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Current Electricity - Drift Velocity

Complete conceptual and exam-oriented guide for CBSE Class 12, NEET, JEE Main, JEE Advanced, Olympiad Physics, AP Physics, IB Physics and A-Level Physics.

CBSENEETJEE MainJEE AdvancedOlympiadAPIBA-Level

Section 1: Drift Velocity Formula Sheet

v_d = eEτ/m

v_d = I/(nAe)

I = nAev_d

J = nev_d

μ = v_d/E

σ = ne²τ/m

ρ = m/(ne²τ)

V = IR

Symbols: vd = drift velocity, e = electronic charge, E = electric field, τ = relaxation time, m = electron mass, I = current, n = electron density, A = area, J = current density, μ = mobility, σ = conductivity, ρ = resistivity. Units: vd in m/s, τ in s, μ in m²V⁻¹s⁻¹, σ in S/m, ρ in Ωm. Dimensions: [vd] = LT⁻¹, [μ] = M⁻¹T²A, [σ] = M⁻¹L⁻³T³A².

Section 2: What Is Drift Velocity?

In a metallic conductor, free electrons move randomly at high thermal speeds. This random motion does not produce current because, on average, equal numbers of electrons move in all directions. When an electric field is applied, electrons acquire a small average velocity opposite to the field. This average directed velocity is called drift velocity.

Random Motion

Very fast, chaotic and directionless. Net average velocity is zero.

Effect of Electric Field

Electric field exerts force on electrons and creates a small bias in their motion.

Physical Meaning

Drift velocity connects microscopic electron motion to macroscopic current.

random motion + slow drift ←conventional current →eeeee

Section 3: Derivation of Drift Velocity

In a metal, a free electron of charge -e is placed in electric field E.

Magnitude of force on electron is F = eE.

Using Newton's law, acceleration magnitude is a = F/m = eE/m.

Electrons collide repeatedly with lattice ions. Let τ be average time between collisions.

Average drift velocity gained between collisions is v_d = aτ.

Therefore v_d = eEτ/m. Direction of electron drift is opposite to E; formula usually gives magnitude.

Section 4: Relaxation Time

Relaxation time is the average time interval between two successive collisions of a conduction electron with lattice ions. Larger relaxation time means fewer collisions, higher drift velocity, higher mobility and higher conductivity.

At higher temperature in metals, lattice vibrations increase, relaxation time decreases and resistance increases.

Section 5: Current in Terms of Drift Velocity

Consider a conductor with area A and free electron density n.

In time dt, electrons in length v_ddt cross the area.

Volume crossing = Av_ddt.

Number of electrons = nAv_ddt.

Charge crossing = neAv_ddt.

Current I = dQ/dt = nAev_d.

Section 6: Current Density

J = I/A

J = nev_d

J vector points along conventional current

Unit: A m⁻²

Current density is useful in advanced problems where area, current flow or material properties change from point to point.

Section 7: Mobility of Electrons

Mobility is drift velocity produced per unit electric field. High mobility means charge carriers respond strongly to the applied field.

Definition: μ = v_d/E.

Using v_d = eEτ/m, μ = eτ/m.

Hence mobility depends on relaxation time and material scattering.

Section 8: Microscopic Form of Ohm's Law

Current density: J = nev_d.

Substitute v_d = eEτ/m.

J = ne(eEτ/m) = (ne²τ/m)E.

Define σ = ne²τ/m, so J = σE.

For a wire, J = I/A and E = V/L. So I/A = σV/L.

V/I = L/(σA) = ρL/A = R. Therefore V = IR.

Section 9: Conductivity and Resistivity

σ = ne²τ/m

ρ = 1/σ

ρ = m/(ne²τ)

J = σE

Metals

Large n, high conductivity, resistance increases with temperature.

Semiconductors

Carrier density increases strongly with temperature, conductivity can increase.

Insulators

Very low mobile carrier density, very high resistivity.

Section 10: Advanced Conceptual Questions

Section 11: Temperature Effects

Temperature affects drift velocity indirectly through relaxation time, resistance and conductivity. In metals, higher temperature increases lattice vibration, reduces τ, decreases mobility, decreases conductivity and increases resistance. For a fixed applied electric field, drift velocity decreases when τ decreases.

Section 12: CBSE Board Questions

Section 13: CBSE Case Study Questions

Section 14: NEET MCQs

Section 15: JEE Main MCQs

Section 16: JEE Advanced Problems

Section 17: Olympiad Challenge Problems

Section 18: Graphs and Diagrams

v_d vs E

For constant τ, v_d is directly proportional to E.

J vs E

Slope of J-E graph is conductivity σ.

R vs T for Metals

Resistance increases approximately linearly with temperature for moderate range.

Section 19: Common Mistakes

Electron Speed vs Drift Velocity

Thermal speed is large and random; drift velocity is small and directed.

Current Direction

Conventional current is opposite to electron drift in metals.

Sign Mistakes

Use magnitude in formulas unless vector direction is explicitly asked.

Relaxation Time

It is average collision time, not total travel time through wire.

Mobility

Mobility is v_d/E, not E/v_d.

Section 20: Exam Strategy

CBSE

Focus on derivations of v_d, I = nAev_d, mobility and J = σE.

NEET

Master direct formulas, units and proportionality tricks.

JEE Main

Practise numerical substitutions and graph-based questions.

JEE Advanced

Focus on variable area, non-uniform current density and temperature effects.

Olympiad

Reason from microscopic motion, collisions and conservation of charge.

AP/IB/A-Level

Connect theory with graphs, uncertainty, experimental circuits and written explanations.

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