Current, Charge & Drift Velocity

Define electric current from first principles, derive the drift velocity equation, and understand charge flow in metals and semiconductors.

AQA A-Level Physics · Unit 5: Electricity

⚡Electric current
Define I = ΔQ/Δt and relate to charge flow

🔬Drift velocity
Derive and apply I = nAvq

➡️Conventional vs electron flow
Explain the historical convention and actual electron flow

🔗Charge carriers in metals
Describe free electron conduction in metallic conductors

🌡️Semiconductors
Compare n-type (electrons) and p-type (holes) charge carriers

💎Charge carrier density
Use n to compare conductors, semiconductors, and insulators

Electric Current and Charge

Electric current is the rate of flow of charge past a given point in a conductor. It is defined as the charge flowing per unit time:

I = ΔQ / Δt

Symbol	Quantity	Unit
I	Electric current	A (amperes)
ΔQ	Charge flowing	C (coulombs)
Δt	Time interval	s

One ampere: One coulomb of charge flowing past a point per second. (1 A = 1 C s⁻¹)

Rearranging: ΔQ = I × Δt. This means that the total charge that flows is the product of current and time. This is consistent with the fact that on a current-time graph, the area under the curve equals the total charge transferred.

The charge on a single electron (or proton) is the elementary charge: e = 1.60 × 10⁻¹⁹ C. For a current I, the number of electrons passing a point per second is I/e.

Charge is quantised — it always comes in multiples of the elementary charge e = 1.60 × 10⁻¹⁹ C.

Conventional Current vs Electron Flow

When electricity was first studied, scientists had no knowledge of electrons. They assumed positive charge carriers flowed from the positive terminal to the negative terminal of a cell. This became the conventional current direction.

Later, when electrons were discovered, it was found that in a metallic conductor the actual charge carriers are electrons — which are negatively charged. Electrons flow from the negative terminal (low potential) to the positive terminal (high potential).

Conventional current: Flows from + to − (positive terminal to negative terminal) — the direction a positive charge would move.

Electron flow: Electrons flow from − to + (negative terminal to positive terminal) — opposite to conventional current.

Although we now know the true physics involves electron flow, conventional current is still universally used in circuit analysis and diagrams. The two are interchangeable mathematically — reversing the charge sign and the direction of flow gives the same result.

In metals, charge is carried by electrons moving in the opposite direction to conventional current. In electrolytes and semiconductors, positive charge carriers (ions or holes) can also contribute to current in the conventional direction.

Drift Velocity: Deriving I = nAvq

In a metal, conduction electrons are in continuous random thermal motion. When a potential difference is applied, electrons experience a net force and slowly drift in one direction, superimposed on the random motion. The average drift speed is called the drift velocity.

Derivation: Consider a conductor of cross-sectional area A with n free charge carriers per unit volume, each carrying charge q, drifting at average speed v.

In time Δt, each electron moves a distance vΔt along the conductor
The volume swept out is A × vΔt
Number of charge carriers in this volume = n × A × vΔt
Total charge: ΔQ = nAvΔt × q
Current: I = ΔQ/Δt = nAvq

I = nAvq

Symbol	Quantity	Unit
n	Number density of charge carriers	m⁻³
A	Cross-sectional area of conductor	m²
v	Drift velocity of charge carriers	m s⁻¹
q	Charge per carrier (e = 1.60 × 10⁻¹⁹ C for electrons)	C

Drift velocity in typical copper wire carrying household current is only about 0.1 mm s⁻¹ — far slower than the random thermal speed (~10⁶ m s⁻¹). Yet electrical signals travel at nearly the speed of light because it is the electric field, not the electrons themselves, that propagates.

Conductors, Semiconductors and Insulators

The key difference between conductors, semiconductors, and insulators is the number density of free charge carriers n:

Conductors (e.g. copper): n ≈ 10²⁸ to 10²⁹ m⁻³ — huge number of free electrons, very low drift velocity for a given current
Semiconductors (e.g. silicon): n ≈ 10¹⁶ to 10²³ m⁻³ — moderate carrier density; much higher drift velocity needed for the same current
Insulators (e.g. rubber): n ≈ 0 (effectively no free charge carriers)

In metals, the charge carriers are conduction electrons — electrons that have left their parent atoms and are free to move through the lattice. The number density is high (roughly 1 free electron per atom).

In semiconductors, charge can be carried by:

Electrons (negative carriers): in n-type semiconductors or thermally excited intrinsic carriers
Holes (positive carriers): spaces left when electrons move — in p-type semiconductors, holes are the majority carrier and drift in the same direction as conventional current

In an electrolyte (e.g. saltwater), both positive ions (moving in conventional current direction) and negative ions (moving in electron current direction) carry charge simultaneously.

From I = nAvq: for the same current, a wider conductor (larger A) or higher carrier density (larger n) requires a smaller drift velocity. This is why thick wires carry current more easily.

A current of 3.5 A flows through a wire for 4 minutes. Calculate the total charge that flows.

1Convert time: 4 min = 240 s

2ΔQ = I × Δt = 3.5 × 240 = 840 C

Charge = 840 C

A copper wire has cross-sectional area 2.0 × 10⁻⁶ m² and carries a current of 5.0 A. The number density of free electrons in copper is 8.5 × 10²⁸ m⁻³. Calculate the drift velocity of the electrons.

1Using I = nAvq: v = I / (nAq)

2v = 5.0 / (8.5 × 10²⁸ × 2.0 × 10⁻⁶ × 1.60 × 10⁻¹⁹)

3Denominator = 8.5 × 10²⁸ × 2.0 × 10⁻⁶ × 1.60 × 10⁻¹⁹ = 2.72 × 10⁴

4v = 5.0 / (2.72 × 10⁴) = 1.84 × 10⁻⁴ m s⁻¹

Drift velocity = 1.84 × 10⁻⁴ m s⁻¹ ≈ 0.18 mm s⁻¹

How many electrons flow past a point in a wire every second when the current is 2.0 A?

1Charge per second = I = 2.0 C s⁻¹

2Number of electrons per second = I / e = 2.0 / (1.60 × 10⁻¹⁹)

3= 1.25 × 10¹⁹ electrons per second

1.25 × 10¹⁹ electrons per second

A wire of diameter 1.5 mm carries the same current as a wire of diameter 3.0 mm, made of the same material. Compare their drift velocities.

1From I = nAvq: v = I/(nAq). For the same I, n, q: v ∝ 1/A ∝ 1/d²

2d₁ = 1.5 mm, d₂ = 3.0 mm → d₂ = 2d₁, so A₂ = 4A₁

3v₁/v₂ = A₂/A₁ = 4

The thinner wire (1.5 mm diameter) has 4 times the drift velocity of the thicker wire.

Q1. A charge of 120 C flows through a lamp in 2 minutes. What is the current?

Q2. In which direction do electrons actually flow in a circuit connected to a battery?

Q3. A semiconductor has a much higher drift velocity than a metal conductor of the same dimensions carrying the same current. Explain why, using I = nAvq.

Q4. A wire carries a current of 0.8 A. The charge carriers are electrons (q = 1.6 × 10⁻¹⁹ C), n = 6.0 × 10²⁸ m⁻³, diameter = 2.0 mm. Find the drift velocity.

Q5. Which of the following best describes the charge carriers in an n-type semiconductor?

Challenge Q1. A wire of diameter 1.0 mm and length 0.5 m carries a current of 3.0 A. Given n = 8.0 × 10²⁸ m⁻³ and q = 1.6 × 10⁻¹⁹ C, calculate (a) the drift velocity and (b) the time for one electron to travel the full length of the wire.

Challenge Q2. Two wires A and B carry the same current. Wire A has twice the diameter and half the number density of charge carriers compared to wire B. Compare their drift velocities.

Challenge Q3. Explain why, despite the very low drift velocity of electrons in a wire (~0.1 mm s⁻¹), a light bulb connected to a battery lights up almost instantly when the switch is closed.