Topic B.5 — Current & CircuitsPaper 1 & 2ρ = RA / L~6 min read
Electrical Resistivity
Copper has a low resistance and makes great wiring — but why? A wire’s resistance isn’t just about what it’s made from. It also depends on how long it is and how thick it is. Let’s untangle all three ingredients.
📘 What You Need to Know
A component’s resistance depends on three things: the material, its length, and its cross-sectional area
Resistance is directly proportional to length — a longer wire has more resistance
Resistance is inversely proportional to cross-sectional area — a thicker wire has less resistance
Resistivity (ρ) strips away the length and area, leaving just the property of the material itself
ρ = RA / L, measured in ohm-metres (Ω m)
Good conductors have low resistivity; good insulators have high resistivity
Why length and thickness matter
Think of a wire like a corridor that electrons have to shuffle through. Make the corridor longer, and there are simply more ions to bump into along the way — so resistance goes up. Now make the corridor wider instead, and suddenly there’s more room for electrons to spread out and pass through side by side — so resistance goes down. Length and width pull in opposite directions.
Doubling a wire’s length doubles its resistance; doubling its cross-sectional area (not shown to scale) halves its resistance — length and width act in opposite directions.
Since a wire’s cross-section is a circle, its area is πr², where r is the radius. That squared term matters: if you double the diameter of a wire, you quadruple its area — and resistance drops to a quarter, not a half.
Introducing resistivity
Length and area tell us about the shape of a wire, but they don’t tell us anything about what it’s made of. That’s where resistivity comes in — it’s a fixed property of a material, completely independent of the size or shape of the sample you’re holding.
Resistivityρ = RA / L
where ρ is resistivity in ohm-metres (Ω m), R is resistance in ohms (Ω), A is cross-sectional area in square metres (m²), and L is length in metres (m). Two wires of the exact same copper will always share the same resistivity — even if one is short and fat and the other is long and thin, and therefore have completely different resistances.
🔧 A Few Typical Resistivity Values
Silver — around 1.6 × 10⁻⁸ Ω m (an excellent conductor)
Copper — around 1.7 × 10⁻⁸ Ω m (why it’s the go-to for wiring)
Aluminium — around 2.6 × 10⁻⁸ Ω m (lighter than copper, slightly more resistive)
Glass — around 10¹² Ω m (a powerful insulator)
Rubber — even higher still, which is exactly why it coats our wires
Quick recap:R ∝ L, R ∝ 1/A, and ρ = RA/L — resistivity is the property of the material alone, stripped of size and shape.
WE 1
A sample of wire has a resistance of 15 Ω, a cross-sectional area of 3.0 × 10⁻⁶ m², and a length of 2.5 m. What is the resistivity of the material?
Write the formula:ρ = RA / LSubstitute the values:ρ = (15 × 3.0 × 10⁻⁶) ÷ 2.5ρ = 1.8 × 10⁻⁵ Ω m
WE 2
Wire A is copper (ρ = 1.7 × 10⁻⁸ Ω m), 4.0 m long, with a diameter of 1.0 mm. Wire B is aluminium (ρ = 2.6 × 10⁻⁸ Ω m), 6.0 m long, with a diameter of 1.5 mm. Which wire has the lower resistance?
Find each cross-sectional area:A = πr² → A_A = π(0.5×10⁻³)² = 7.85 × 10⁻⁷ m²A_B = π(0.75×10⁻³)² = 1.77 × 10⁻⁶ m²Find each resistance:R_A = (1.7×10⁻⁸ × 4.0) ÷ 7.85×10⁻⁷ ≈ 0.0866 ΩR_B = (2.6×10⁻⁸ × 6.0) ÷ 1.77×10⁻⁶ ≈ 0.0883 ΩWire A (copper) has the lower resistance, even though it’s shorter and thinnerCopper’s lower resistivity wins out over aluminium’s larger area.
💡 Top Tips
Always convert millimetres to metres before squaring them for area — a small slip here throws your answer off by a factor of a million
Remember diameter and radius aren’t the same thing — halve the diameter before you square it
You’ll always be given resistivity values in an exam, so focus on knowing the formula and using it confidently rather than memorising numbers
⚠ Common Mistakes
Confusing resistance and resistivity — resistance depends on shape and size, resistivity doesn’t
Forgetting that area is proportional to radius squared, so doubling the diameter quarters the resistance, not halves it
Assuming the thicker or shorter wire always wins — as WE2 shows, the material’s resistivity can outweigh a size advantage
Up next: I-V Characteristics — now we can predict a component’s resistance, let’s see how that resistance can actually change as current and voltage change.
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