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)