Charge Motion in Metals

Written by William Rountree

Mobile Electron Sea

Metals, like all matter, are made of atoms. These atoms consist of a nucleus surrounded by electrons. The majority of metals have few electrons in the outer orbitals, and these valence electrons aren't tightly bound to the nucleus. As a result they are "free" and able to move through the material. The electrons aren't shared or transferred between atoms; they are available to all nuclei in the metal. Often there is only one free electron per atom, but that is all it takes to create a "sea" of electrons surrounding the atoms. Due to every atom lacking a negatively charged electron, the atoms are positively charged and remain bound together by the "sea." There is an even distribution of positive and negative charges, so the net electric field inside of metal is zero.

Charge Motion

Electrons naturally repel each other. When an electric field is applied to a metal, the mobile electrons begin to experience a force and accelerate. The electrons continue to accelerate until they collide with other objects in the mobile electron sea. This process continues to propagate throughout the metal, and is the reason for the relatively quick polarization of metal in an electric field.

An electron's average speed as it moves through the metal, v, is described as it's drift speed. This speed can be found by dividing the momentum of the electron, p, by its mass, m:

   v = p/m

Assuming that the momentum is zero when the electrons collide, the momentum of the electron is equivalent to the force on the electron multiplied by the time between collisions, Δt . Remember, the force on a point charge is qE, where q is the charge of the particle (e for an electron) and E is the net electric field. Therefore, the drift speed can rewritten as:

    v = (eEΔt)/m

The drift speed is proportional to the electric field, so to simplify the relationship a new term is introduced. The electron mobility, μ, describes how easily an electron can move through a material, and varies with temperature and material.

    μ = (eΔt)/m

Substituting this term back in to the equation for drift speed clarifies the relationship between the speed of a mobile charge, the electric field, and the material's electrical properties.

    v = μE

History

This model for the motion of electrons in metal is credited to physicist Paul Drude, who first proposed the model in 1900 three years after J.J. Thompson discovered the electron. Dubbed the Drude Model, this theory was expanded by Hendrik Lorentz five years after it was introduced. The model was updated once more by Arnold Sommerfeld in 1933 after the development of quantum theory.

Examples

Simple Problem

The electron mobility in a certain metal is .02 (m/s)/(N/C). If the drift speed measured in the material is .001 m/s, what is the magnitude of the net electric field inside the metal?

Solution

Use the equation relating drift speed, mobility, and electric field to solve this problem.

E = v/μ = (.001(m/s))/(.02(m/s)/(N/C)) = .05 N/C

Difficult Problem

If the drift speed of mobile electrons in a material is measured to be .0072 m/s and the net electric field applied to the material is known to be .05 N/C everywhere, what would the drift speed be if the magnitude of the net electric field increased to .07 N/C?

Solution

First, find the electron mobility in the mystery material. This property is constant at a consistent temperature.

μ= v/E = (.0072 m/s)/(.05 N/C) = .144 (m/s)(N/C)

Next, use the electron mobility with the updated electric field to find the new drift speed.

v = μE = (.144 (m/s)(N/C))(.07 N/C) = .01008 (m/s)

See also

Electric Field

Electric Force

Polarization

Charge Transfer

Further reading

https://www.sciencedaily.com/terms/electrical_conduction.htm

https://www.scienceclarified.com/Di-El/Electrical-Conductivity.html

https://pfnicholls.com/physics/current.html

References

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/conins.html#c1 http://physics.bu.edu/~okctsui/PY543/1_notes_Drude_2013.pdf