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Current Electricity | Drift Velocity | Mobility

Current Electricity - Drift Velocity and Mobility Relation

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1. Complete Formula Sheet

Start here. Most exam questions are solved by connecting current, current density, drift velocity, mobility, conductivity and resistivity.

I = Q/tElectric current

J = I/ACurrent density

v_d = I/(nAe)Drift velocity from current

J = nev_dCurrent density relation

μ = v_d/EMobility definition

v_d = eEτ/mDrift velocity from relaxation time

μ = eτ/mMobility relation

σ = neμConductivity

ρ = 1/σResistivity

V = IROhm's law

J = σEMicroscopic Ohm's law

ρ = 1/(neμ)Resistivity from mobility

Symbols, Units and Dimensions

I: current, A, [A]

J: current density, A m⁻², [A L⁻²]

v_d: drift velocity, m s⁻¹, [L T⁻¹]

μ: mobility, m² V⁻¹ s⁻¹, [M⁻¹ T² A]

n: number density, m⁻³, [L⁻³]

e: electronic charge, C, [A T]

E: electric field, V m⁻¹, [M L T⁻³ A⁻¹]

τ: relaxation time, s, [T]

m: mass/effective mass, kg, [M]

σ: conductivity, S m⁻¹, [M⁻¹ L⁻³ T³ A²]

ρ: resistivity, Ω m, [M L³ T⁻³ A⁻²]

A: area, m², [L²]

2. Introduction to Drift Velocity

Inside a conductor, free electrons move randomly at high thermal speeds. This random motion has no preferred direction, so the average displacement in any direction is zero and it does not produce current. When an external electric field is applied, a small average motion is superposed on the random motion. This small average velocity is called drift velocity.

Random motion: fast, chaotic, direction changes after collisions, no net current.

Drift motion: slow average motion caused by electric field, responsible for current.

3. Derivation of Drift Velocity

An electron placed in electric field E experiences electric force. For magnitude, F = eE.

The electron charge is negative, so the actual force is opposite to E, but the magnitude remains eE.

Using Newton's second law, a = F/m.

Substitute F = eE: a = eE/m.

The electron accelerates only between collisions. The average time between two successive collisions is relaxation time τ.

Average drift speed gained in this time is v_d = aτ.

Substitute acceleration: v_d = (eE/m)τ.

Therefore v_d = eEτ/m.

4-5. What Is Mobility and Its Derivation?

Mobility is the drift velocity acquired by a charge carrier per unit electric field. It measures how easily charge carriers move inside a material.

Definition: μ = v_d/E. Unit = m² V⁻¹ s⁻¹. Higher mobility means larger drift velocity for the same electric field.

Start from drift velocity: v_d = eEτ/m.

Mobility is defined as μ = v_d/E.

Substitute v_d: μ = (eEτ/m)/E.

Cancel E: μ = eτ/m.

Relaxation time increases mobility; larger effective mass and stronger collision mechanism decrease mobility.

Mean free path is the average distance travelled between collisions. Temperature increases lattice vibrations, which usually reduces relaxation time in metals and hence reduces mobility.

6. Relation Between Drift Velocity and Mobility

Core relation: v_d = μE

Change	Effect on drift velocity
Electric field increases	v_d increases directly if μ remains constant.
Voltage increases at fixed length	E = V/L increases, so v_d increases.
Length increases at fixed voltage	E decreases, so v_d decreases.
Temperature increases in metals	μ decreases, so v_d decreases for same E.
Mobility increases	v_d increases for same E.

7-8. Conductivity, Resistivity and Mobility

Current through a conductor is I = nAev_d.

Divide both sides by A: I/A = nev_d.

Since J = I/A, J = nev_d.

Use v_d = μE: J = neμE.

Microscopic Ohm's law is J = σE. Comparing gives σ = neμ.

Resistivity is reciprocal of conductivity: ρ = 1/σ = 1/(neμ).

Why mobility lowers resistivity: larger μ means carriers drift faster for the same E, producing more current, so ρ decreases.

Metals vs insulators: metals have huge free electron density and useful mobility; insulators have extremely low carrier concentration.

9. Temperature Effect

Material	Microscopic effect	Observed effect
Metals	Temperature increases lattice vibrations, relaxation time decreases and mobility decreases.	Resistance increases; conductivity decreases.
Semiconductors	Carrier concentration increases strongly with temperature, even if mobility may decrease.	Conductivity usually increases; resistance decreases.

10-15. Exam Question Bank With Accordion Solutions

Click any question to reveal the answer and detailed explanation.

16. Common Student Mistakes

17. Exam Strategy

CBSE

Master definitions, units, dimensions, derivations of v_d, μ, σ and ρ.

NEET

Practise quick formula selection and proportional reasoning for v_d, E, A, n and μ.

JEE Main

Connect V/L, I = nAev_d, J = σE and numerical unit conversion.

JEE Advanced

Focus on graphs, variable cross-section, multiple carriers and microscopic assumptions.

Olympiad

Build model-level intuition: random motion, average drift, collisions and relaxation time.

AP Physics

Use conceptual explanations linking circuit current to microscopic carrier motion.

IB Physics

Prioritise explanation, dimensions, assumptions and graph interpretation.

A-Level

Practise drift velocity, number density, resistivity and microscopic Ohm's law derivations.

FAQ

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