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4.4 — Current, Resistance and What Electrons Actually Do
Flick a light switch and the bulb lights instantly, even if the wire runs thirty metres. Almost everyone's mental picture is electrons racing down the wire at something near the speed of light.
They are not. In a household wire carrying a normal current, an individual electron drifts along at roughly 0.1 millimetres per second — slower than a snail, slower than the hour hand on a clock. It would take an electron over three hours to travel one metre. This chapter derives that number, explains why the light still comes on instantly, and builds Ohm's law from the motion of individual electrons rather than quoting it as a rule.
Current
Current is the rate at which charge passes a point:
I = \frac{dQ}{dt}
measured in coulombs per second, called the ampere (A) after André-Marie Ampère. Since 2019 the ampere is defined by fixing the elementary charge, so one ampere is exactly 1/1.602176634\times10^{-19} = 6.24\times10^{18} elementary charges per second.
Conventional current flows from positive to negative. In a metal wire the actual carriers are electrons, which are negative and therefore move the other way. This is Franklin's guess from Chapter 4.1 and everybody has lived with it since. It genuinely does not matter for any calculation: a positive charge moving right and a negative charge moving left transport charge in the same direction, so they are the same current.
It does matter in one place. In a semiconductor, current is carried by both electrons and by holes — vacancies in the electron sea that behave exactly like positive particles — and the two move in opposite directions. Volume III, Chapter 2 builds transistors out of exactly that fact.
Current is not used up. A common misconception has the bulb "consuming" current so that less returns to the battery. It does not. Charge is conserved (Chapter 4.1), the same current flows all the way round the loop, and what the bulb consumes is energy, not charge. The electrons come back; they come back with less potential energy.
Drift velocity, and why it is so slow
A metal contains free electrons — roughly one per atom, released because the outermost electron of a metal atom is barely held. With no applied field they are already moving, and moving fast: they are a gas at room temperature, so from Chapter 3.2 you might expect \sqrt{3k_BT/m}, which for an electron is about 1.2\times10^5 m/s. (The real number is around 1.6\times10^6 m/s for reasons that need quantum mechanics — the electrons obey the Pauli principle of Chapter 7.7 and are forced into much higher energies than thermal motion alone would give. The classical estimate is wrong in detail but right that the speeds are enormous.)
That motion is completely random, so it transports no net charge. Now apply a field. Each electron feels a force -e\vec{E} and accelerates — but only until it collides with something, at which point its direction is randomised again. The result is a tiny average velocity superimposed on the huge random motion, like a very slight breeze in a room full of frantically bouncing balls.
That average is the drift velocity v_d, and it is what carries the current.
Deriving the current from the drift
Take a wire of cross-sectional area A containing n free electrons per cubic metre, each of charge e, drifting at v_d.
In time \Delta t, every electron moves v_d\Delta t along the wire. So every electron within that distance of a chosen cross-section will pass through it. The volume containing them is A v_d\Delta t, and the charge in it is:
\Delta Q = n\,(Av_d\Delta t)\,e
Divide by \Delta t:
\boxed{I = nAv_de}
Read aloud: I equals n A v-d e — current equals the number density of carriers, times the area, times the drift speed, times the charge on each.
Worked example: how slow, exactly
A copper wire of 1.0 mm² cross-section carries 5.0 A. Find v_d.
First, n for copper. Copper has density 8960 kg/m³ and molar mass 63.5 g/mol, and contributes one free electron per atom:
n = \frac{8960\ \text{kg/m}^3}{0.0635\ \text{kg/mol}}\times6.022\times10^{23}\ \text{mol}^{-1} = (1.411\times10^{5})(6.022\times10^{23})
n = 8.50\times10^{28}\ \text{electrons per m}^3
Now solve for v_d with A = 1.0\times10^{-6} m²:
v_d = \frac{I}{nAe} = \frac{5.0}{(8.50\times10^{28})(1.0\times10^{-6})(1.602\times10^{-19})}
v_d = \frac{5.0}{1.362\times10^{4}} = 3.7\times10^{-4}\ \text{m/s}
0.37 mm per second. An electron takes 45 minutes to cross a one-metre wire.
So why does the light come on instantly?
Because the signal travels at nearly the speed of light, and the signal is not an electron.
Three ways to see it, all correct and all worth having.
The water-pipe picture. A pipe already full of water delivers water at the far end the instant you open the tap at the near end, even though any individual molecule takes a long time to make the trip. The wire is already full of electrons everywhere, including inside the bulb filament.
The field picture. Closing the switch establishes an electric field in the wire, and that field propagates at close to c — it is an electromagnetic disturbance, and Chapter 4.7 shows those travel at light speed. Within a few nanoseconds every electron along the whole wire feels the field and starts drifting simultaneously. Nothing had to travel from the switch to the bulb except the news.
The energy picture, which is the deepest one. The energy does not travel inside the wire at all. It travels in the electromagnetic field in the space around the wire, and the wire's job is to guide it. Chapter 4.7 makes this precise with the Poynting vector. It sounds outlandish and it is the correct account: the copper is a waveguide, not a pipe.
Why the huge n matters
Look again at I = nAv_de. The drift velocity is minuscule and the current is large, and the reconciliation is n = 8.5\times10^{28}. There are about a hundred billion billion free electrons in every cubic millimetre of copper. Each one crawls, and there are so many of them that the total flow is enormous.
This also explains why semiconductors behave so differently. Pure silicon has n \approx 10^{16} per m³, twelve orders of magnitude fewer, which is why it barely conducts — and why adding one impurity atom in a million (doping) changes its conductivity by a factor of a million. Volume III, Chapter 2 builds on this.
Ohm's law, derived
Georg Ohm published in 1827 that current is proportional to voltage:
V = IR
This is an empirical observation about certain materials, not a law of nature, and plenty of things disobey it. A diode does not. A filament bulb does not, once it heats up. But metals at fixed temperature obey it beautifully, and here is why, from the electron picture.
Step 1: the electron accelerates between collisions. Force = eE, so by Newton's second law:
a = \frac{eE}{m_e}
Step 2: it collides after an average time \tau. This is the mean free time between collisions, the electronic version of the mean free path from Chapter 3.2. A collision randomises the electron's direction completely, so after each one the drift starts again from zero on average.
Step 3: average the velocity gained. Starting from zero and accelerating for time \tau gives a final velocity a\tau, and the average over that interval is half of it. Averaging properly over the distribution of times between collisions gives:
v_d = \frac{eE\tau}{m_e}
This is the crucial result: the drift velocity is proportional to the field, not the acceleration. Left alone, a charge in a field would accelerate forever. Collisions convert that runaway into a steady drift, exactly the way air resistance turns a falling object's acceleration into a terminal velocity (Chapter 1.11). Resistance is a terminal-velocity phenomenon.
Step 4: put it into the current formula.
I = nAv_de = nAe\cdot\frac{eE\tau}{m_e} = \frac{ne^2\tau A}{m_e}E
Step 5: convert to voltage. For a wire of length L with a uniform field, E = V/L from Chapter 4.3:
I = \frac{ne^2\tau A}{m_e}\cdot\frac{V}{L} = \frac{ne^2\tau}{m_e}\cdot\frac{A}{L}V
Rearranged:
V = I\cdot\frac{m_e}{ne^2\tau}\cdot\frac{L}{A}
Compare with V = IR and read off:
\boxed{R = \rho\frac{L}{A}, \qquad \rho = \frac{m_e}{ne^2\tau}}
Ohm's law is derived, and its constant is explained. The symbol \rho (rho) is the resistivity, a property of the material alone, measured in ohm-metres. The resistance R of a particular piece is the resistivity times its length over its area — long and thin means high resistance, short and fat means low, exactly as intuition says and now with a reason.
And the microscopic formula for \rho says something specific: resistivity is high when there are few carriers (n small) or when they collide often (\tau small). Everything about a material's conductivity comes down to those two numbers.
Some values at 20 °C, in Ω·m:
| Material | \rho |
|---|---|
| Silver | 1.59\times10^{-8} |
| Copper | 1.68\times10^{-8} |
| Aluminium | 2.65\times10^{-8} |
| Iron | 9.71\times10^{-8} |
| Nichrome | 1.1\times10^{-6} |
| Silicon (pure) | 2.3\times10^{3} |
| Glass | 10^{11} to 10^{15} |
The range from copper to glass is twenty-three orders of magnitude, which is the largest span of any physical property of ordinary materials. Nothing else in physics varies by 10^{23} between two things you can hold.
Silver beats copper by 5 %, and copper is used everywhere anyway because it costs a hundredth as much. Aluminium is worse per metre but far lighter, which is why overhead power lines are aluminium — for a line held up by pylons, resistance per kilogram matters more than resistance per metre.
What the electrons collide with
The obvious guess is that electrons bounce off atoms. That guess gives a resistivity about a hundred times too high, and it fails a sharper test: it cannot explain why a perfect crystal has essentially no resistance at all.
The quantum answer, which needs Chapter 7 to do properly, is that an electron in a perfectly periodic lattice does not scatter at all — it propagates as a wave through the repeating structure exactly as light propagates through glass. Resistance comes entirely from departures from perfect periodicity, and there are three:
1. Thermal vibration. At any temperature above absolute zero the atoms vibrate about their sites, so the lattice is not quite periodic. This is the dominant source in a pure metal at room temperature.
2. Impurities. A foreign atom breaks the pattern.
3. Defects. Missing atoms, dislocations, grain boundaries.
Sources 2 and 3 do not care about temperature; source 1 grows with it. That immediately predicts the temperature behaviour.
Resistivity and temperature
Hotter atoms vibrate over a larger amplitude, so they present a bigger target and \tau falls, so \rho rises. Over a moderate range the rise is linear:
\rho = \rho_0\left[1 + \alpha(T-T_0)\right]
where \alpha is the temperature coefficient of resistivity, about 0.0039 per K for copper. So copper's resistance rises about 0.4 % per degree.
Worked example: the light bulb. A 100 W filament bulb runs at 230 V, so its hot resistance is:
R_{\text{hot}} = \frac{V^2}{P} = \frac{230^2}{100} = 529\ \Omega
Measure the same bulb cold with a meter and you find about 40 Ω. Thirteen times less.
The filament is tungsten, running at about 2800 K. With \alpha_{\text{W}} \approx 0.0045:
\frac{R_{\text{hot}}}{R_{\text{cold}}} = 1 + 0.0045(2800-293) = 1 + 11.3 = 12.3
which matches the measurement.
This explains why bulbs blow when you switch them on and almost never in the middle of the night. At the instant of switching, the cold filament draws 230/40 = 5.75 A instead of its normal 0.43 A — a thirteenfold surge. That surge heats the thinnest point of the filament fastest, and the thinnest point is wherever the tungsten has slowly evaporated most over the bulb's life. The switch-on surge is what finally opens it.
And it explains why a filament bulb is self-regulating. Push more current through and it gets hotter, which raises its resistance, which limits the current. That negative feedback is why a bulb settles at a stable temperature instead of running away.
Semiconductors do the opposite. Heating silicon shakes more electrons loose from their bonds, so n rises steeply — exponentially — and this swamps the fall in \tau. Silicon's resistance drops as it warms. That has a dangerous consequence called thermal runaway: a hot spot conducts more, so it carries more current, so it gets hotter, and power semiconductors need careful thermal design to stop the cycle. It also has a useful one: a thermistor is a deliberately temperature-sensitive resistor, and it is what measures the temperature in your oven, your car and your 3D printer.
Superconductivity
Cool some materials below a critical temperature and the resistance does not merely get small — it becomes exactly zero, as far as anyone can measure. A current started in a superconducting loop has been observed to persist for years without measurable decay.
Heike Kamerlingh Onnes found it in mercury at 4.2 K in 1911, three years after he became the first person to liquefy helium. He expected resistance to fall smoothly towards zero; instead it fell off a cliff at a definite temperature.
Zero is not "very small". A normal metal cooled towards absolute zero approaches a residual resistivity set by its impurities and never gets below it. A superconductor goes to zero exactly, which means it is not a matter of fewer collisions — it is a different state of matter. The explanation, given by Bardeen, Cooper and Schrieffer in 1957, is that below the critical temperature electrons pair up (through a subtle interaction with the lattice) and the pairs behave as bosons, which can all occupy one quantum state and move as a single coherent object that has nothing to scatter off individually. Chapter 7.7 explains what a boson is and why that matters.
Superconductors also expel magnetic fields entirely, which is called the Meissner effect and is what makes a magnet levitate over one. MRI scanners are the biggest commercial application: their magnets are superconducting coils carrying hundreds of amps in a permanently closed loop, using no power at all to maintain the field, and needing only the liquid helium that keeps them cold.
Power and heating
Charge \Delta Q falling through potential difference V releases \Delta Q\,V of energy. Divide by time:
\boxed{P = IV}
and combined with V = IR, the two forms that get used constantly:
P = I^2R = \frac{V^2}{R}
In a resistor that energy becomes heat, and the mechanism is now visible: the field accelerates each electron, the electron gains kinetic energy, and the next collision dumps that energy into the lattice as vibration — which is temperature (Chapter 3.2). This is Joule heating, and it is the same Joule whose paddle wheel appeared in Chapter 3.3.
Which formula to use depends on what is held fixed, and getting this wrong is a classic error.
When the current is fixed, use P = I^2R. More resistance means more heat. This is the case in a series circuit, and it is why a loose or corroded connection is a fire hazard: the current is set by the rest of the circuit, the bad joint adds resistance, and all the extra power appears at that one spot.
When the voltage is fixed, use P = V^2/R. More resistance means less heat. This is the case for anything plugged into a wall socket, and it is why a 2 kW heater has a lower resistance than a 100 W bulb.
Worked example: why power lines run at high voltage
A town needs 1 MW delivered over a line with total resistance 2 Ω.
At 10 kV:
I = \frac{P}{V} = \frac{10^6}{10^4} = 100\ \text{A}
P_{\text{lost}} = I^2R = (100)^2(2) = 20{,}000\ \text{W} = 2\ \%
At 400 kV:
I = \frac{10^6}{4\times10^5} = 2.5\ \text{A}
P_{\text{lost}} = (2.5)^2(2) = 12.5\ \text{W} = 0.001\ \%
A factor of 1600 less loss, because the loss goes as I^2 and raising the voltage forty-fold cuts the current forty-fold.
This is the entire reason transmission lines run at hundreds of kilovolts and the entire reason alternating current won the "war of the currents" against Edison's DC in the 1890s. Changing voltage requires a transformer, a transformer requires a changing magnetic field, and a changing field requires alternating current. Chapter 4.6 derives how a transformer works; the point here is that the economics of I^2R made it necessary.
Where this shows up in your life
A fuse is a deliberately weak link. It is a thin wire chosen so that at the rated current its I^2R heating just balances the heat it can shed, and above that it melts. A circuit breaker does the same job with a bimetallic strip (Chapter 3.1) or an electromagnet.
An electric hob, a toaster and a hair dryer all use nichrome, whose resistivity is 65 times copper's and which does not oxidise away when red hot. The whole design problem is picking a length and thickness that gives the right R for V^2/R to be the wattage you want.
Extension leads must be uncoiled when running a heater. Coiled, the cable cannot shed its I^2R heat, and the insulation can reach its melting point even at a current the cable handles easily when laid out flat.
Your phone charger is warm because every conversion stage has resistance somewhere, and every ohm carrying current makes heat. Efficiency in a power supply is largely the art of keeping I^2R small.
Strain gauges measure force through R = \rho L/A. Stretch a wire and L rises while A falls, so R rises measurably. Bond one to a beam and you have a load cell — which is what every digital scale in every shop contains.
What the next chapter fixes
Charges have been sitting still and then flowing in straight lines, and all of it has been electric. But a compass needle placed near a current-carrying wire swings, which Hans Christian Ørsted discovered by accident in front of a lecture class in 1820 and which nothing in the last four chapters predicts. Moving charge produces a completely different kind of field, one that pushes sideways rather than along, and Chapter 4.5 builds it: the magnetic force, the two laws that give the field of any current, and the machines that fall out of them.