IB Physics HLCurrent & CircuitsPaper 1 & 2ρ = RA / L~9 min read
Resistivity
Take two wires made of the same metal — one long and thin, one short and fat. They’ll have completely different resistances. So a wire’s resistance isn’t just about what it’s made of; its shape matters too. Resistivity is the clever idea that separates the two: it captures how “resistive” a material is by nature, once you strip the shape away. Let’s build it up.
📘 What you need to know
A wire’s resistance depends on three things: its material, its length and its cross-sectional area
Resistance is directly proportional to length — longer wire, more resistance
Resistance is inversely proportional to cross-sectional area — fatter wire, less resistance
Resistivity (ρ) measures how strongly a material opposes current, regardless of shape
The equation is ρ = RA / L, measured in ohm-metres (Ω m)
Conductors have low resistivity; insulators have very high resistivity
What changes a wire’s resistance?
Three things decide how much a particular wire resists:
The material — copper resists less than steel, for example
The length — how long the wire is
The cross-sectional area — how thick the wire is
The material we’ll handle in a moment with resistivity. The other two follow a clean pattern:
How shape affects resistanceResistance is directly proportional to length (↑ length → ↑ resistance)Resistance is inversely proportional to area (↑ area → ↓ resistance)
Longer wire = more resistance. Fatter wire = less resistance. Think of it like water through a pipe.
The water-pipe picture nails this. A long pipe is harder to push water through than a short one — that’s the length effect. A wide pipe lets far more water flow than a narrow one — that’s the area effect. Wire and charge behave exactly the same way. Long and thin resists; short and fat lets charge race through.
Bringing in resistivity
Length and area explain the shape — but what about the material? That’s the job of resistivity, given the symbol ρ (the Greek letter “rho”). It’s a single number that tells you how much a material naturally opposes current, no matter what shape you cut it into.
Resistivity — definition
The resistance of a material of unit length and unit cross-sectional area
Put the material, length and area together and you get the resistivity equation:
Resistivity equationρ = RA / L
Where:
ρ = resistivity, in ohm-metres (Ω m)
R = resistance, in ohms (Ω)
A = cross-sectional area, in square metres (m2)
L = length, in metres (m)
You’ll often want to find the resistance instead, so it’s worth rearranging:
Rearranged for resistanceR = ρL / A
Look at that and the two shape rules jump straight out: R grows with length L (on top) and shrinks with area A (on the bottom). Exactly what we said.
WE 1
A copper wire has a length of 2.0 m and a cross-sectional area of 5.0 × 10−7 m2. The resistivity of copper is 1.7 × 10−8 Ω m. Calculate its resistance.
Step 1 — use R = ρL / AR = ρL / AStep 2 — substituteR = (1.7 × 10⁻⁸ × 2.0) ÷ (5.0 × 10⁻⁷)R = 0.068 ΩTiny — and that’s the point. Copper’s low resistivity makes it a brilliant wire.
WE 2
A wire of length 3.0 m and cross-sectional area 2.0 × 10−8 m2 has a resistance of 15 Ω. Calculate the resistivity of the material, and say whether it is more like a conductor or an insulator.
Step 1 — use ρ = RA / Lρ = RA / LStep 2 — substituteρ = (15 × 2.0 × 10⁻⁸) ÷ 3.0ρ = 1.0 × 10⁻⁷ Ω mThat’s a small resistivity, close to a metal — so it behaves like a conductor.
The cross-section is a circle
One thing to watch: a wire’s cross-section is a circle, so its area is A = πr2, where r is the radius. If a question gives you a diameterd instead, halve it first:
Cross-sectional area of a wireA = πr2 = πd2 / 4
Here’s a neat consequence. Because area depends on the radius squared, doubling the thickness has a big effect. Double the diameter and the area goes up four times — so the resistance drops to a quarter.
Radius is squared in the area, so a wire twice as wide has four times the room for charge — and a quarter of the resistance.
WE 3
A copper wire of length 1.2 m has a diameter of 0.50 mm. Taking the resistivity of copper as 1.7 × 10−8 Ω m, calculate its resistance.
Step 1 — find the area (circle!)
convert d: 0.50 mm = 0.50 × 10⁻³ m
A = π(d/2)² = π(0.25 × 10⁻³)² = 1.96 × 10⁻⁷ m²Step 2 — use R = ρL / AR = (1.7 × 10⁻⁸ × 1.2) ÷ (1.96 × 10⁻⁷)R = 0.10 Ω (2 s.f.)The trap is the area. Halve the diameter, then square it — miss that and every answer is wrong.
Resistivity values: conductors vs insulators
Resistivity is a property of the material, so each substance has its own fixed value. You never need to memorise them — they’re given in questions — but it’s worth seeing the huge gulf between conductors and insulators:
Type
Material
Resistivity / Ω m
Conductor
Copper
1.7 × 10−8
Conductor
Aluminium
2.6 × 10−8
Insulator
Glass
≈ 1012
Insulator
Rubber
≈ 1013
Look at those powers of ten. A conductor sits around 10−8; an insulator is up near 1012 or more. That’s a difference of about twenty orders of magnitude — which is exactly why copper carries current beautifully and rubber blocks it almost completely. Low resistivity makes great wires; sky-high resistivity makes great safety coatings.
Material resistivity ρ
× length ÷ area
Resistance R = ρL / A
💡 Top tips
R = ρL / A — length on top (R grows with it), area on the bottom (R shrinks with it).
Area is a circle:A = πr2. Given a diameter? Halve it before squaring.
Double the diameter → quarter the resistance (area goes up by 4).
Resistivity is fixed for a material — it’s given in the question, never memorised.
Keep everything in SI units: metres for length, m2 for area — convert mm and mm2 first.
⚠ Common mistakes
Using the diameter as the radius — always halve the diameter first
Forgetting to square the radius in A = πr2
Flipping the equation — it’s R = ρL/A, not ρA/L when solving for R
Leaving length in mm or area in mm2 — convert to metres and m2
Confusing resistivity (ρ, a material property) with resistance (R, depends on shape too)
Quick recap: A wire’s resistance depends on material, length and area. Resistance rises with length and falls with area: R = ρL/A. Resistivity ρ (in Ω m) captures the material alone. The cross-section is a circle, A = πr2. Conductors have tiny resistivity; insulators enormous.
So far every component we’ve met has had a fixed, steady resistance. But not everything plays fair — a filament lamp’s resistance changes as it heats up, and a diode only lets current one way. To spot these behaviours we plot current against voltage. That’s exactly where we head next, in I–V Characteristics & Ohm’s Law, where the shape of a graph tells you everything.
Want resistivity calculations to feel easy?
Book a free meeting and we’ll practise the area trick and the equation together, step by step.
