Why “current” needs a deeper story
You already know I = q/t. That’s the engineering definition of current — it tells you how much charge crosses a section per second, and it’s all you need for circuit calculations. But it hides a strange fact: inside a copper wire carrying current to light your room, the electrons responsible are drifting along at a speed of roughly 0.1 millimetre per second — slower than a snail. Yet the light switches on almost instantly.
This note builds the microscopic model that explains both things at once: why individual electrons crawl, and why the effect of flipping a switch is (almost) instant anyway.
Free electrons: chaos first, drift second
In a metal, free electrons are in constant, random thermal motion, colliding with vibrating lattice ions roughly every 10^−14 seconds. At room temperature this thermal speed is enormous ⁻ — of the order of 10 m/s. But it’s ⁵ random: at any instant, roughly equal numbers of electrons move in every direction. Pick any cross-section of the wire, and the number crossing left-to right due to thermal motion alone is (on average) equal to the number crossing right-to-left. Net motion due to thermal agitation is zero. If this were the whole story, there would be no current at all — just electrons jostling in place.
Now apply an electric field E (by connecting a cell). Between collisions, each electron feels a small force −eE and accelerates slightly in the direction opposite to E. It doesn’t get far before it collides with a lattice ion and its velocity is randomized again — but each electron gains a little extra velocity, in the same direction, between each collision. Averaged over the huge number of electrons and collisions, this small directional bias survives even though the random thermal component averages to zero. That surviving average velocity is the drift velocity, vd.
Drift velocity is the average velocity gained by free electrons in a conductor due to an applied electric field, superimposed on their much larger random thermal motion.
This is why vd is so small compared to the thermal speed: it’s not that electrons move slowly — it’s that only a tiny, systematic sliver of their motion is common to all of them and therefore shows up as net charge transport.
Deriving I = nAevd
Consider a conductor of cross-sectional area A, with free-electron number density n (electrons per unit volume — a material property). Suppose the electrons drift with average speed vd.
Step 1 — how far does the “front” of charge travel in time dt? In a small time interval dt, an electron drifting at v_d moves a distance v_d·dt along the wire.
Step 2 — what volume of the conductor does this correspond to? All electrons within a distance vd· dt “upstream” of a chosen cross-section will cross that section within dt. This defines a cylindrical volume:
Volume = A × (vd · dt)
Step 3 — how many electrons are in that volume? Number of electrons = n × Volume = n × A × vd × dt
Step 4 — how much charge is that? Each electron carries charge e (magnitude), so: dq = n × A × vd × dt × e
Step 5 — divide by dt to get current.
I = dq/dt = n A e vd
I = n A e vd where n = free-electron number density (m ³),
A = cross-sectional area (m²),
e = electron charge (1.6×10-19 C),
vd = drift velocity (m/s).
Notice what this derivation actually used: not the speed of any individual electron, but the number of electrons swept through a cross-section per second. This is why current can be a large, steady, easily-measured quantity even though v_d itself is tiny — n is enormous (of the order of 10²8 –10²9 m–³ in a typical metal).
Worked Example 1
A copper wire has a cross-sectional area 1×10-6 m² and carries a current of 2 A. Given the free-electron density of copper, n = 8.5×10²8m ³, find the drift velocity.
Solution: Rearranging I = nAev_d:
vd = I / (nAe)
= 2 / (8.5×10²⁸ × 1×10⁻⁶ × 1.6×10⁻¹⁹)
= 2 / (1.36×10⁴)
vd ≈ 1.47×10⁻⁴ m/s
That’s about 0.15 mm/s — confirming the “slower than a snail” claim, and showing why this is never the mechanism you’d invoke to explain how fast a circuit “responds.”
So why does the bulb light up instantly?
Because what propagates quickly isn’t an electron travelling the length of the wire — it’s the electric field. The moment you close a switch, an electric field is established throughout the (already electron-filled) wire at a speed close to the speed of light. Every free electron in the
wire — not just the ones near the switch — starts drifting almost simultaneously, because the field reaches them almost simultaneously. The light comes on because electrons everywhere in the filament start drifting together, not because a single electron sprints from the switch to the bulb.
Common misconception: “Electrons travel from the switch to the bulb at the speed the light turns on.” This confuses the propagation speed of the electric field ( c) with ∼ the drift velocity of individual electrons (10^−4 m/s) — they differ by roughly twelve ∼ ⁻ orders of magnitude.
Mobility (μ)
Since drift velocity is caused by the field, it’s natural to ask: how much drift velocity do you get per unit field strength? That ratio is called mobility:
μ = vd / E
Mobility is a property of the material (and temperature) — it tells you how “responsive” the charge carriers are to an applied field, independent of how strong that field happens to be. Its SI unit is m²V –¹s –¹.
Combining with the earlier result, & using E = V/L for a wire of length L:
vd = -μE I = nAeμE = nAeμ(V/L) ⟹
Rearranging: V = I × (L / (nAeμ)). Compare this with Ohm’s law, V = IR. This shows resistance itself has a microscopic origin: R = L / (nAeμ) — resistance is high when mobility is low (electrons are “sluggish” between collisions) and low when mobility is high.
Current density (J)
Current density describes current per unit cross-sectional area, and unlike current itself, it’s a vector — it has a direction (the direction of conventional current flow at that point):
J = I / A = n e vd
SI unit: A/m². Current density is the more fundamental quantity when the cross-section isn’t uniform, or when you want to describe current flow at a point inside a three-dimensional conductor rather than a total current through a wire.
Relaxation time (τ) and the microscopic form of Ohm’s law
Relaxation time is the average time between successive collisions of a free electron with the τ lattice. Between collisions, an electron of mass m accelerates under force −eE, so it gains an extra velocity:
vd = (eEτ) / m
(This is the same v_d as before, now expressed in terms of the microscopic quantities and m τ rather than measured directly.) Substituting into I = nAev_d:
I = nAe × (eEτ/m) = (nAe²τ/m) × E
Using E = V/L again and comparing to V = IR:
R = mL / (nAe²τ)
This is the deepest form of the result: resistance depends on the collision rate (1/ ) at the τ microscopic level. A material where electrons collide more often (shorter ) has higher τ resistance — which is exactly the mechanism behind temperature dependence of resistance: heating a metal increases lattice vibration, increases collision frequency, decreases , and τ increases R.
Worked Example 2 — connecting the pieces
Two wires of the same material and same length, but wire B has twice the cross-sectional area of wire A, carry the same current I. Compare their drift velocities.
Solution: From I = nAevd, for fixed I, n, e: vd ∝ 1/A.
vd(B) / vd(A) = A(A) / A(B) = 1/2
Wire B, with double the area, has half the drift velocity of wire A for the same current — matching the fluid-flow intuition that a wider “pipe” carrying the same flow rate has slower moving “fluid.”
Quick Practice
Q1. If the number density of free electrons in a conductor were somehow doubled (same current, same cross-section), what would happen to the drift velocity?
Answer: vd would halve, since vd ∝ 1/n for fixed I and A.
Q2. Two conductors of the same material carry the same current density J. Does this mean they carry the same total current I?
Answer: Not necessarily — I = J × A, so equal current density with different cross-sectional areas gives different total currents.
Q3. A student claims that increasing the applied voltage across a wire increases the number density n of free electrons available to carry current. Is this correct?
Answer: No. n is a property
of the material (how many electrons per unit volume are “free” in the metal’s structure) and does not depend on the applied voltage. Increasing voltage increases E, which increases vd — not n.
Summary
Quantity Symbol Relation Depends on
Drift velocity vd = I/(nAe) = μE
Field, mobility
Mobility μ = vd/E Material, temperature
Current density J = I/A = nevd Local cross section
Relaxation time vd = eE /m τ Collision frequency
Resistance (microscopic)
R = mL/(nAe² ) τ n, , geometry τ
The chain field → drift velocity → current — and its inverse, current → microscopic collision behaviour → resistance — is the bridge between Ohm’s law as a circuit rule and Ohm’s law as a consequence of what electrons are actually doing inside the metal.
| Quantity | Symbol | Key relation | Depends on |
| Drift velocity | vd | vd=μE=I/nAe | Electric field, mobility |
| Mobility | μ | μ=vd/E | Material, temperature |
| Current density | J | J=I/A=nevd | Carrier density, drift velocity |
| Relaxation time | τ | vd=eEτ/m | Scattering processes |
| Resistance | R | R=mL/ne^2Aτ | Material, geometry, relaxation time |
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