Microscopic View of Ohm's Law
The current density (electric current per unit area, J=I/A) can be expressed in terms of the free electron density as
The number of atoms per unit volume (and the number free electrons for atoms like copper that have one free electron per atom) is
The next step is to relate the drift velocity to the electron speed, which can be approximated by the Fermi speed:
The drift speed can be expressed in terms of the accelerating electric field E, the electron mass, and the characteristic time between collisions.
The conductivity of the material can be expressed in terms of the Fermi speed and the mean free path of an electron in the metal.
Microscopic View of Copper Wire
As an example of the microscopic view of Ohm's law, the parameters for copper will be examined. With one free electron per atom in its metallic state, the electron density of copper can be calculated from its bulk density and its atomic mass.
The Fermi energy for copper is about 7 eV, so the Fermi speed is
The measured conductivity of copper at 20°C is
The mean free path of an electron in copper under these conditions can be calculated from
The drift speed depends upon the electric field applied. For example, a copper wire of diameter 1mm and length 1 meter which has one volt applied to it yields the following results.
For 1 volt applied this gives a current of 46.3 Amperes and a current density
This corresponds to a drift speed of only millimeters per second, in contrast to the high Fermi speed of the electrons.
Caution! Do not try this at home! Dr. Beihai Ma of Argonne National Laboratory wrote to point out that the current density of 5900 A/cm2 in this example is over ten times the current density of 500 A/cm2 that copper can normally withstand at 40°F. So doing this in the laboratory might be too exciting. Thanks for the sanity check Dr. Ma.
(If you scale down the voltage applied so that the current is just 3 Amperes, current density 382 A/cm2, so that the copper wire will remain intact, the calculated drift velocity is just 0.00028 m/s. This would be more typical for working conditions in this wire. )
Free Electron Density in a Metal
The free electron density in a metal is a factor in determining its electrical conductivity. It is involved in the Ohm's law behavior of metals on a microscopic scale. Because electrons are fermions and obey the Pauli exclusion principle, then at 0 K temperature the electrons fill all available energy levels up to the Fermi level. Therefore the free electron density of a metal is related to the Fermi level and can be calculated from
The number of atoms per unit volume multiplied by the number of free electrons per atom should agree with the free electron density above.
While these two approaches should be in agreement, it may be instructive to examine both for self-consistency.
Consider the element zinc with a tabulated Fermi energy of 9.47 eV. This leads to a free electron density of
From the Periodic Table, the density of zinc is 7140 kg/m3 and its atomic mass is 65.38 gm/mole. The number of atoms per unit volume is then
The number of free electrons per zinc atom to make these consistent is
This number is what we would expect from the electron configuration of zinc, (Ar)3d104s2 , so these two approaches to the free electron density in a metal are consistent.