Why cables warm up

1. Every connecting lead and every mains cable is made of metal, and every metal has some resistance, so charge passing along a cable always leaves a little energy behind in the cable itself.
2. That energy is transferred by the drifting electrons colliding with the vibrating ions of the metal, and it ends up in the thermal store of the cable and then of the surroundings.
3. Energy dissipated in a cable is wasted twice over. It never reaches the appliance, and it raises the temperature of the cable, which can damage the cable's covering.
4. Cable design therefore aims to make the energy dissipated in the cable as small as is reasonably possible, and there are four separate levers available: the metal chosen, the thickness of the wire, the length of the wire, and the size of the current it carries.
5. The first three levers all work by lowering the resistance of the wire. The fourth works by reducing how much charge is pushed through that resistance every second.
6. None of the four levers can remove the heating completely. A real cable always dissipates some energy, and the target is to make that amount small enough to ignore.

Choosing the metal

1. A good electrical conductor holds a very large number of delocalised electrons that are free to move, so a small potential difference is enough to drive a large current through it.
2. Copper is the standard choice for connecting leads and mains cables. It has a very low resistance for its size, it can be drawn into thin flexible strands, and it does not corrode quickly.
3. Silver is a slightly better conductor than copper, but it is far too expensive to use for ordinary wiring, so it appears only in specialised contacts.
4. Iron and steel have a much higher resistance than copper of the same length and thickness, so an iron lead dissipates far more energy for the same current in it.
5. Nichrome has a deliberately high resistance and is used for heating elements, which makes it precisely the wrong metal for a connecting lead.
6. Aluminium is a poorer conductor than copper but much less dense, so it is chosen where the mass of the cable matters more than the last part of its resistance.
7. Changing the metal lowers the resistance without changing the size or shape of the cable at all, which is why it is usually the first design decision made.

Thickness and length

1. The cross-sectional area of a wire is the area of the circular face that would be exposed if the wire were cut straight across.
2. A wire of larger cross-sectional area has a lower resistance, because the drifting electrons have a wider channel and more routes past the lattice ions, so a smaller fraction of them is obstructed.
3. For the same metal and the same length, resistance is inversely proportional to cross-sectional area, so doubling the area halves the resistance.
4. A longer wire has a higher resistance, because each electron must pass a greater number of vibrating lattice ions on its way through and so makes more collisions.
5. For the same metal and the same thickness, resistance is directly proportional to length, so a $4.0m$ lead has twice the resistance of a $2.0m$ lead cut from the same reel.
6. Halving the resistance of a cable halves the energy dissipated in it each second, provided the current through it is unchanged.
7. This is why the flex on a kettle is short and made of thick multi-strand copper, while the hair-thin tracks inside a piece of electronics are acceptable only because the currents in them are tiny.
8. Thickness and length cannot be pushed as far as a designer might like. Thick cable costs more, uses more copper and bends less easily, and a cable must be long enough to reach the socket.

Keeping the current small

1. The energy dissipated in a cable rises steeply as the current in it rises, because a larger current means more charge passing every second and therefore more collisions with the lattice every second.
2. Doubling the current in a given cable raises the rate of heating in it by considerably more than double, which makes current the most powerful of the four levers.
3. A cable feeding several appliances at once carries the sum of their currents, so plugging a fan heater and a kettle into the same lead can make that lead noticeably warm.
4. Thin extension leads sold for lamps and radios are built for small currents only. Running a $13A$ appliance through such a lead forces far more charge through its thin wires each second than they were designed for, and the lead gets hot.
5. Cables are therefore marked with a current rating in amperes, and that rating must be at least as large as the current the appliance draws through them.
6. Two leads of the same thickness behave quite differently in use if one carries $0.50A$ to a phone charger and the other carries $10A$ to an electric heater. The heater lead needs the thicker copper.

Insulation and cable ratings

1. An electrical insulator has almost no free charges to move, so charge cannot flow through it. PVC, rubber and dry air are good insulators.
2. A cable pairs the two kinds of material deliberately: a conductor core of copper to carry the charge with as little dissipation as possible, wrapped in an insulator of coloured plastic to keep that charge inside the cable.
3. The insulation does nothing to reduce the heating. The energy is dissipated in the copper, and the plastic simply surrounds it and slows the escape of that energy to the air.
4. Insulation is damaged by heat. A cable that runs too hot softens, then melts its plastic, after which the conductors inside can touch each other or be touched by a person.
5. An undersized cable is therefore a fire risk. The energy dissipated in its thin conductors raises the temperature of the insulation and of anything packed around the cable.
6. An undersized cable has a second and quieter drawback. A larger share of the supply potential difference is used up along the cable, so less is left across the appliance and the appliance runs below its intended output.
7. A well-designed cable is a compromise between four demands: low dissipation, low cost, enough flexibility to be used, and enough length to reach.