Treat the electrons like billiard balls
In 1900, just three years after the electron was discovered, Paul Drude asked a simple question: what if the electron gas in a metal obeys the same plain mechanics as a swarm of billiard balls rolling on a table? No quantum strangeness, no fancy forces — just balls that get pushed, speed up, and occasionally smack into something and bounce off in a random direction. That picture is the Drude model, and it was the first real theory of how metals conduct.
Take the electron gas from the last guide. With no battery connected, the electrons fly around in every direction at high speed, like gas molecules in a warm room. But they go nowhere on average: for every electron heading left, another heads right, so there is no net flow and no current. The metal just sits there, full of frantic but balanced motion.
Push, then crash: the rhythm of conduction
Now connect a battery. It sets up an electric field inside the wire, a steady push on every electron. Each electron starts to accelerate, gaining a little extra speed in the push direction. But it does not get to keep speeding up forever. After a short while it collides — with a vibrating atom, an impurity, a flaw in the crystal — and the crash scatters it off in a random direction, wiping out the extra speed it had gained.
Then the push begins again, the electron accelerates again, and crashes again. Over and over: speed up, crash, speed up, crash. The average time an electron coasts freely between two collisions has a name — the relaxation time. It is usually unimaginably short, around a hundred-trillionth of a second, but it is the single most important number in the whole model.
A slow river inside a fast storm
Here is the crucial subtlety. The electrons' random thermal motion is blisteringly fast — far faster than anything the battery does. The push only nudges things ever so slightly. So on top of all that wild zig-zagging, the whole crowd gains a tiny, steady lean in the push direction. That gentle average shift is called the drift velocity, and it is astonishingly slow — often less than a millimetre per second.
If electrons drift so slowly, why does a lamp light the instant you flip the switch? Because the field that pushes them spreads through the wire almost at the speed of light, so every electron everywhere starts drifting at once. Think of a long pipe already full of water: push at one end and water spurts from the far end immediately, even though each water molecule itself barely creeps along.
Out comes Ohm's law
Now turn the crank. A bigger push gives a bigger drift, and so a bigger current — and they grow in exact proportion. Double the push, double the current. That straight-line relationship is Ohm's law, the rule every student of electricity meets first, and the Drude model derives it from scratch instead of just asserting it.
current ∝ push (voltage) → Ohm's law conductivity ∝ (electrons per volume) × (charge²) × (relaxation time) ÷ mass longer relaxation time → higher conductivity → lower resistance
The formula tells a clear story about electrical conductivity. A metal conducts better when it has more electrons to carry charge, and when each electron can coast longer between crashes. Resistance is simply the price of all that crashing. Heat a wire and the atoms jiggle harder, so electrons crash more often, the relaxation time shrinks, and resistance climbs — which is exactly what real metals do when they warm up.
Mobility: how nimble is each electron?
There is one more handy number worth meeting. Mobility answers the question: for a given amount of push, how fast does an electron drift? A high-mobility electron is nimble — a small nudge sends it gliding briskly. A low-mobility electron is sluggish, hemmed in by frequent collisions. Mobility bundles together the relaxation time and the electron's mass into one easy figure of how responsive the charge carriers are.
Drude's billiard-ball picture is a triumph of imagination, and it gets Ohm's law and the rough size of conductivity right. But it has a hidden flaw, one that took quantum mechanics to expose. Drude assumed all the electrons share the heat the way air molecules do — and that assumption gives badly wrong predictions for how metals store warmth. Fixing it is the job of the next guide, where the electron gas finally has to obey the rules of the quantum world.