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Microscopic View of Ohm's Law (courtesy of GSU)
The current density (electric current per unit area, J=I/A) can be expressed in terms of the free electron density as:
 
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The number of atoms per unit volume (and the number free electrons for atoms like copper that have one free electron per atom) is:
 
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From the standard form of Ohm's law and resistance in terms of resistivity:
 
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The next step is to relate the drift velocity to the electron speed, which can be approximated by the Fermi speed:
Fermi Speed
$$v_F = \sqrt{2E_F \over m}$$
where:
vF = Fermi speed
EF = electric field
m = mass
The drift speed can be expressed in terms of the accelerating electric field E, the electron mass, and the characteristic time between collisions.
Drift Velocity of Electron
vd =
eE
m
τ = eE
m
d
vF
where:
vd = drift velocity
e = charge of electron
E = electric field
m = mass of electron
τ = electromagnetic decay
d = distance between atoms
vF = Fermi speed
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.
Conductivity
σ =
ne2d
mvF
where:
σ = conductivity
n = free electron density
e = charge of electron
d = distance between atoms
m = mass of electron
vF = Fermi speed

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