What is electric current in a metal?
Electric current is the rate at which charge flows: I = q / t. Its unit is the ampere (A); 1 A means 1 coulomb of charge passing a point every second.
A metal has free electrons. These are outer electrons that are not tied to any one atom. The atoms that lost them become fixed positive ions. In copper there are about 8.5 × 1028 free electrons in every cubic metre. This number per volume is called n, the number density.
By convention, current direction is the direction positive charge would move. Electrons are negative, so they move the opposite way to the current.
Current density j = I / A is the current per unit area (unit A/m²). It has a direction, so it is a vector.
Random motion: why no current without a cell
At room temperature free electrons move very fast, around 105 m/s, in random directions. They keep hitting ions and changing direction. The average time between two collisions is called the relaxation time, τ (about 10−14 s in copper).
Because the directions are random, the average velocity of all electrons is zero. As many electrons cross any slice left-to-right as right-to-left, so the net current is zero.
Drift velocity: the slow average shift
When a field E acts, each electron feels a force eE and gets an acceleration a = eE/m (opposite to E). Between collisions it gains extra velocity a·t. Averaged over all electrons, the extra velocity is a·τ.
Drift velocity = the average velocity with which free electrons move in a conductor when a field is applied:
vd = eEτ / m (direction: opposite to E)
It is tiny, about 10−4 to 10−3 m/s.
Deriving I = n e A vd
Take a wire of area A. In time Δt, every electron drifts a distance vdΔt. So all electrons inside a cylinder of length vdΔt cross a slice.
- Volume of that cylinder = A vd Δt
- Number of electrons = n A vd Δt
- Charge crossing = e n A vd Δt
- Current = charge ÷ time = I = n e A vd
Dividing by A: current density j = n e vd. Putting vd = eEτ/m gives j = (ne²τ/m) E, which is Ohm's law in its basic form, j = σE.
Mobility
Mobility (μ) is the drift velocity per unit electric field:
μ = vd / E = eτ / m
SI unit: m²V−1s−1. It is always taken as positive. Larger τ (fewer collisions) means greater mobility. In semiconductors holes and electrons have different mobilities. Using mobility: I = n e A μ E and σ = n e μ.
Why does a bulb glow at once if electrons are so slow?
The wire is already full of free electrons. When the switch closes, the electric field spreads along the wire at nearly the speed of light. Electrons everywhere, including those inside the bulb filament, start drifting at the same time. No electron has to travel from the switch to the bulb.
Try it: the 3D and at home
In the 3D (predict, then check): set E = 1, then E = 2. Guess what happens to vd before you move the slider. Then make A = 3 at fixed E and guess the current.
At home: line up 10 coins touching each other on a table. Flick one end coin gently. The coin at the far end moves almost at once, but each coin moved only a tiny bit. That is how the push travels fast while each electron drifts slowly.
Board exam focus
Common questions: define drift velocity and relaxation time (1–2 marks); derive I = n e A vd or j = σE (3 marks); numericals on vd and μ; effect of doubling length or voltage on drift velocity.
Key formulas and definitions
- I = q / t
- j = I / A
- vd = eEτ / m
- I = n e A vd
- j = n e vd = σE, σ = ne²τ/m
- μ = vd / E = eτ / m
- σ = n e μ
Worked examples
1. A charge of 120 C passes through a wire in 1 minute. Find the current.
Step 1: t = 1 min = 60 s. Step 2: I = q/t = 120/60. Answer: I = 2 A.
2. How many electrons pass a point each second when the current is 1.6 A? (e = 1.6 × 10⁻¹⁹ C)
Step 1: charge per second = 1.6 C. Step 2: number = q/e = 1.6 / 1.6 × 10⁻¹⁹. Answer: 10¹⁹ electrons per second.
3. A copper wire of area 1.0 × 10⁻⁶ m² carries 1.7 A. n = 8.5 × 10²⁸ m⁻³. Find the drift velocity.
Step 1: vd = I/(n e A). Step 2: n e A = 8.5 × 10²⁸ × 1.6 × 10⁻¹⁹ × 1.0 × 10⁻⁶ = 1.36 × 10⁴. Step 3: vd = 1.7 / 1.36 × 10⁴ = 1.25 × 10⁻⁴ m/s. Answer: about 0.125 mm/s.
4. In the wire above, how long would one electron take to drift 1 m?
Step 1: t = distance / vd = 1 / 1.25 × 10⁻⁴. Step 2: t = 8000 s ≈ 2.2 hours. This shows the drift is very slow, even though the lamp lights instantly.
5. Electrons drift at 2 × 10⁻⁴ m/s in a field of 0.5 V/m. Find the mobility and the relaxation time. (m = 9.1 × 10⁻³¹ kg)
Step 1: μ = vd/E = 2 × 10⁻⁴ / 0.5 = 4 × 10⁻⁴ m²V⁻¹s⁻¹. Step 2: μ = eτ/m, so τ = μm/e = 4 × 10⁻⁴ × 9.1 × 10⁻³¹ / 1.6 × 10⁻¹⁹. Step 3: τ ≈ 2.3 × 10⁻¹⁵ s.
6. A wire is stretched so that its length doubles. The same potential difference is kept across it. What happens to the drift velocity?
Step 1: E = V/l. Length doubles, so E halves. Step 2: vd = eEτ/m, so vd ∝ E. Step 3: vd becomes half. (Area also halves, so the current becomes one quarter, but vd depends only on E.)
7. Two wires of the same material, areas A and 2A, are joined end to end and carry a current I. Compare the drift velocities.
Step 1: The same current flows through both (series). Step 2: vd = I/(neA), with n and e the same. Step 3: vd ∝ 1/A, so the thin wire has twice the drift velocity of the thick wire: v₁ : v₂ = 2 : 1.
Common mistakes
- Thinking electrons race from the switch to the bulb. They only drift about a millimetre per second; the field spreads fast.
- Mixing up thermal speed (~10⁵ m/s, random) with drift speed (~10⁻⁴ m/s, ordered).
- Forgetting that electrons drift opposite to E and opposite to the conventional current.
- Saying vd depends on the area at fixed voltage. At fixed V and l, vd = eVτ/(ml) does not depend on A; only the current does.