# Derive expression for drift speed using drude lorentz theory

1
by spmetta

2014-11-06T12:43:33+05:30

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The following is my solution.   see  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 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.

Then,
I = current flowing across the wire
= number of charged particles * their charge crossing a particular cross section P' of wire in one second.

Let v = Average drift speed is v meters/sec.  An electron travels (on an average) vt meters in t seconds.

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

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

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

Resistance of a conductor = R = ρL / A = ρ₀ (1+α T) L / A

current = I = V/R = V / [ ρ₀ L (1+α T) L / A ] = V A /  [ ρ₀ L (1+α T)]

n = electron density = N atoms * f free electrons per atom / molar volume
= N f / (M/d) = N f d / M

So drift velocity = v = I / n A e =  {V A / [ρ₀ L (1+αT) ]  } /  (N f d /M) (A) e

v = V M / N f d e [ρ₀ L (1+αT) ]