Resistance and resistivity: what resistance depends on
Resistance of a wire is directly proportional to its length and inversely proportional to its area of cross-section. Resistivity ρ depends only on the material and temperature.
Four factors that decide resistance
- Length (l): a longer wire has more resistance. R ∝ l. Double the length, double the resistance.
- Area of cross-section (A): a thicker wire has less resistance. R ∝ 1/A. Double the area, half the resistance.
- Nature of the material: for the same size, copper has a low resistance and nichrome a high one.
- Temperature: the resistance of a metal wire usually increases as it gets hotter.
- Putting length and area together: R = ρl/A, where ρ (rho) is the resistivity of the material.
What is resistivity?
Resistivity ρ is the resistance of a piece of the material that is 1 m long with a cross-section of 1 m². Its SI unit is the ohm metre (Ω m). Resistivity is a property of the material. It does not change when you cut or stretch the wire, but it does change with temperature. Conductors such as metals have very low resistivity. Insulators such as rubber and glass have very high resistivity. An alloy usually has a higher resistivity than the metals it is made from.
Choosing the right material
| Material | Resistivity | Used for | Why |
|---|---|---|---|
| Copper, aluminium | Very low | House wiring and power lines | Very little energy is wasted as heat |
| Nichrome (an alloy) | High | Heating coils of irons, toasters and heaters | High resistance, and it does not oxidise (burn) easily when red-hot |
| Tungsten | Low, but higher than copper | Filament of an electric bulb | Very high melting point (3380 °C), so it glows white-hot without melting |
| Rubber, PVC, glass | Extremely high | Covering on wires, handles of tools | Current cannot pass, so they keep us safe |
Worked example: finding resistance
A copper wire is 50 m long and has an area of cross-section of 1 × 10⁻⁶ m² (1 mm²). Take the resistivity of copper as 1.6 × 10⁻⁸ Ω m. Find the resistance of the wire.
- Given: l = 50 m, A = 1 × 10⁻⁶ m², ρ = 1.6 × 10⁻⁸ Ω m.
- Formula: R = ρl/A.
- R = (1.6 × 10⁻⁸ Ω m × 50 m) ÷ (1 × 10⁻⁶ m²).
- R = 80 × 10⁻⁸ ÷ 10⁻⁶ = 80 × 10⁻² Ω = 0.8 Ω.
R = 0.8 Ω
Worked example: cutting and stretching a wire
A wire has a resistance of 10 Ω. (a) It is cut into two equal halves. What is the resistance of each half? (b) An identical wire is stretched to twice its length, and its volume stays the same. What is its new resistance?
- (a) R ∝ l. Each half has half the length, so R = 10 Ω ÷ 2 = 5 Ω.
- (b) Volume = A × l stays the same. If the length becomes 2l, the area becomes A/2.
- New R = ρ(2l) ÷ (A/2) = 4 × (ρl/A).
- New R = 4 × 10 Ω = 40 Ω.
(a) 5 Ω each; (b) 40 Ω
The heating coil of an electric iron or an immersion rod is made of nichrome, while the wires inside the walls are copper. The cable that feeds a geyser is thicker, because a thicker wire has less resistance and stays cooler.
2 marks: “Why are alloys, not pure metals, used in electric heating devices?” Alloys have a higher resistivity, so they produce more heat, and they do not oxidise (burn) readily at high temperatures.
3 marks: “A wire of length 4 m and area of cross-section 2 × 10⁻⁷ m² has a resistance of 10 Ω. Find the resistivity of its material.” ρ = RA/l = (10 Ω × 2 × 10⁻⁷ m²) ÷ 4 m = 5 × 10⁻⁷ Ω m.
3 marks: “List the factors on which the resistance of a conductor depends. How does the resistance change if the length is doubled and the area is halved?” Length, area of cross-section, material and temperature. New R = ρ(2l)/(A/2) = 4R: four times.
Long and thin: high resistance. Short and thick: low resistance. In R = ρl/A, length sits on top and area sits below.