Drift velocity

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 n q} ,

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:

:u = {m ; \sigma \Delta V \over \rho e f \ell} ,

where

• u is again the drift velocity of the electrons, in m⋅s−1
• m is the molecular mass of the metal, in kg
• σ is the electric conductivity of the medium at the temperature considered, in S/m.
• ΔV is the voltage applied across the conductor, in V
• ρ is the density (mass per unit volume) of the conductor, in kg⋅m−3
• e is the elementary charge, in C
• f is the number of free electrons per atom
• ℓ is the length of the conductor, in m

Numerical example

Electricity is most commonly conducted through copper wires. Copper has a density of and an atomic weight of , so there are . In one mole of any element, there are atoms (the Avogadro number). Therefore, in of copper, there are about atoms (). Copper has one free electron per atom, so n is equal to electrons per cubic metre.

Assume a current , and a wire of diameter (radius = ). This wire has a cross sectional area A of π × ()2 = = . The elementary charge of an electron is . The drift velocity therefore can be calculated: \begin{align} u &= {I \over nAe}\ &= \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)}\ &= 2.3 \times 10^{-5} \text{m}/\text{s} \end{align}

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 . At 60Hz alternating current, this means that, within half a cycle (1/120th sec.), on average the electrons drift less than 0.2 μm. In context, at one ampere around electrons will flow across the contact point twice per cycle. But out of around movable electrons per meter of wire, this is an insignificant fraction.

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 .

References

References

1. Griffiths, David. (1999). "Introduction to Electrodynamics". Prentice-Hall.
2. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html Ohm's Law, Microscopic View, retrieved 2015-11-16