Microscopic View of Ohm's Law

When electric current in a material is proportional to the voltage across it, the material is said to be "ohmic", or to obey Ohm's law. A microscopic view suggests that this proportionality comes from the fact that an applied electric field superimposes a small drift velocity on the free electrons in a metal. For ordinary currents, this drift velocity is on the order of millimeters per second in constrast to the speeds of the electrons themselves which are on the order of a million meters per second. Even the electron speeds are themselves small compared to the speed of transmission of an electrical signal down a wire, which is on the order of the speed of light, 300 million meters per second.

The current density (electric current per unit area, J=I/A) can be expressed as

The number of free electrons, assuming one per atom, is

From the standard form of Ohm's law and resistance in terms of resistivity:

The next step is to relate the drift velocity to the electron speed, which can be approximated by the Fermi speed:

Table

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.

Table
Numerical example for copper.Table of resistivities
Table of free electron densities
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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.

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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

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Alternatively, if you can identify the number of electrons per atom that participate in conduction, then the free electron density can just be implied from the atomic mass and mass density of the material.

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.

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