Drift of Electrons and the Origin of Resistivity

The average velocity acquired by free electrons in a conductor under the influence of an electric field is called the drift velocity.

The average time interval between two successive collisions of a free electron with the ions of the metallic lattice is called the relaxation time and is denoted by τ.

Current density is the current flowing per unit cross-sectional area of the conductor. The source material connects current density with the drift motion of electrons.

Conductivity, denoted by σσ, measures how easily current flows through a material. From the microscopic model, conductivity depends on free electron density and relaxation time.

Using the average time between collisions ττ, the source derives the drift velocity as: 

Random Motion Without Electric Field

• Free electrons in a metal move randomly due to thermal energy. 
• Their velocities are in different directions. 
• The average velocity of electrons becomes zero when no electric field is present. 
• Therefore, net current is zero in the absence of an electric field. 

Motion With Electric Field

• Suppose an electric field E is applied to the conductor. 
• An electron of charge −e experiences a force opposite to the field. 
• Its acceleration is given by: a = −\[\frac {eE}{m}\]​
  where e is the electronic charge, and mm is the mass of an electron. 
• Between two collisions, the electron accelerates under the electric field. 
• Because collisions occur repeatedly, the electron does not keep accelerating indefinitely. Instead, it acquires a constant average drift velocity. 

If nn is the number of free electrons per unit volume and A is the area of cross-section of the conductor, then the current is obtained from electron drift as: 

This links microscopic electron motion with macroscopic current in a wire. 

Current Density

The source further gives the relation:

Substituting the expression for drift velocity, the current density becomes: 

This is the microscopic form of Ohm’s law. 

Comparing

with the microscopic expression,

We get conductivity as:

Interpretation

• A material conducts better if it has more free electrons per unit volume.
• A material conducts better if collisions are less frequent, that is, if the relaxation time is larger.
• Resistivity arises because collisions oppose the smooth drift of electrons.

The source explains that in a conductor, electrons do not move freely in a perfectly straight path for long intervals. They continuously collide with fixed positive ions of the lattice and with imperfections present in the material. 

These collisions interrupt the acceleration produced by the electric field. Because of this interruption, electrons acquire only a small average drift speed, and the material shows resistance to current flow. 

Simple Analogy

Imagine students walking randomly in a crowded school corridor. If the exit gate opens at one end, the crowd still moves irregularly, but there is a slow net movement toward the exit. The crowd’s interruptions due to people bumping into one another are like electron collisions inside a conductor.

Question idea: Copper wire, area 1.0 × 10⁻⁷ m², current 1.5 A. Find the drift speed of electrons. Then compare it with the thermal speed and the speed of the electric signal.

(a) Finding drift speed

• Use formula: vd = \[\frac {I}{neA}\]

• For copper, assume each atom gives 1 free electron.

• Use given data to find the number density of electrons n ≈ 8.5 × 10²⁸ m⁻³.

• Substitute I = 1.5 A, e = 1.6 × 10⁻¹⁹ C, A = 1.0 × 10⁻⁷ m², and this n.

• You get v_d ≈ 1.1 × 10⁻³ m s⁻¹, i.e. about 1.1 mm per second.

Meaning: Electrons drift forward very slowly, only a few millimetres each second, even when the current is 1.5 A.

(b) Comparing speeds

1. With the thermal speed of atoms

• Copper atoms at room temperature move randomly with speeds of about 2 × 10² m s⁻¹.
• This is about 10⁵ times larger than the electron drift speed.

2. With the speed of the electric field (signal)

• The electric field (signal) travels along the wire at almost 3.0 × 108 m s−1 (the speed of light).
• This is about 10¹¹ times larger than the drift speed.

Key takeaway: Electrons move forward very slowly (with a small drift speed), but the electrical signal travels extremely fast, close to the speed of light.

(a) If the drift speed is only a few mm/s, how is the current established almost instantly?

• The electric field is set up in the whole circuit almost instantly at light speed.
• So electrons everywhere start drifting at the same time; current appears quickly without waiting for electrons to travel from one end to the other.

Short idea: Signal is fast, electrons are slow. Current starts when the signal reaches, not when electrons travel across the wire.

(b) If force causes acceleration, why do electrons have a steady drift speed?

• An electron accelerates only between two collisions.
• At each collision, it loses the drift speed it gained and starts again.
• On average, due to repeated collisions, it settles to a constant average drift velocity.

Short idea: Accelerate → collide → lose drift → accelerate again. The average becomes a steady, small drift speed.

(c) If drift speed and electron charge are small, how can current be large?

• Because there are an enormous number of free electrons per unit volume (about 10²⁹ m⁻³).
• Even with a small drift speed, so many electrons moving together give a large current.

Short idea: Huge number of electrons × low speed = big current.

(d) When electrons drift from low to high potential, do all free electrons move in the same direction?

• No. Each electron still moves randomly in all directions.
• The electric field only adds a small net drift in one direction on top of this random motion.

Short idea: Random motion continues; drift is just a tiny bias in one direction.

(e) Are electron paths straight between collisions in (i) no field, (ii) with a field?

1. Without an electric field, paths between collisions are straight lines.
2. With an electric field, paths are generally curved (the field bends their motion between collisions).

Short idea: No field → straight paths; with field → curved paths.