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# SPEED OF "ELECTRICITY"1996 Bill Beaty

How fast does electricity flow? Well, it depends on what you mean by "electricity." The word Electricity has more than one contradictory meaning, so before we can talk about its flow, we have to decide on which of several "electricities" we really mean. For a discussion of electric current, see below. But for articles about fast-flowing electromagnetic energy, see the FAQ, this energy article, or an old email.

OK, then how about this. When we turn on a flashlight, something called an "electric current" begins to happen. Inside the flashlight bulb, the thin filament-wire gets hot because there is electric current in the metal. This current is a motion of something. How fast does this "something" move? This question can be answered.

Inside the wires, the "something" moves very, very slowly, almost as slowly as the minute hand on a clock. Electric current is like slowly flowing water inside a hose. Very slow, so perhaps a flow of syrup. Even maple syrup moves too fast, so that's not a good analogy. Electric charges typically flow as slowly as a river of warm putty. And in AC circuits, the moving charges don't move forward at all, instead they sit in one place and vibrate. Energy can only flow rapidly in an electric circuit because metals are already filled with this "putty." If we push on one end of a column of putty, the far end moves almost instantly. Energy flows fast, yet an electric current is a very slow flow.

Within all metals there is a substance which can move. This stuff has several different names: the Sea of Charge, or the Electron Sea, or the Electron Gas, or "charge." We often call it "electricity," and state that electric currents are flows of electricity. Calling it "electricity" can be misleading because many people believe that electricity is a form of energy, yet charge is not energy, and currents are not flows of energy. Also it can be misleading because the Sea of Charge exists within in all metal objects, all the time, even when the metal hasn't been made into a wire and is not part of an electric device. If the Electron Sea is "electricity," then we must say that all metals are always full of electricity, and that batteries are simply electricity-pumps. Better to call it by the name "charge-sea," and avoid the misleading word "electricity" entirely.

During an electric current, the metal wire stays still and the sea of charge flows along through it. When the flashlight switch is turned off and the lightbulb goes dark, the charge-sea stops moving forward. Even though it stops moving, the charge-sea is still inside of that wire. If the flashlight is again turned on, but then two light bulbs are connected in parallel instead of one, the electric current will have twice as large a value, and twice as much light will be created. And most important, the charge-sea within the battery's wires will flow twice as fast. In other words, the speed of the charges is proportional to the value of electric current; small current means slow charge-flow, large current means high speed. Zero current means the charges have stopped in place. Note however that an electric current does not have just one speed within any circuit. Charges speed up whenever they flow into a thinner wire. The high current in a large flash-lantern's lightbulb will be much faster than the same current in the other conductors in the lantern. Even though an electric current is a very slow flow of charges, we can't know the actual speed of flow unless first we know the thickness of the wires, as well as the *value* (the amperes) of the current in the wires.

If a thin wire is connected in a circuit end to end with a thick wire, it turns out that the charges in the thin wire move faster. This makes sense: it works just like water in rivers. If a huge wide river moves into a narrow channel, the water speeds up. When the channel opens out again downstream, the river slows down again. The electric current inside a very thin wire will be tend to be fast, even if the value of current is fairly low. This means that we can't know the speed of the flowing charge-sea unless we know how thick the wires are.

If a copper wire is connected into a series circuit with an aluminum wire of the same diameter, the charges in the copper will flow slower. This occurs because there is one movable charge per each atom in the metals, but there are more atoms packed into the copper than into the aluminum, so there is more charge in each bit of copper. When the charge-sea flows into the copper, it gets packed together and slows down. When it flows out into the aluminum, it spreads out a bit and speeds up. This means that we cannot know how fast the charges flow unless we know how dense the charge-sea is within the metal.

### The speed of electric current

Since nothing visibly moves when the charge-sea flows, we cannot measure the speed of its flow by eye. Instead we do it by making some assumptions and doing a calculation. Let's say we have an electric current in normal lamp-cord connected to bright light bulb. The electric current works out to be a flow of approximatly 3 inches per hour. Very slow!

Here's how I worked out that value. First, I know that:

• Value for electric current: I = 1 ampere
• Wire diameter: D = 2/10 cm, radius R=.1cm (that's #12 gauge AWG)
• Mobile electrons per cc (for copper, if 1 per atom): Q = 8.5*10^+22
• Charge per electron: e = 1.6*10^-19
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## The equation:

```
= cm/sec
=   ________I_______
Q * e * R^2 * pi

= .0023 cm/sec  or  8.4 cm/hour

[ This is for 1.0A in #12 copper wire. ]
[ Higher currents give faster drift,   ]
[ and thinner wire gives faster drift. ]

```
This is for DC. Chris R. points out that for a particular value of frequency of AC, the "skin effect" can cause the flow of charges at the center of a wire to be reduced while the current on the surface becomes stronger. There then are fewer charges flowing overall, and hence the ones near the surface must flow faster. ("Skin Effect" is stronger at high frequencies and with thick wires. The effect can USUALLY be ignored in thin wires under 5mm diameter, at 60Hz power-line frequencies.)

### The size of the wiggle

And about AC... how far do the electrons move as they vibrate back and forth? Well, we know that a one-amp current in 1mm wire is moving at 8.4cm per hour, so in one second it moves:
```  8.4cm / 3600sec = .00233 cm/sec
```
And in 1/60 of a second it will travel back and forth by
``` = .00233cm/sec * (1/60)
= .0000389cm
or around .00002 in.

```
This simple calculation is for a square wave. For a sine wave we'd integrate the velocity to determine the width of electron travel.

So for a typical AC current in a typical lamp cord, the electrons don't actually "flow," instead they vibrate back and forth by about a hundred-thousandth of an inch.

### The width of one Coulomb

On thinking along these lines I notice something interesting: in copper, one coulomb of movable electrons has a certain size! There are about 14,000 coulombs of free electrons per cubic centimeter of copper.
``` = 8.5*10+22 elect/cc * 1.6*10-19 coul/elect
= 13600 Coul./cc
```
Therefore one coulomb would form a cube approximately 0.4mm across...
```  1/(13600cc^(1/3)) = 0.042 cm
```
HA! A coulomb in copper is about the size of a grain of sand! We can now discuss electric currents within wires as if they were cc-per-second fluid-flows inside of small hoses. If an Ampere is one coulomb per second, we're REALLY saying that an Ampere is "one saltgrain-sized blob, moving each second, squeezing itself into whatever sized wire." So, for the usual sizes of wires used in electric circuitry, if we deliver one salt-grain per second (one amp,) that's a very slow flow. The tiny saltgrains are going by: bip, bip, bip, once per second.. In 16-gauge wire the saltgrain blobs would be morphed to fill the cross-section, so they would resemble very thin stacked pancakes. In 30-gauge wire the saltgrains would be almost undistorted, and so the charges would move at about 0.4 mm/sec during a 1-amp current.

### Visible motion of charges

Here's another way to look at it. During electroplating, for each metal ion deposited on the metal surface, one or more electrons must move up to the surface to produce the electrically neutral metal. The layer of metal is slowly growing, while electrons and ions flow in towards that growing surface. Ah, but look closely: if each incoming atom needs one electron, then as the metal atoms stack up, the electrons must flow at a particular speed. This speed is exactly twice as fast as the growing electroplated metal! In other words, if we're electroplating just the tip of a metal wire (making the wire grow slowly longer,) then the flowing charges in that wire are moving very slowly: twice as fast as the advancing wave of newly plated metal. Pretty cool, eh?

One thing's not certain in the above calculations: the charge density for copper. My above value for Q assumes that each copper atom donates a single movable electron. The email from the person below points out that this might not be true. For example, if only one in ten conduction electrons are movable, while the rest are "compensated" and frozen, then the speed of the charge flow will be ten times greater than 8.4cm/hour.

One final point. Electrons in metals do not hold still. They wiggle around constantly even when there is zero electric current. However, this movement is not really a flow, it is more like a vibration, or like a high-speed wandering movement. How should we picture this? Well, remember that we can speak of wind or of flowing water as if they had a genuine velocity... yet a similar type of rapid wandering motion is found in the atoms of all normal liquids and gases. Even when the wind is less than one MPH, the air molecules are zooming around at hundreds of MPH. Even in still air the molecules still wiggle around at the same high speeds. We usually ignore this when discussing the motions of air, and instead take the average velocity of all molecules in a certain small volume. We call it "thermal vibration," and we see the fast movements as a separate issue. Therefore we should do the same with circuitry: the electric current is akin to wind, while the high speed wandering motions of individual electrons is akin to thermal vibrations of individual air molecules. In the above article I concentrate on the slow "electron wind" which is measured by electric current meters, and I ignore the electrons' high speed "thermal vibration."

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I've seen one way to directly observe and measure the drift velocity of charges in a (non-metal) conductor. Connect metal electrodes to the ends of a large salt crystal (NaCl), then heat it to 700 degrees C and apply high voltage to the electrodes. At this temperature the salt becomes conductive, but as electrons flow through it they discolor the crystal, and a wave of darkness moves across the clear crystal. The velocity of this slow-moving wave can be measured. (And if you double the current, the speed of the wave doubles.) This demonstration appears in:
Physics Demonstration Experiments (two volumes)
H. F. Meiners, ed. Ronald Press Co 1970

Date: Tue, 17 Oct 95 09:53:00 PDT
From: O. Quist

On Fri. 13 Oct 1995 Bill Beaty Wrote:
> Very interesting! All the sources I've encountered state that
> each atom in a conductor contributes one (or two?) electrons to
> the conduction band. Might you know a rough figure for the
> actual number of electrons/atom in a copper lattice? How much
> smaller is it than 1.0?

The number of electrons in the conduction band is indeed as you say. But, that is not what I was saying (below). The actual number of electrons which contribute to the electrical current is not equal to the number of electrons in the conduction band.

The electrons which contribute to electrical conduction are those electrons within the Fermi Surface which are "uncompensated." From symmetry, these electrons lie on, or near the surface, and result as the Fermi Surface is "shifted" by the electric field. The fraction of electrons that remain uncompensated is approximately given by the ratio (drift velocity)/(Fermi velocity). The result is the number of electrons which produce an observed current being considerably less than Avagadro's number.

The number of electrons producing current being thus reduced, produces an increase in their average velocity. Average electron velocities are more probably in the meters/sec range rather than the 10ths of a millimeter/sec as is predicted by the free-electron theory.

Date: Tue, 16 Jun 1998 00:31:01 -0500
From: Roy M.
To: William Beaty <>
Subject: Re: Electron drift velocity in metals
Newsgroups: sci.physics.electromag

Its a minor point, but, drift velocity is an average. If some of those conduction electrons are "stuck", they still contribute to the average.

If you want to exclude the slowest 99% then the average of those you do allow will be higher. But, its probably an unnecessary refinement in this context, which is to treat electrons like classical particles and calculate average drift velocities.

Anyway, the effect of which you refer involves the fermi theory, Pauli exclusion and conservation of energy. In effect fewer electrons participate in conduction, but their mean free path is longer.

The explanation is something like: no more than two (with opposite spins) electrons can occupy a given state. When two electrons collide, their final states must have the same total energy and the final states must have been vacant. Thus, if all the states which can be reached at a given energy level are already filled, then the two electrons cannot collide. Net result is that electrons in low energy states are "stuck" in those states. So only the relatively few electrons in high energy states are really available to participate, but most of the other electrons are not available to collide with the high energy electrons so that those electrons that do participate go futher (mean free path) than you might expect.

Subject: Re: Electron drift velocity in metals
Newsgroups: sci.physics.electromag
From:
Organization: Eskimo North (206) For-Ever
Distribution:

Interesting. If part of the conduction band is excluded from conducting, then the average drift velocity of all of the conduction band electrons is unaffected.

However, the average drift velocity of the "non-stuck" electrons becomes much greater. The "stuck" electrons are not "conducting" and are not part of the drifting population, even though they are in the conduction band, right?

After all, for purposes of calculating the drift velocity we could have counted all the valence electrons in every copper atom too (since they are all "stuck") and then claimed that the average drift velocity for electrons was even slower than if each atom contributed only one electron to the current.

I wonder what the real percentage of "free" electrons might be. If it was tiny, then perhaps the drift velocity is in fact very large. If it was REALLY tiny, then perhaps the velocity of the non-stuck electrons rivals the thermal/quantum random motion speeds, and therefor electric current is not a tiny average motion of a fast-moving random cloud.

Wouldn't it be interesting if electric currents in metals tended to create a few relativistic electrons, rather than a large number of slightly drifting "trajectories."

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