Mean Drift Velocity

In the previous lesson we defined electric current as the rate of flow of charge: I = ΔQ/Δt. But this does not tell us anything about what is happening inside the wire. How fast are the electrons actually moving? How many are there? What controls the current for a given applied voltage?

This lesson derives and applies the drift velocity equation I = nAve, which links the macroscopic current to the microscopic motion of charge carriers. It is a required OCR H556 topic in Module 4.2.1, and it appears in exam questions every single year.

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A Wire Full of Electrons

A typical metal conductor — say a copper wire — contains a vast number of free electrons. Each copper atom contributes about one electron to a "sea" of delocalised charge carriers that move throughout the lattice. The positive ions remain fixed; only the electrons drift.

In the absence of a potential difference, these free electrons zoom around randomly at very high thermal speeds (~10⁵ m s⁻¹ at room temperature) but with no net direction. The average velocity is zero, so there is no net current.

When a potential difference is applied across the ends of the wire, the random motion continues — but now there is a superimposed slow drift in the direction opposite to the electric field (because electrons are negative). This slow average drift is what we call the mean drift velocity, symbol v.

flowchart LR
  subgraph Wire
    e1((e)) --> e2((e)) --> e3((e)) --> e4((e))
  end
  EF[Electric field E] -.->|pushes + charges this way| R[Right]
  EL[Electrons drift opposite to E] -.-> L[Left]

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Deriving I = nAve

Consider a cylindrical wire of cross-sectional area A, containing n free charge carriers per cubic metre, each with charge e. Suppose these carriers drift along the wire at a mean velocity v.

In a time Δt, each carrier moves a distance v · Δt along the wire. Therefore every carrier currently inside the small cylinder of length v Δt will cross the right-hand end in that time.

• Volume of cylinder = A · v · Δt
• Number of carriers in this volume = n · A · v · Δt
• Charge passing the end = (n · A · v · Δt) · e

By definition, current is charge per unit time, so:

I = (n A v Δt · e) / Δt = n A v e

Hence the drift velocity equation:

I = n A v e

where:

• I = current (A)
• n = number density of free charge carriers (m⁻³)
• A = cross-sectional area of the wire (m²)
• v = mean drift velocity (m s⁻¹)
• e = charge per carrier, = 1.60 × 10⁻¹⁹ C for electrons

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Worked Example 1 — Copper Wire

A copper wire of cross-sectional area 1.0 mm² carries a current of 2.0 A. The number density of free electrons in copper is n = 8.5 × 10²⁸ m⁻³.

Calculate the mean drift velocity of the electrons.

Step 1 — Convert units.

• A = 1.0 mm² = 1.0 × 10⁻⁶ m²

Step 2 — Rearrange for v.

• v = I / (nAe)
• v = 2.0 / (8.5 × 10²⁸ × 1.0 × 10⁻⁶ × 1.60 × 10⁻¹⁹)
• v = 2.0 / (1.36 × 10⁴)
• v ≈ 1.5 × 10⁻⁴ m s⁻¹

That is about 0.15 mm per second. Less than a millimetre per second! This is a famously counter-intuitive result. How can a light bulb switch on "instantly" when the electrons are moving so slowly?

The answer is that the electric field that pushes every electron propagates along the wire at close to the speed of light. Every electron in the entire circuit starts drifting almost simultaneously. Your lamp lights up because the electrons already inside the filament begin to drift immediately; you do not have to wait for a particular electron to travel from the battery to the bulb.

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Worked Example 2 — Varying Cross-Section