# Speed of electrons in a wire

What is the speed of electrons in a copper wire, used to charge a device? If there is a fixed speed, how is it determined?

• So where have you looked yourself regarding electron speeds? – user163104 Aug 6 '17 at 16:31
• Search for drift velocity – AHusain Aug 6 '17 at 16:49
• Very roughly the speed of electrons for DC is an inch or two per second. This however does not affect the speed of the signal that is close to the speed of light. To understand, imagine a mile long straight line of billiard balls touching each other. Then move the first ball forward at 1 inch per second. Because the balls all touching each other, they all start moving at once and the last ball will deliver your force at the end almost instantly, so the speed of your signal would be closer to a mile per second even though the balls are moving at an inch per second. This is just an illustration. – safesphere Aug 6 '17 at 17:55
• I think OP is concerned about the speed of a signal thru a copper wire. But maybe I'm wrong. I answered that specific question but deleted it because it wasn't worded well. – jmh Aug 6 '17 at 17:56

There is no point in speaking about the "speed of an electron in a copper wire". You may ask the drift velocity of the electron under an applied potential. The electrons are randomly scattered by phonons (lattice vibrations) as well as the metal ions. Under an applied field, in addition to the thermal random motion, the electron moves from region of negative potential to the region of positive potential with an average velocity, a motion which is known as drift. The drift velocity of an electron is very low: about $1 mm/s$. However, the Fermi velocity is as higher as several ten or hundred thousands of meters per second for a metal. The drift velocity of an electron in a metal is given by

$$v_{d}=\frac{eE\tau}{m}$$

where $e$ is the electronic charge, $m$ is the electron rest mass, $E$ is the applied electric field and $\tau$ is the relaxation time.

To know the drift velocity of electrons in copper, all you have to do is just measure the resistance of copper.

Then the conductivity of copper is given by

$$\sigma=\frac{l}{RA}$$

where $R$ is the resistance of the copper wire, $l$ and $A$ are the length and cross-sectional area of the wire. If the applied potential difference across the length of the wire is $V$, then the electric field can be approximated as

$$E=\frac{V}{l}$$

Now, the Fermi velocity of copper can be found out if you know the Fermi energy of copper ($7.00 eV$):

$$v_F=\sqrt{\frac{2E_F}{m}}$$

Next, you need the mean free path length, which is given by

$$\lambda=\frac{mv_F\sigma}{ne^2}$$

where, $n$ is the electron density in copper, and is given by

$$n=\frac{N_A\rho}{A}$$

where, $N_A$ is the Avogadro number, $\rho$ is the density of copper and $A$ is its atomic weight. Knowing $\lambda$, you can calculate the relaxation time as

$$\tau=\frac{\lambda}{v_F}$$

Substitute all these results in the first expression and you are done.

Note:

1. This is an experimental way of doing the job. Standard values are available on textbooks and internet.
2. Caution: Use a very long copper wire (10 m or above) or use mV range potential. The current density of copper is very large.