# Drift velocity

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In physics a drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an average velocity of zero. Applying an electric field adds to this random motion a small net flow in one direction; this is the drift.

Drift velocity is proportional to current. In a resistive material it is also proportional to the magnitude of an external electric field. Thus Ohm's law can be explained in terms of drift velocity. The law's most elementary expression is:

$u=\mu E,$ where u is drift velocity, μ is the material's electron mobility, and E is the electric field. In the MKS system these quantities' units are m/s, m2/(V·s), and V/m, respectively.

When a potential difference is applied across a conductor, free electrons gain velocity in the direction opposite to the electric field between successive collisions, thus acquiring a velocity component in that direction in addition to its random thermal velocity. As a result there is a definite small drift velocity of electrons, which is superimposed on the random motion of free electrons. Due to this drift velocity, there is a net flow of electrons opposite to the direction of the field.

## Experimental measure

The formula for evaluating the drift velocity of charge carriers in a material of constant cross-sectional area is given by:

$u={j \over nq},$ where u is the drift velocity of electrons, j is the current density flowing through the material, n is the charge-carrier number density, and q is the charge on the charge-carrier.

This can also be written as:

$j=nqu$ But the current density and drift velocity, j and u, are in fact vectors, so this relationship is often written as:

$\mathbf {J} =\rho \mathbf {u} \,$ where

$\rho =nq$ is the charge density (SI unit: coulombs per cubic metre).

In terms of the basic properties of the right-cylindrical current-carrying metallic ohmic conductor, where the charge-carriers are electrons, this expression can be rewritten as[citation needed]:

$u={m\;\sigma \Delta V \over \rho ef\ell },$ where

## Numerical example

Electricity is most commonly conducted in a copper wire. Copper has a density of 8.94 g/cm3, and an atomic weight of 63.546 g/mol, so there are 140685.5 mol/m3. In one mole of any element there are 6.02×1023 atoms (Avogadro's constant). Therefore, in 1 m3 of copper there are about 8.5×1028 atoms (6.02×1023 × 140685.5 mol/m3). Copper has one free electron per atom, so n is equal to 8.5×1028 electrons per cubic metre.

Assume a current I = 1 ampere, and a wire of 2 mm diameter (radius = 0.001 m). This wire has a cross sectional area of 3.14×10−6 m2 (A = π × (0.001 m)2). The charge of one electron is q = −1.6×10−19 C. The drift velocity therefore can be calculated:

{\begin{aligned}u&={I \over nAq}\\u&={\frac {1{\text{C}}/{\text{s}}}{\left(8.5\times 10^{28}{\text{m}}^{-3}\right)\left(3.14\times 10^{-6}{\text{m}}^{2}\right)\left(1.6\times 10^{-19}{\text{C}}\right)}}\\u&=2.3\times 10^{-5}{\text{m}}/{\text{s}}\end{aligned}} Dimensional analysis:

$u={\dfrac {\text{A}}{{\dfrac {\text{electron}}{{\text{m}}^{3}}}{\cdot }{\text{m}}^{2}\cdot {\dfrac {\text{C}}{\text{electron}}}}}={\dfrac {\dfrac {\text{C}}{\text{s}}}{{\dfrac {1}{\text{m}}}{\cdot }{\text{C}}}}={\dfrac {\text{m}}{\text{s}}}$ Therefore, in this wire the electrons are flowing at the rate of 23 μm/s. At 60 Hz alternating current, this means that within half a cycle the electrons drift less than 0.2 μm. In other words, electrons flowing across the contact point in a switch will never actually leave the switch.

By comparison, the Fermi flow velocity of these electrons (which, at room temperature, can be thought of as their approximate velocity in the absence of electric current) is around 1570 km/s.