Drift Velocity of Electrons: Derivation, mobility, and its direct relation with electric current (Ohm's Law derivation)

📘 1. What is Drift Velocity?

In a conductor, free electrons are in continuous random motion. When an external electric field is applied, electrons acquire a small average velocity opposite to the direction of the electric field. This average velocity is called drift velocity.

v_d = Drift Velocity of Electrons

⭐ Electrons move opposite to the electric field, while conventional current flows in the direction of the electric field.

⚡ 2. Derivation of Drift Velocity

Consider an electron of charge −e and mass m placed in an electric field E.

F = eE

Using Newton's second law:

F = ma

Therefore,

a = eE/m

If τ is the average relaxation time between two successive collisions:

v_d = aτ

Hence,

v_d = eEτ/m

Since the electron moves opposite to E, the vector form is:

⃗v_d = −eτ⃗E/m

🔬 3. Interactive Drift Velocity Practical

Electric Field E 5

Relaxation Time τ 5

Electron Density n 5

Animation

Electric Field E →

Conventional Current I →

Relative Drift Velocity: 25
Relative Current: 125
Mobility: 5
Relation: v_d ∝ Eτ

🔋 4. Drift Velocity and Electric Current

Suppose a conductor has cross-sectional area A and contains n free electrons per unit volume.

In time Δt, electrons travel an average distance:

v_dΔt

Volume swept by electrons:

Av_dΔt

Number of electrons:

N = nAv_dΔt

Charge passing through the cross-section:

Q = neAv_dΔt

Current:

I = Q/Δt

Therefore,

I = neAv_d

📊 5. Current Density

Current per unit cross-sectional area is called current density.

J = I/A

Using:

I = neAv_d

we obtain:

J = nev_d

🚀 6. Mobility of Electrons

Mobility is defined as the drift velocity acquired by a charge carrier per unit electric field.

μ = v_d/E

Using:

v_d = eEτ/m

we get:

μ = eτ/m

⭐ Greater relaxation time means greater mobility.

🔗 7. Direct Relation Between Current and Electric Field

Starting with:

I = neAv_d

and

v_d = eEτ/m

Substitution gives:

I = neA(eEτ/m)

Therefore,

I = ne²AτE/m

💡 8. Derivation of Ohm's Law

For a conductor of length L:

E = V/L

Substituting in:

I = ne²AτE/m

we get:

I = ne²AτV/mL

Rearranging:

I = [ne²Aτ/mL]V

The term in brackets is constant for a given conductor at constant temperature. Therefore,

I ∝ V

Hence,

V = IR

or

I = V/R

This is Ohm's Law.

⭐ Thus, the microscopic theory of electron drift leads to Ohm's Law.

📐 9. Expression for Resistance

Comparing:

I = ne²AτV/mL

with:

I = V/R

we obtain:

1/R = ne²Aτ/mL

Therefore:

R = mL/(ne²Aτ)

Since:

ρ = m/(ne²τ)

we get:

R = ρL/A

📚 10. Important Formula Summary

Drift Velocity

v_d = eEτ/m

Current

I = neAv_d

Current Density

J = nev_d

Mobility

μ = v_d/E

Mobility Formula

μ = eτ/m

Resistance

R = mL/(ne²Aτ)

Resistivity

ρ = m/(ne²τ)

Ohm's Law

V = IR

🧮 11. Solved Numerical

Question:

A conductor has free electron density n = 8.5 × 10²⁸ m⁻³, cross-sectional area A = 1 mm² and drift velocity v_d = 1 × 10⁻⁴ m/s. Find the current.

I = neAv_d

I = (8.5×10²⁸)(1.6×10⁻¹⁹) (1×10⁻⁶)(1×10⁻⁴)

I ≈ 1.36 A

📝 12. MCQ Practice

1. Drift velocity of electrons is directly proportional to:

A. 1/E

B. E

C. 1/τ

D. m

✔ Answer: B

2. Drift velocity of electrons is:

A. Along electric field

B. Opposite to electric field

C. Perpendicular to electric field

D. Zero always

✔ Answer: B

3. Current in a conductor is:

A. neA/v_d

B. neAv_d

C. ne/v_d

D. nA/e

✔ Answer: B

4. Mobility is defined as:

A. E/v_d

B. v_d/E

C. I/V

D. V/I

✔ Answer: B

5. Mobility of electrons is:

A. eτ/m

B. m/eτ

C. e/mτ

D. mτ/e

✔ Answer: A

6. Current density is:

A. ne/v_d

B. nev_d

C. neAv_d

D. n/e

✔ Answer: B

7. Ohm's law is obtained from drift theory when:

A. τ changes continuously

B. Temperature is constant

C. n becomes zero

D. A becomes zero

✔ Answer: B

8. Resistance according to microscopic theory is:

A. mL/ne²Aτ

B. ne²Aτ/mL

C. mA/ne²Lτ

D. neAτ/mL

✔ Answer: A

9. If electric field is doubled, drift velocity becomes:

A. Half

B. Same

C. Double

D. Four times

✔ Answer: C

10. Conventional current is opposite to:

A. Electric field

B. Electron drift velocity

C. Positive charge motion

D. Potential gradient

✔ Answer: B

🟢 13. 2 Marks — 6 Questions

Q1 Define drift velocity.

Q2 Write the expression for drift velocity of an electron.

Q3 Define mobility of a charge carrier.

Q4 Write the relation between current and drift velocity.

Q5 What is relaxation time?

Q6 Write the microscopic expression for resistance.

🟡 14. 3 Marks — 6 Questions

Q1 Derive v_d = eEτ/m.

Q2 Derive I = neAv_d.

Q3 Define mobility and derive μ = eτ/m.

Q4 Explain the direction of electron drift velocity and conventional current.

Q5 Derive the relation J = nev_d.

Q6 Explain the role of relaxation time in drift velocity.

🟠 15. 4 Marks — 6 Questions

Q1 Derive the expression for drift velocity of electrons in a conductor.

Q2 Derive the relation I = neAv_d.

Q3 Explain mobility and obtain μ = eτ/m.

Q4 Derive the relation between current density and electric field.

Q5 Derive the microscopic expression for resistance.

Q6 Explain the microscopic basis of Ohm's law.

🔴 16. 5 Marks — 6 Questions

Q1 Derive the expression for drift velocity and explain all physical quantities involved.

Q2 Derive I = neAv_d and discuss the factors affecting current.

Q3 Derive the expression for mobility of electrons.

Q4 Using drift velocity, derive the microscopic form of Ohm's law.

Q5 Derive R = mL/(ne²Aτ).

Q6 Explain the relation between current density, electric field and conductivity.

🔵 17. 6 Marks — 6 Questions

Q1 Derive the expression for drift velocity of electrons and explain its dependence on electric field and relaxation time.

Q2 Starting from drift velocity, derive I = neAv_d.

Q3 Derive the mobility of electrons and discuss its physical significance.

Q4 Using the electron drift model, derive Ohm's law V = IR.

Q5 Derive expressions for conductivity and resistivity from the microscopic theory of conduction.

Q6 Starting with F = eE, derive v_d, current I, conductivity σ, resistivity ρ and finally Ohm's law.

🚀 Quick Revision

Drift Velocity: v_d = eEτ/m

Current: I = neAv_d

Current Density: J = nev_d

Mobility: μ = v_d/E = eτ/m

Conductivity: σ = ne²τ/m

Resistivity: ρ = m/(ne²τ)

Resistance: R = ρL/A

Ohm's Law: V = IR