Resistivity

Resistance is a property of a component: it depends on the material the component is made of, its length, its cross-sectional area and its temperature. Resistivity is the property of a material: it is what you get if you strip away geometric factors and ask "how resistive is copper, intrinsically?"

This lesson derives the resistivity equation, discusses how to measure resistivity experimentally, and looks at the resistivity of metals, alloys and semiconductors. It is a key OCR H556 topic (Module 4.2.2 (h)) that appears on almost every A-Level paper.

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Motivating the Equation

Imagine a uniform rectangular block of material with length L and cross-sectional area A, and a pd V applied across its ends producing a current I. Intuitively:

• Doubling L should double the resistance (the charges have twice as far to go).
• Doubling A should halve the resistance (twice as much "room" for the charges to flow).

Writing this as a proportionality:

R ∝ L / A

To turn the proportionality into an equation, we introduce a constant of proportionality called the resistivity, symbol ρ (Greek rho):

R = ρL / A

• R in ohms (Ω)
• L in metres (m)
• A in square metres (m²)
• ρ in ohm metres (Ω m)

The resistivity ρ is a property of the material alone. It is independent of the geometry, but it does depend on temperature.

Rearranging:

ρ = RA / L

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Typical Resistivities

Here is an order-of-magnitude hierarchy:

Material	Type	ρ at 20 °C (Ω m)
Silver	Metal	1.6 × 10⁻⁸
Copper	Metal	1.7 × 10⁻⁸
Aluminium	Metal	2.7 × 10⁻⁸
Gold	Metal	2.4 × 10⁻⁸
Iron	Metal	~1.0 × 10⁻⁷
Nichrome (heating alloy)	Alloy	1.1 × 10⁻⁶
Constantan (precision alloy)	Alloy	4.9 × 10⁻⁷
Sea water	Electrolyte	~0.2
Pure (intrinsic) silicon	Semiconductor	~2000
Glass	Insulator	10¹⁰–10¹⁴
PTFE	Insulator	~10²³

The span from silver to PTFE is thirty orders of magnitude — easily the largest range of any physical property in classical physics. Resistivity is the electrical fingerprint of a material.

Note:

• Silver has the lowest resistivity, but copper is almost as low and much cheaper — hence copper's dominance in wiring.
• Nichrome has a resistivity nearly 100× larger than copper, which is why it is used for heating elements: a useful amount of heat dissipates in a practical length of wire.
• Constantan is chosen for precision resistors because its resistivity changes only very slightly with temperature.

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Worked Example 1 — Length of a Resistor

You want to make a 5.0 Ω resistor from nichrome wire of cross-sectional area 0.20 mm². The resistivity of nichrome is ρ = 1.1 × 10⁻⁶ Ω m. What length of wire do you need?

• A = 0.20 mm² = 0.20 × 10⁻⁶ m² = 2.0 × 10⁻⁷ m²
• L = RA / ρ = (5.0 × 2.0 × 10⁻⁷) / (1.1 × 10⁻⁶)
• L = 1.0 × 10⁻⁶ / 1.1 × 10⁻⁶
• L ≈ 0.91 m

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Worked Example 2 — Copper Wire for a Mains Cable

A 10 m length of copper cable carries the current to a kettle. The wire must have a resistance no greater than 0.10 Ω to avoid excess heating. What minimum cross-sectional area is required? (ρ = 1.7 × 10⁻⁸ Ω m)

• A = ρL / R = (1.7 × 10⁻⁸ × 10) / 0.10
• A = 1.7 × 10⁻⁶ m²
• A = 1.7 mm²

A typical UK mains cable has a conductor area of 2.5 mm² — comfortably above this minimum.

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Worked Example 3 — Comparing Two Wires

Two wires are made of the same material and have the same length. Wire P has twice the diameter of wire Q. What is the ratio of their resistances?

• Diameter doubles → cross-sectional area quadruples (A ∝ d²).
• R ∝ 1/A, so R_P = R_Q / 4.
• Ratio R_P : R_Q = 1 : 4.

Exam Tip: Always think about the area, not the diameter or radius, when applying R = ρL/A. A factor of 2 in diameter is a factor of 4 in area.

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Factors Affecting Resistivity

Resistivity is not a universal constant — it depends on:

1. Temperature