Drift velocity and mobility

Drift Velocity and Mobility: Derivation and Dependence

Drift velocity and mobility are key concepts in understanding the flow of electric current through a conductor. This article explains these concepts starting from the basic idea of charge flow, leading to drift velocity, and then to mobility.

Electric Field and Potential Difference

When a potential difference V is applied across a conductor of length l, it produces an electric field E inside the conductor:

E = V / l

Total Charge in the Conductor

Consider a conductor with cross-sectional area A and length l. Let n be the number of free electrons per unit volume, and e the charge of an electron.

The total charge q contributed by electrons in the volume Al can be expressed as:

q = n × e × A × l

Derivation of Drift Velocity

If this charge q flows in time t, then current I is:

I = q / t = (n × e × A × l) / t

Since velocity is distance over time, and here electrons drift a length l in time t:

v_d = l / t

Substitute into the equation:

I = n × e × A × v_d

⇒ v_d = I / (n × e × A)

This shows that:

• The drift velocity increases proportionally with the electric current I.
• It is inversely proportional to the number density n and area A.

Definition of Mobility

Mobility μ is defined as the drift velocity per unit electric field:

μ = v_d / E

Substituting E = V / l and v_d = l / t gives:

μ = (l / t) / (V / l) = l² / (V × t)

Alternatively, using microscopic theory, mobility is given by:

μ = e × τ / m

• e = charge of electron
• τ = mean relaxation time (average time between collisions)
• m = mass of electron

Factors Affecting Mobility

Mobility depends on the following factors:

• Relaxation time (τ): Longer τ means fewer collisions, so higher mobility.
• Mass of the charge carrier (m): Particles with lower mass, such as electrons, tend to exhibit greater mobility.
• Temperature: As temperature increases, lattice vibrations increase, reducing τ and hence mobility.
• Material type: Conductors, semiconductors, and insulators all have different n, τ, and structural properties affecting mobility.

Understanding drift velocity and mobility is crucial for analyzing current flow in conductors, semiconductors, and nanomaterials—making these concepts highly important for exams.