Drift Velocity of Electrons in a Copper Wire – Complete Theory and Detailed Numerical Solution

Concept Used

The current flowing through a conductor is related to the drift velocity by

I = neAv_d

where

• I = Current
• n = Number of free electrons per unit volume
• e = Charge of an electron (1.6 × 10⁻¹⁹ C)
• A = Cross-sectional area
• v_d = Drift velocity

Once the drift velocity is known, the time taken to travel a distance of 1 m is simply

t = Distance / Drift Velocity

Step 1: Convert Density into SI Units

Density of copper is given as

8.92 g/cm³

Converting into SI units,

ρ = 8920 kg/m³

Step 2: Calculate the Number Density of Free Electrons

The number of copper atoms per unit volume is

n = (ρ/M) × N_A

Substituting the values,

n = (8920 / 0.0635) × (6.02 × 10²³)

n ≈ 8.46 × 10²⁸ electrons/m³

Since each copper atom contributes one free electron, this is also the free electron density.

Step 3: Calculate the Drift Velocity

Using

I = neAv_d

Therefore,

v_d = I/(neA)

Substituting the values,

v_d = 20 / [(8.46 × 10²⁸) × (1.6 × 10⁻¹⁹) × (6.62 × 10⁻⁶)]

v_d ≈ 2.23 × 10⁻⁴ m/s

Step 4: Calculate the Time Required to Travel 1 m

Using

t = Distance / Drift Velocity

t = 1 / (2.23 × 10⁻⁴)

t ≈ 4477 s

Converting seconds into minutes,

t = 4477 / 60 ≈ 74.6 min

Why Is the Drift Velocity So Small?

Many students are surprised that electrons take more than an hour to travel just one metre. This happens because electrons move randomly in all directions due to thermal motion. The applied electric field produces only a very small average drift in one direction. Although the drift velocity is small, the electric signal travels through the conductor almost instantaneously because the electric field propagates nearly at the speed of light.

Important Formulae

Current: I = neAv_d

Number Density: n = (ρ/M)N_A

Drift Velocity: v_d = I/(neA)

Time: t = Distance/v_d

Final Answer

The time required for the electrons to travel a distance of 1 m is

74.6 minutes (approximately)