On what factors does resistance depend?
If you test many wires, you find that the resistance of a wire depends on four things:
- Length (L): R ∝ L. Double the length, double the resistance.
- Area of cross-section (A): R ∝ 1/A. A thicker wire has less resistance. Since A = πr², doubling the radius makes A four times, so R becomes one-fourth.
- Nature of the material: copper and silver have low resistance, nichrome and manganin much higher.
- Temperature: for metals, resistance increases when they get hotter.
Putting the first two together: R = ρ L / A.
What is resistivity?
The constant ρ (rho) in R = ρL/A is the resistivity (or specific resistance) of the material. From ρ = RA/L, it equals the resistance of a piece of that material 1 m long with 1 m² cross-section.
SI unit: Ω m (ohm metre). Resistivity depends on the material and its temperature, but not on the length or thickness of the wire. Resistance depends on all of these.
Conductors, alloys and insulators
Metals have very low resistivity, about 10−8 to 10−6 Ω m: silver (about 1.6 × 10−8) and copper (about 1.7 × 10−8) are the best, then aluminium, tungsten and iron. Insulators such as glass and rubber have resistivity of 1012 to 1017 Ω m.
Alloys such as nichrome, manganin and constantan have much higher resistivity than pure metals, change little with temperature and do not oxidise (burn) easily when hot. That is why heater, toaster and iron coils use nichrome. Copper and aluminium are used for transmission and house wiring because their low resistivity wastes little energy.
Exam tips for resistivity numericals
Always change units: mm → m (1 mm = 10−3 m) and mm² → m² (1 mm² = 10−6 m²). When a wire is stretched, its volume stays the same, so if L becomes n times, A becomes 1/n times and R becomes n² times. When a wire is cut into equal parts, each part has 1/n of the resistance but the same resistivity.
Key formulas and definitions
- R = ρ L / A
- ρ = R A / L, SI unit Ω m
- A = π r² = π d² / 4
- Stretched wire (same volume): L → nL gives R → n² R
Worked examples
1. A wire of resistance 4 Ω is doubled in length (same thickness). New resistance?
R ∝ L, so R = 2 × 4 = 8 Ω.
2. Two wires of the same material and length: wire B has twice the radius of A. Compare their resistances.
A ∝ r², so B's area is 4 times. R ∝ 1/A, so R_B = R_A / 4.
3. A copper wire is 2 m long with an area of 1 mm². ρ = 1.7 × 10⁻⁸ Ω m. Find R.
A = 1 × 10⁻⁶ m². R = ρL/A = 1.7 × 10⁻⁸ × 2 / 10⁻⁶ = 0.034 Ω.
4. A 1 m wire of area 0.5 mm² has a resistance of 2 Ω. Find its resistivity.
ρ = RA/L = 2 × 0.5 × 10⁻⁶ / 1 = 1 × 10⁻⁶ Ω m (nichrome-like).
5. What length of nichrome wire (ρ = 1 × 10⁻⁶ Ω m, area 0.2 mm²) is needed for a 20 Ω coil?
L = RA/ρ = 20 × 0.2 × 10⁻⁶ / 10⁻⁶ = 4 m.
6. A 10 Ω wire is stretched to three times its length. Find the new resistance.
Volume is fixed, so L → 3L and A → A/3. R → 3 × 3 × 10 = 90 Ω.
Common mistakes
- Saying resistivity changes when a wire is cut or stretched. Only resistance changes; resistivity is a property of the material.
- Forgetting to convert mm² to m² (× 10⁻⁶) in R = ρL/A.
- Taking resistance ∝ 1/r instead of 1/r². Area depends on radius squared.
- Writing the unit of resistivity as Ω/m. It is Ω m (ohm metre).