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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:
 
Current Density from Free Electrons
J = nevd
where:
J = current density
n = free electron density
e = electric charge
vd = drift velocity
 
The number of atoms per unit volume (and the number free electrons for atoms like copper that have one free electron per atom) is:
 
Atoms per Volume
n = NAρ/A
where:
n = number of atoms
NA = Avogadro's number
ρ = density (kg / m3)
A = atomic mass (kg / mole)
 
From the standard form of Ohm's law and resistance in terms of resistivity:
 
Electric Resistance
R = ρL/A
where:
R = resistance (ohms)
ρ = resistivity
L = length
A = area
 
Electric Current
I = V / R
where:
I = current (amps)
V = potential (volts)
R = resistance (ohms)
 
Current Density from Volts & Ohms
J =
V
RA
  =   V
(ρLA)/A
  =   EL
ρL
  =   E
ρ
  =   σE
where:
J = current density (joules)
V = electric field (volts)
R = resistance (ohms)
A = area (meters2)
ρ = resistivity
L = length (meters)
E = electric field
σ = conductivity
 
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 =
E = electric field
m =
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 =
d = distance between atoms
m =
vF = Fermi speed

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