What Resistance Depends On - Physics for Pilot Assessments

Resistance is not an arbitrary number attached to a component: it follows from what the component is made of and what shape it has been made into. By the end of this lesson you should be able to say how a conductor's length and thickness each affect its resistance and why, predict what doubling either of them does, explain why two wires of the same size can still have different resistances, and say which way a metal's resistance moves when it gets warmer.

Length: more wire to get through

Take a length of thin wire and measure its resistance, then measure a piece of the same wire twice as long, and the second reading is twice the first. The mechanism is the one from the last lesson. Resistance arises because the drifting electrons keep colliding with the atoms of the lattice they are moving through, and each collision costs the flow a little energy; a longer wire simply contains more lattice to get through, so there are proportionally more collisions and proportionally more opposition. Three times the length gives three times the resistance, half the length gives half, and the relationship is exact rather than approximate, provided the wire is uniform along its length.

That is worth holding on to because it explains something practical: the resistance of the connecting wires in a circuit is usually ignored in these questions, and it can be ignored because they are short and thick. Make them long enough and it stops being negligible, which is why a very long extension lead delivers less to a tool at the far end than a short one does.

Thickness: more room to get through

Now compare two wires of the same length and the same material, one twice as thick as the other in the sense of having twice the cross-sectional area — the area of the circle you would see if you cut it straight across. The thick one has half the resistance of the thin one. The reasoning that trips people up here is the tempting one: a thicker wire contains more metal, so surely it puts more in the way? It does contain more metal, but the extra metal is alongside the path rather than along it, and it provides extra room for charge to pass. Think of two lanes of a road rather than a longer road. The current does not have to fight its way through the extra metal; it gets to spread out through it, and spreading out means less opposition.

So resistance is proportional to the length and inversely proportional to the cross-sectional area — meaning that whatever you multiply the area by, you divide the resistance by. Double the area and you halve the resistance; a quarter of the area gives four times the resistance. One caution for round wires, because it is a favourite: doubling a wire's diameter does not double its area, it quadruples it, since the area of a circle grows with the square of its diameter. So twice the diameter gives a quarter of the resistance, not half.

Same material, different shapes. Double the length and you double the resistance; double the cross-sectional area and you halve it. The upper conductor is twice as long and half as thick, so both changes push its resistance up, while the lower one is short and fat — which is exactly what a good connecting wire is.

Material: resistivity

Length and area between them still do not settle the question, because a steel wire and a copper wire of identical size do not have the same resistance. What is left over is a property of the material itself, called its resistivity: how much resistance a standard piece of it would have. Written with the Greek letter rho, ρ, it is quoted in ohm metres, and the smaller it is the better the material conducts. Silver has the lowest resistivity of any metal, copper is very slightly worse and enormously cheaper, which is why wiring is made of copper; aluminium is worse than copper but much lighter, which is why overhead power lines and a good deal of aircraft wiring use it; steel is several times worse than copper; and nichrome, an alloy of nickel and chromium, is worse still, which is precisely why heating elements are made of it. Put all three factors together and the resistance of a uniform conductor is fully determined.

Resistance of a uniform conductor

R = \frac{\rho L}{A}

$R$: the resistance, in ohms (Ω)
$\rho$: the resistivity of the material, in ohm metres (Ω·m) — a property of the substance, not the shape
$L$: the length of the conductor, in metres (m)
$A$: the cross-sectional area, in square metres (m²)

One note on notation, since it would otherwise be a trap: the same Greek letter rho was used for density earlier in the course, and here it means resistivity. The two are unrelated quantities and nothing distinguishes them but context. In practice you will not be asked to calculate with resistivity values — no battery expects you to remember that copper is about 1.7 hundred-millionths of an ohm metre — but you are expected to know that the material matters as a third independent factor, and to read the formula for what it says about proportions.

Temperature

The last factor is the one that changes while a circuit is running. Warm a metal and its atoms vibrate more violently about their fixed positions in the lattice; a more violently vibrating atom is a bigger obstacle to an electron trying to get past, so there are more collisions and the resistance rises. In a metal, then, hotter means more resistive — and the effect is large enough to matter in a lamp, whose filament runs white hot and has several times the resistance when lit that it has when cold. That is why the graph of a filament lamp bends over rather than staying straight, which was the one non-ohmic case mentioned in the previous lesson, and it is why a lamp draws a brief surge of current at the instant you switch it on, before the filament has had time to heat up.

Example 1 — changing the shape

A wire has a resistance of 6 Ω. What is the resistance of a wire of the same material that is twice as long? And of one that is the original length but with twice the cross-sectional area?

Resistance is proportional to length, so doubling the length multiplies the resistance by 2: $R = 6 \times 2 = 12\ \Omega$
Resistance is inversely proportional to area, so doubling the area divides the resistance by 2: $R = \frac{6}{2} = 3\ \Omega$
Notice the two changes pull in opposite directions, so a wire that is both twice as long and twice as thick has the same 6 Ω as the original: $R = \frac{6 \times 2}{2} = 6$
That is exactly what $R = \frac{\rho L}{A}$ predicts, with L and A each doubled and the fraction unchanged.

Example 2 — working backwards to a shape

A 2-metre length of a particular wire has a resistance of 4 Ω. What length of the same wire would have a resistance of 10 Ω? And what would happen to that 4 Ω if the wire were replaced by one of half the cross-sectional area?

First find the resistance per metre: 4 Ω over 2 metres is $\frac{4}{2} = 2$ Ω for every metre.
To reach 10 Ω you need $\frac{10}{2} = 5$ metres.
Check it forwards: 5 metres at 2 Ω per metre is 10 Ω, as required.
For the second part, halving the area doubles the resistance, because resistance is inversely proportional to area: $R = 4 \times 2 = 8\ \Omega$
And if that thinner wire were also cut to 1 metre, the length would be halved as well: $R = \frac{8}{2} = 4\ \Omega$ That is back where it started.

Example 3 — a filament, cold and hot

A filament lamp draws 2 A from a 12 V supply once it is running at full brightness. Measured cold, its filament has a resistance of 1.5 Ω. What is the filament's resistance when lit, and what current flows at the instant the lamp is switched on?

When lit, Ohm's law gives the resistance from the working figures: $\text{resistance} = \frac{\text{voltage}}{\text{current}} = \frac{12}{2} = 6\ \Omega$
At the instant of switch-on the filament is still cold, so the supply sees only 1.5 Ω. $\text{Current} = \frac{\text{voltage}}{\text{resistance}} = \frac{12}{1.5}$ , and multiplying both numbers by 10 to clear the decimal gives $\frac{120}{15} = 8$ amperes.
So the lamp briefly draws four times its running current — this is the switch-on surge described above, and it lasts only as long as the filament takes to heat up.
Read the two resistances together and the temperature effect has been measured rather than merely asserted: heating the filament from cold to white hot multiplied its resistance by four, from 1.5 Ω to 6 Ω, with nothing about its length, thickness or material changed at all.

Why it matters in aviation

These geometry rules are why aircraft wiring is a weight problem as much as an electrical one. Resistance grows with length and falls with cross-sectional area, and an airframe imposes long runs, so holding the resistance of a run down means either a shorter route or a fatter conductor, and the fatter conductor is bought with mass. It is also part of why aluminium is used for wiring at all: its resistivity is worse than copper's, but it is so much less dense that for a given resistance the aluminium conductor can come out lighter even though it has to be thicker.

The temperature dependence is exploited rather than merely tolerated. Because a metal's resistance rises with temperature in a predictable way, a small coil of wire becomes a thermometer: pass a known current through it, measure the voltage across it, and the resistance you calculate tells you how hot the coil is. Resistance thermometers of exactly that kind are how quantities such as outside air temperature and oil temperature are measured and displayed.

What the test asks

The distractor that catches people is the thick-wire one: an option claiming that a thicker wire has more resistance because there is more metal in it. There is more metal, and it is alongside the current rather than in front of it, so it lowers the resistance rather than raising it — which is also why a thin wire is the one that overheats. Expect an item pairing two changes at once, where the safe method is to apply them one at a time in either order rather than guessing at the combination. Expect the diameter trap, where doubling a round wire's diameter is offered as halving its resistance instead of quartering it. And expect one item on temperature, where the wrong option has a metal conducting better as it warms up.