coming online and contribututing to the forum.   It is always a pleasure to read your posts.

                   

[gregc]

i was always under the impression that current travels ~80% the speed of light, ~2.4 10^8 m/sec.  

if you assume the speed is correct, then it's a question of how many electrons crossing some point on the wire are needed to carry the current.     i assume 10x as many electrons to carry 10A as 1A.

i think your value is based on the total number of electrons on the cross sectional area of a wire conducting any current.

[ADCX Rob]

gregc

i was always under the impression that current travels ~80% the speed of light, ~2.4 10^8 m/sec.

The current, yes, but the electron flow is apparently a diferent story.

[Flintlock76]

This is a pretty good explanation of why some LEDs keep their glow (although diminishing) when you turn off the power, they're "feeding" off the latent power left in the wiring. 

[lionelsoni]

Alan, thanks for your kind words.

Your comments about Niagara are interesting to me, in that the word "current" came to be used in science to describe the flow of electric charge, even before the electron itself was known.  The resemblance between the new electrical current and the flow of water familiar from ancient times was so strong that using the same word for them must have seemed so natural to the early experimenters as it was for you, that my writing about one inspired your thoughts about the other.

[lionelsoni]

Greg, the number of electrons past a particular point in a unit of time is indeed proportional to the current.  The number of electrons in the wire doesn't change, but the electrons do move ten times faster when the current is 10 amperes instead of 1.  The fascinating thing to me about it is that, even when rather large currents are considered, the increased speed of the electrons is still glacial.

I chose to examine the case of 14 AWG wire because the electron speed actually increases for fully loaded wire (15 amperes in that case) as the size of the wire decreases.  It's as if the skinny wire acts like a nozzle on a garden hose, speeding up the water or electrons by narrowing their path.  For example, 10 AWG can carry twice the current of 14 AWG, but its cross sectional area is 2.5 times as great, so the electron speed is 80 percent that in 14 AWG.

[Flintlock76]

lionelsoni

The resemblance between the new electrical current and the flow of water familiar from ancient times was so strong that using the same word for them must have seemed so natural to the early experimenters as it

If I remember my high school German correctly one of the words for electricity in German is Strom,  the same word used for current in bodies of water.

[Laurastom]

Bob, your posts are always enjoyable and sometimes they hurt my head, even as a former EE. This one is interesting. I know nothing about speed and density of electrons flowing in a circuit. I do know a few things about speed of a control signal. Years ago I was responsible for a unit of the company that had projects designing and building particle accelerators for basic physics experiments. One particular accelerator was about a mile in diameter, its job was to accept protons generated in another device, accelerate and "clump" them into a small packet, then inject them into another ring to hit a target for a physics experiment. The ring has 4, 6 and 8 axis magnets for control of the proton beam. The beam speed is about .999 time the speed of light, the speed of the control signal is about .7 times the speed of light through the copper wire and control devices. The challenge is to control the magnet power using a signal that moves much slower than the item that must be controlled. An interesting challenge for the physicists and engineers. My only point is that signals riding on electron flow in a copper wire move somewhat less than .8 speed of light, add in some silicon devices and whatnot and the effective signal speed was just over .7 speed of light.

[lionelsoni]

The speed in a cable is often less than in a vacuum, usually because of the permittivity of the insulation.  Transmission lines that use mostly air as insulation can come quite close to light speed.

Measuring the speed of signals can be tricky.  For many years long ago, before GPS, I worked in design of navigation receivers using the now-defunct Omega system.  Omega compared the carrier phase among 10200-hertz signals from 8 transmitters around the world.  The signal traveled on a zig-zag path in the waveguide formed between the Earth and the ionosphere.  This slightly diagonal path stretched the points of constant phase in the horizontal direction, making the wavelength appear greater than it would otherwise be.  Since the frequency was unchanged, the phase velocity would have been greater than the speed of light, except that the refractive index of the atmosphere had a very slightly greater effect.

I was responsible for telling the programmers what phase velocity to use in their software.  I happened to mention how close the system had come to having a greater-than-light velocity.  They thought I was putting them on, because they had been taught that nothing goes faster than light.  They were unaware that the speed limit applied to information, like the modulation on the carrier, which did indeed travel substantially slower than light, and not to carrier phase, which approaches an infinite velocity around 1 kilohertz in the VLF-radio case.  They used the numbers I gave them, but they never did believe my explanation.

[gregc]

lionelsoni

when rather large currents are considered, the increased speed of the electrons is still glacial

why are you assuming the current (i.e. 1A) is carried by all available electrons?

what if there are different size wires in the path?

[lionelsoni]

Only one of the 29 electrons in a copper atom is bound loosely enough to the atom to be available for current.  Those "conduction electrons" constitute a veritable atomic pinball machine, ricocheting every which way, whether there is a current flowing or not.  It is actually their velocity and charge motion, averaged across the zillions of copper atoms, that compose the current and electron speed that is going on in the wire.  I didn't mention this averaging quirk, to try to keep things simple, since it makes no difference in the result.

Two or several different-size wires connected in series will have to conduct the same current, so they will have different (average) electron speeds.  Wires in parallel will split the current among themselves according to Ohm's law, and then will have (average) electron speeds according to their wire sizes.  In fact, because the wires' conductances and therefore their individual currents are proportional to their cross-sectional areas, and the individual electron velocities are inversely proportional to the same areas, all of however many copper wires are connected in parallel will have the same electron speed!  The total current will still depend on the sum of the conductances and the configuration of the circuit overall, but the speeds will be the same from wire to wire.

[gregc]

lionelsoni

Two or several different-size wires connected in series will have to conduct the same current

so somehow, the charge of one or more electrons going from one wire size gets distributed to a different number of electrons.   

how does that happen?

[lionelsoni]

The number of electrons in the conducted current doesn't change.  They just move faster in the smaller wire or slower in the larger wire.

Imagine a garden hose with a nozzle.  The water moves slowly in the large-diameter hose, but speeds up when it enters the small-diameter nozzle.  But the same volume of slow water flows past any fixed point per second in the hose, as the volume of fast water that flows past another fixed point per second in the nozzle.  But all the water that flows in the hose flows in the nozzle--no water is gained or lost at the transition between the hose and the nozzle. 

[gregc]

then there's no change in the number of electrons

weren't your calculations based on the total number of conductive electons being in motion within the wire.   what if the wire size is only half the size and there are only half the number of conductive electrons.   how can half the number of conductive electrons now carry the current?   

does one electron from the larger wire cause 2 electrons to travel twice as fast thru the smaller wire

"imagine a garden hose" -- there's a constant number of molecules of water moving thru the hoeses

there's a constant number of electrons

[lionelsoni]

"then there's no change in the number of electrons"  Right

"weren't your calculations based on the total number of conductive electons being in motion within the wire.  what if the wire size is only half the size and there are only half the number of conductive electrons.  how can half the number of conductive electrons now carry the current?"  By moving twice as fast.

"does one electron from the larger wire cause 2 electrons to travel twice as fast thru the smaller wire"  No.  That would double the current at the joint.  As each electron from the larger wire crosses the joint, it speeds up and then continues at twice its former speed in the smaller wire.

"'imagine a  garden hose'--there's a constant number of molecules of water moving thru the hoeses"  Right.

"there's a constant number of electrons"  Right.

 

[Postwar Paul]

Hi Bob,

let me put this out there, and maybe you have an answer:

have you noticed, or can explain why this occurs.....

with A.C. trains, Lionel, American Flyer, and similar, as you get away from the lock on, speed drops dramatically, and additional lock-ins are required at intervals. 
With D.C. operated trains, one lock on is sufficient, with very little decrease in speed on the back side of the layout?

I love'em both, understand, but trying to find out why this is....

Paul

[gregc]

lionelsoni

gregc

"weren't your calculations based on the total number of conductive electons being in motion within the wire.  what if the wire size is only half the size and there are only half the number of conductive electrons.  how can half the number of conductive electrons now carry the current?" 

By moving twice as fast.

then the number of electrons in a volume of wire is not limited by the number of conduction electrons and doesnt' that throw off your calculations?

what is constant?

what if a very thin wire is connected to the source and is connected to a very very thick wire.    why would the number increase to the number of conduction electrons and slow down?    or would there be the same number of eletrons travelling thru both wires at the same speed?

 

[lionelsoni]

Every copper atom has 29 electrons, one of which is a conduction electron, so there are as many conduction electrons as there are atoms, and the number of condution electrons in a wire is proportional to the volume of the wire.  These numbers are constant.

The electrical current in a series circuit is the same through all the circuit elements wired together in series.  When the wire is small, the electron speed is high; when the wire is large, the electron speed is small.  The current is proportional to the number of electrons passing a particular point in a unit of time.  This is accomplished in the smaller wire by the electrons' moving faster past some point in that wire and in the larger wire by a wider stream of electrons' moving slower past some point in that wire.

But the number of electrons per second passing the point in either wire is the same, regardless of the size of the wire.

[gregc]

you did a calculation based on a particular wire size that determined the number of electrons carrying the charge was the number of conduction electrons in the wire.

when the different wire sizes are connected, the basis for that calculation changes.

you start with a calculation of speed based on a 14g wire carrying a some current.   then you say if the 14g wire is connected to a thinner 22g wire the speed is higher.    if the 14g wire is connected to a 10g wire the speed is slower.

so it seems like your calculation of the speed of electrons would be different depending on the wire gauge and current

 

you assume all conduction electrons are in motion.    i could make a similar calcuation based on a specific number of electrons carrying a charge

the problem for me is what is the speed of the current?

lionelsoni

Let us call the speed of any particular electron past any particular place on the wire, S.  In one second, a cylindrical volume of electrons with length S and with a cross-sectional area a passes the point of interest on the wire.

So the rate of the volume of electrons flowing in the wire is  

S * a

what exactly is S?   a point, a distance (m, cm, um)?  what is "rate of the volume"?

determining the number of conduction electrons in a volume doesn't describe the speed of the current

[lionelsoni]

The factor I / a in the expression for the speed S accounts for the speed's being proportional to the current and inversely proportional to the wire's cross-sectional area a.  This expression applies not just to 14 AWG but to any wire of area a.  I put in the area of 14 AWG to get the numerical example.

What is S?  "Let us call the speed of any particular electron past any particular place on the wire, S."  It is the electron speed.  It has the units of distance per unit of time.  The metric unit of speed is meter/second.

The rate of the volume of electrons past the point of interest is how much volume of electrons passes that point per second, which is the cross-sectional area of the volume (the same as the wire volume) multiplied by the electrons' speed past that point.

To calculate the electron speed S, I assumed a speed S, computed the current from that, then finally solved that expression for the speed as a function of current.

[gregc]

i think you S is calculating the amount of charge past a point per unit time (i.e. coulumb / sec)

but this isn't the speed across some distance.

what is the speed of current?

i once had to measure the optical power across 18 km of fiber (spool) and had to make the measurements 60 usec apart.

if i were to apply a voltage to a 18 km spool of wire, is there a delay between when applied and measured at the far end?    is this the speed of current?

or would it take 

    28 571 sec = 1 m / 0.000 035 m/sec

for a current of 1A to be detectable across 1m of wire

hope you see what i'm suggesting

[lionelsoni]

Greg, I don't think we're getting anywhere.  If you want to continue this, I suggest that you find the first item in my original post that you disagree with and that we dispose of that first, then go on to the next, and so on.  If you would like, we can stop posting and go to e-mail.  My address is my forum name followed by @aol.com .

[gregc]

that's too bad

i think the last point i made hit the nail on the head

[Erik_Mag]

Postwar Paul

with A.C. trains, Lionel, American Flyer, and similar, as you get away from the lock on, speed drops dramatically, and additional lock-ins are required at intervals. 

With D.C. operated trains, one lock on is sufficient, with very little decrease in speed on the back side of the layout?

If your using steel rails, the most likely explanation is skin effect as the magnetic permeability dramatically decreases the effective conductor area. This effect does not occur with DC.

As for how slow electrons flow in a typical conductor - that's a fantastic thing because I would want several feet of concrete and/ot lead between me and any conductor where the electrons were traveling a significant fraction of the speed of light.

One analogy is thinking of hydraulic brakes, where the pressure pulse travels very fast through the hydraulic tubing, but the fluid itself may be moving hundreds of times slower than the pressure pulse.

[lionelsoni]

Paul and Erik, I'm sorry that I completely missed Paul's post.  I do agree with Erik that the skin effect is a likely culprit.  It affects prototype railroads as well, making iron or steel third rails practical only for DC.  The skin depth at 60 hertz is about a quarter of a millimeter, which isn't a showstopper for toy trains; but would be serious for prototype railroads, even at 25 or 16 hertz.

[Postwar Paul]

Erik, Bob,

thank you for your reply. The reason I brought this up was the" Dream, Plan, Build" video series offered by Kalmbach about 12 years back. I was basically a garden railroader at that time, and this series introduced me to the world of Toy Trains-- it planted the seeds that have taken deep root!

Two railroads in particular in the videos " Stan Roy's Lionel Layout", and 

" Dick Robinson's American Flyer Layout" make mention of running these Postwar trains on D.C. They do not go into any depth about the benefit other than mentioning the Lionel trains running at a constant speed. This made me curious, hence my question.

Thanks again!

Paul

[phrankenstign]

Despite having some of the slowest, laziest, no-account electrons on my temporary layouts over the years, my little Crayola diesel is still able to pull its consist of tiny cars fairly quickly.