Current electricity class 12 notes: Drift Velocity & Mobility

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Current electricity class 12 notes: Drift Velocity & Mobility

Unit 2 • Current Electricity

Understanding the directed motion of electrons in conductors and their response to electric fields

Drift Velocity (v_d)

Definition:

Drift velocity is the average velocity attained by charged particles (like electrons) in a conductor due to an applied electric field, superimposed on their random thermal motion.

Mathematical Expression

\[ v_d = \frac{eE\tau}{m} \]

Where:
e = Electron charge (1.6×10^-19 C)
E = Electric field (V/m)
τ = Mean free time between collisions (s)
m = Electron mass (9.1×10^-31 kg)

Relation with Current

\[ I = n e A v_d \]

Where:
n = Charge carrier density (m^-3)
A = Cross-sectional area (m²)
I = Current (A)

Key Characteristics

Typical values: ~10^-4 m/s (very slow!)
Directly proportional to applied electric field
Inversely proportional to resistivity of material
Independent of time for constant electric field

Electron Mobility (μ)

Definition:

Mobility measures how quickly an electron can move through a conductor when pulled by an electric field, defined as drift velocity per unit electric field.

Mobility Equation

\[ \mu = \frac{v_d}{E} = \frac{e\tau}{m} \]

Units: m²/V·s

Relation with Conductivity

\[ \sigma = n e \mu \]

Where σ = conductivity (Ω^-1m^-1)

Material	Mobility (m²/V·s)	Resistivity (Ω·m)
Copper	0.0032	1.68×10^-8
Silver	0.0056	1.59×10^-8
Silicon	0.15 (electrons)	2.3×10³

Derivation of Drift Velocity

Step-by-Step Derivation

Force on electron in electric field:

\[ F = eE \]

Acceleration between collisions:

\[ a = \frac{F}{m} = \frac{eE}{m} \]

Velocity gain before collision:

\[ \Delta v = a\tau = \frac{eE\tau}{m} \]

Average drift velocity:

\[ v_d = \frac{1}{2} \Delta v = \frac{eE\tau}{2m} \]

(Exact derivation gives same form with different τ interpretation)

Worked Example

Problem:

A copper wire (μ = 0.0032 m²/V·s, n = 8.5×10²⁸ m^-3) with 1 mm² cross-section carries 5 A current. Calculate:

Electric field required to maintain this current
Drift velocity of electrons
Time taken by an electron to drift 1m

Solution:

(1) Electric field:

\[ \sigma = ne\mu = 8.5\times10^{28} \times 1.6\times10^{-19} \times 0.0032 = 4.35\times10^7 \, \Omega^{-1}m^{-1} \] \[ J = \frac{I}{A} = \frac{5}{10^{-6}} = 5\times10^6 \, A/m^2 \] \[ E = \frac{J}{\sigma} = \frac{5\times10^6}{4.35\times10^7} \approx 0.115 \, V/m \]

(2) Drift velocity:

\[ v_d = \mu E = 0.0032 \times 0.115 \approx 3.68 \times 10^{-4} \, m/s \]

(3) Time for 1m drift:

\[ t = \frac{d}{v_d} = \frac{1}{3.68 \times 10^{-4}} \approx 2717 \, s \approx 45 \, \text{minutes} \]

Practice Problems

Problem 1

Calculate the drift velocity in a silver wire (μ = 0.0056 m²/V·s) when subjected to an electric field of 0.5 V/m.

Problem 2

A germanium sample (μ = 0.39 m²/V·s, n = 1.5×10¹⁹ m^-3) has 2V applied across 5mm length. Find the current density.

Problem 3

If the mean free time between collisions for electrons in copper is 2.5×10^-14s, calculate their mobility (m_e = 9.1×10^-31kg).

Problem 4

A wire carries 3A current with electron drift velocity of 0.1 mm/s. If the electron density is 6×10²⁸ m^-3, find its diameter.

Problem 5

An aluminum wire (μ = 0.0012 m²/V·s) has resistivity 2.65×10^-8 Ω·m. Calculate its charge carrier density.

Key Takeaways

Essential Relationships

Drift velocity: \( v_d = \mu E = \frac{I}{neA} \)
Mobility: \( \mu = \frac{v_d}{E} = \frac{\sigma}{ne} \)
Conductivity: \( \sigma = ne\mu = \frac{1}{\rho} \)
Current density: \( J = \sigma E = nev_d \)

Common Pitfalls

Confusing drift velocity with electron thermal velocity (~10⁶ m/s)
Forgetting that mobility is material-dependent
Mixing up conductivity (σ) and resistivity (ρ) relationships
Using incorrect units (e.g., cm instead of m in calculations)

Next Topic: Ohm’s Law and V-I Characteristics

In the next section, we’ll explore the fundamental relationship between voltage and current in conductors and semiconductors.

Continue to Next Topic →

Drift Velocity (vd)