Question

Derive expression for drift speed by using drude Lorentz theory?

Answer 1

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

Let A be cross section of a conductor wire of length L.  Let its resistivity be ρ. Let a current I flow through it. Let V be voltage difference across the conductor. R be the resistance of wire. 

Let T be the temperature of the wire. Let α be the thermal coefficient of resistance. Let e be the charge on an electron. Let there be n electrons per unit volume of the conductor. Let m be mass of the wire. Let M be molar mass of the conductor. Letd be volume density of the conductor.

N = Avogadro number (number of atoms in a mole of the conductor).
Let us say that there are f free electrons in each atom.

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

Suppose an electron travels (on an average) x meters in t seconds. Then average drift speed v of an electron is x/t meters/sec.  

Let us take volume x * A to one side of P'.  All the electrons in the volume x * Awill cross P' in t seconds. 

So the charge crossing P' in one second is = I = x*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, the  drift velocity (speed) = v =
       = I / n A e
       =  {V A / [ρ₀ L (1+αT) ]  } /  (N f d /M) (A) e
 
drift speed
     v = V M / N f d e [ρ₀ L (1+αT) ]