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	=	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