## What is the speed of electricity? – Cheng Luyu

We shall start off with the famous figure: 299792458. Of course people know (at least physicists) what this number represents at a glance. Yes, the speed of light, the supposed limit to the velocity of objects. (I’m quite curious about what will happen if speed of light is exceeded. What are the implications besides the denominator in Lorentz’s transformation being a imaginary number.)

Wait! That is wrong… The reason is… a physics quantity without unit is incorrect! The speed of light is 299792458 m/s.

OK. I’ll stop being lame.

The problem here is, how fast does electricity travel?

In real life, we always have the practical experience that once we plug in an electrical appliance, we get electrical current and electrical power instantly, no matter how long the wire is.

Perhaps for this reason, people always think that electricity travels at speed of light. This is partially correct. The reason is simple (and maybe a bit lame again)… The word electricity can have a meaning more than one.

If we qualify the term electric current, then we are talking about the flow of mobile charges. They can be gaseous or liquid ions, as shown in following figure.

These things are less common in real life. Usually we think of charge carriers as the mobile delocalised electrons in metals, like copper and aluminium. The electrons, as opposed to intuition, moves fairly slowly. I suppose a snail moves much faster than the electrons.

Here we can establish an approximation to model the situation. (If I am wrong here, please correct me.)

Imagine there is a light bulb of 60 watts, and it is connected to a DC source of 50 V with copper wire of radius 1 mm. The use of DC source here is to simply the maths, so the bulb operates at 50 V DC. The number of delocalised electrons per copper atom is 1 (the reason is that although copper atom has two 4s valence electrons, after losing one valence electron, it will have a 3d10 configuration, which is rather stable. This is the case at room temperature whereby the lattice has a fairly low energy level.)

Now let’s calculate the number of atoms per volume in copper. Some data are helpful: copper has density 8.93 g/cm3, and its X-ray diffraction wavelength 1.54 angstroms (1 Å=10-10 m) at first order, which has an angle of 11.53°.

Using Bragg’s Law for crystal diffraction, , we can get that the interatomic spacing of copper is 3.85 Å. Suppose copper atoms arrange themselves in nice cubes in a lattice, and from the density, one gram of copper takes up 1/8.93=0.112 cm3 of space. Since copper has relative atomic mass of 63.5, 1 g is equivalent to atoms. Hence in 1 cm3 of copper, there are copper atoms inside, so there are mobile electrons inside 1 cm3.

Now come back to the problem. I am abusing maths notation here. Let be an infinitesimal amount of time, and current . Meanwhile, in this case .

Consider a portion of wire with length , and let be the number of charged particles per unit volume, we have , where is the charge of electron, , and is the radius of the wire.

Hence , i.e. .

Suppose the voltage of the source is 50 V. Then by , the current is 1 A, assuming the light bulb behaves perfectly as an Ohmic resistance.

.

This means the electrons will move by around 2 metres daily. Very slow indeed!

The above calculated velocity is called the drift velocity. It is given by . (There is another derivation using current density. Interested readers can find out more).

Let’s take a closer look at what is happening within the metal. From our knowledge, electrons vibrate within the metal even if there is no electric current. Hence the drift of electrons can be loosely understood as a more concerted vibration towards one direction. Electrons themselves can have quite high velocity. Drift velocity is only a net velocity in the direction of electric field given by outside source. The actual motion can be like the figure shown below.

Another point to consider: according to relativity, the momentum and energy of a particle will increase by a significant proportion once it moves at a speed approaching the speed of light. If an electron travels at a speed close to speed of light, it would require a lot a lot of energy to achieve that. Not very possible.

So why does electric current appear to travel so fast? Does electric current propagate at speed of light?

The answer is most likely no. Any accelerating electric charge, and therefore any changing electric current, produces 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, and is therefore many times faster than the drift velocity of the electrons. Hence in power lines, especially AC power lines with changing current, the waves of electromagnetic energy propagate through the space between the wires, moving from thesource to an appliance. The electrons in the wire are, however, travelling very slowly at drift velocity in the direction of current flow.

How fast electric current travels depends on varying factors, mainly the electromagnetic properties of the conductor and the insulating materials surrounding it, and on their shape and size as well.

Hence usually electric current will appear to travel at a speed close to 299792458 m/s, but not actually reaching that.