Question

Derive an expression for drift velocity of electrons in a conductor.Hence deduce ohm's law.

Answer 1

    We calculate the drift velocity of conduction band electrons in a conductor.  The conductor is connected across a potential of V.

    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 


Then, 
      I  = current flowing across the wire in unit time
         = (number of electrons crossing a particular 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 =  [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)

   Ohm's law states that current flowing in a conductor is proportional to the potential across it. In equation (5) we have drift velocity being proportional to the current I. So clearly voltage is proportional to the current.  We find the ration from above to get the resistance, which is constant of proportionality.

Answer 2

$\underline \bold{Drift\: velocity}$

Electrons inside the conductor move along straight line and and there remains randomly oriented. The net moment remains zero. But when electric field is set up inside the conductor, when the electrons accelerated uniformly in the direction opposite to electric field.

Thus,

The average of velocities attained by various electrons in the influence of electric field is called as Drift velocity.

$\underline \bold{Derivation}$

Refer to the above attachment.

$\underline \bold{Ohm's \:law}$

Ohm's law state that when the the physical conditions remain same then the potential difference across a conductor is directly proportional to the current flowing through it.