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Drift velocity is expressed in the following equations: 
where is the current density, is charge density (in units C/m3), and is the drift velocity, and where is the electron mobility (in units (m^2)/V.s) and is the electric field (in units V/m)...

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We find the expression for calculating the drift speed of electrons along the length of a conductor, across the ends of which, a potential difference is applied.   The expression involves constants related to the material of the conductor.

See the  picture enclosed.

Drift speed of electrons: Average speed of electrons over the length of a conductor when a potential difference is applied to the ends of the conductor. Electrons move under the influence of electric and magnetic effects of atoms and particles inside conductor.

A = cross section of a conductor wire of length L.
ρ = resistivity of material of wire.
I = current flow
V = Voltage difference applied across the conductor.
R = resistance of wire.

T = temperature of the wire.
 α = Linear thermal coefficient of resistance.
 e = charge on an electron.
n = number of electrons per unit volume of the conductor.
m = mass of the wire.
M = molar mass of the conductor.
d = volume density of the conductor.
N = Avogadro number (number of atoms in a mole of the conductor).
f = number of free electrons in each atom.
ρ = Resistivity of the conductor

      I  = current flowing across the wire in unit time
         = (number of electrons crossing any cross section P' of wire in 1 sec.)
                 * (charge of an electron)

Let  v = Average drift speed.   So an electron travels v t meters in t seconds.

Let us take a volume (v t * A) in the conductor on one side of cross section P'.   All the electrons in the volume (v t A) will cross P' to the other side in t seconds.

So the charge crossing P' in one second is = current = v t A * n e / t
    =>          I = n A e v
    =>        v = I / (n A e)        --- (1)

Resistance of a conductor = R = ρ L / A
Current = I = V / R = V / [ ρ L / A ] = V A /  [ ρ L ]        -- (2)
Molar volume = Molar mass / density = M kg/mole / d  = M / d   m^3/mole

n = electron density  = number of electrons in a mole / volume of a mole
    = (f  free electrons per atom * N atoms/mole) / molar volume
=>  n  = N f / (M / d) = N f d / M        --- (3)

So drift velocity = v = I / n A e =  [V A / (ρ L) ] /  [ (N f d / M) A e ]
     v = V M / [ N f d e ρ L  ]          ---- (4)

Resistivity of a conductor = ρ = ρ₀ (1+αT)  taking into account the thermal increase of resistance.

=>     v = V M / [ N f d e ρ₀ L (1+αT) ]       --  (5)

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