Drift Velocity Derivation

As each free electron in the conductor experiences a force,

F = -eE

and the acceleration of each electron is

a = F / m

From both the above equations, we get

∴ a = -eE / m (where m is the mass of the electron)

At any instant of time, the velocity acquired by an electron having thermal velocity u₁ will be given using v = u + at as,

v₁ = u₁ + aτ₁

Where, 

• τ₁ is Time taken by Electron to collide with Positive Ion
• v₁ is Initial Velocity

Similarly, velocity acquired by electron, v₂ = u₂ + aτ₂ , v₂ = u₂ + aτ₂ , . . . . . . , v_n = u_n + aτ_n

The average velocity of all the free electrons in the conductor under the effect of the external electric field is the drift velocity v_d of the free electrons.

v_d = ( v₁ + v₂ + . . . . + v_n ) / n

⇒ v_d  = ((u₁ + aτ₁) + (u₂ + aτ₂) + . . . . + (u_n + aτ_n)) / n

⇒ v_d  = ((u₁ + u₂ + . . . +u_n)/n) + a((τ₁ + τ₂ + . . . . . + τ_n) /n)   . . . (1)

when an Electric field is not applied to the conductor then there is no current in the conductor.

∴ (( u₁ + u₂ + . . . + u_n )/n) =0  . . . (2)

Putting equation 2 in equation 1, we get

v_d  = 0 + aτ                                                                [where τ = ((τ₁ + τ₂ + . . . . . + τ_n) /n)]

where τ is the average relaxation time and it refers to the length each electron takes to recover from its most recent interaction with a conductor atom.

we know, a = -eE / m [from force experienced by charge in an electric field]

∴ v_d = -eEτ / m

where the negative sign shows the opposite direction of the electric field.

Drift Velocity, v_d = eEτ / m  …(3)

Alternate Formula for Drift Velocity

If the length of the conductor and time taken by the electron to travel throughout the length of the conductor is given then drift velocity can be calculated using the following formula

• Due to the frequent collisions that electrons have, their drift velocity is small.
• If the cross-section is constant, i ∝ J i.e. for a given cross-section area, the greater the current density, the larger will be current.
• In the presence of an exceptionally large number of free electrons in a conductor, a tiny amount of drift velocity produces a large amount of electric current.
• The electric bulb glows immediately when the switch is turned on because current transmission is practically as fast as light and involves electromagnetic processes.
• In the presence of an electric field, the path of electrons between successive collision are curved and In the absence of an electric field, the path of electrons between successive collision is a straight line.
• Free electron density in metal is

n = (N_A × d) / A 

Where, 

• N_A is Avogadro number
• x is number of free electrons per atom
• d is density of metal
• A is Atomic weight

Relaxation time (τ) 

The time interval between two successive collisions of electrons with the positive ions in the metallic lattice is defined as relaxation time.

τ = mean free path / r.m.s. velocity of electrons 

τ = λ / v_rms

Thus, τ is inversely proportional to v_rms

Mobility (μ)

Electron mobility is defined as “drift velocity per unit electric field.”

μ = v_d / E

The unit of Mobility is m² / volt-sec and using the above formula ohm’s law can be explained in terms of drift velocity as,

v_d = μE

The magnitude of charge drift velocity per unit electric field applied is specified as the mobility of the charge carrier responsible for current.

μ = drift velocity / Electric field 

⇒ μ = v_d / E

⇒ μ = (eEτ / m) / E   [from equation 3]

Thus, mobility of electron, 

μ = eτ / m

Drift Velocity as the name suggests refers to the slow movement of electrons in the conductor when an emf is introduced. Electrons do not move in a straight line in the conductor, but they move randomly in the conductor colliding with the other electrons and atoms exchanging energy, this exchange of energy moves forward in the direction opposite to the current and made the flow of electricity possible.