Resistivity

Resistivity is a material property that quantifies how strongly a material opposes the flow of electric current. Unlike resistance, which depends on the shape and size of a conductor, resistivity is an intrinsic property of the material itself.

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The Resistivity Equation

$\rho = \frac{RA}{L}$ρ=LRA​

Rearranged to give resistance:

$R = \frac{\rho L}{A}$R=AρL​

where:

• ρ = resistivity (Ω m)
• R = resistance (Ω)
• A = cross-sectional area (m²)
• L = length (m)

Key Point: The SI unit of resistivity is the ohm-metre (Ω m), NOT ohm per metre. Check the dimensions: R × A / L = Ω × m² / m = Ω m ✓

Typical Resistivity Values

Material	Resistivity (Ω m) at 20°C	Classification
Silver	1.59 × 10⁻⁸	Metal (conductor)
Copper	1.68 × 10⁻⁸	Metal (conductor)
Aluminium	2.65 × 10⁻⁸	Metal (conductor)
Nichrome	1.10 × 10⁻⁶	Alloy (higher resistivity)
Silicon (pure)	6.40 × 10²	Semiconductor
Glass	10¹⁰ – 10¹⁴	Insulator
PTFE (Teflon)	10²³	Insulator

Note the enormous range: from about 10⁻⁸ Ω m for good conductors to over 10²³ Ω m for excellent insulators — a factor of more than 10³⁰.

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Worked Examples

Worked Example 1 — Finding Resistance from Resistivity

Calculate the resistance of a copper wire of length 2.0 m and diameter 0.50 mm. The resistivity of copper is 1.68 × 10⁻⁸ Ω m.

Solution:

L = 2.0 m

Diameter = 0.50 mm = 0.50 × 10⁻³ m, so radius r = 0.25 × 10⁻³ m

Cross-sectional area: A = πr² = π × (0.25 × 10⁻³)² = π × 6.25 × 10⁻⁸ = 1.963 × 10⁻⁷ m²

R = ρL/A = (1.68 × 10⁻⁸ × 2.0) / (1.963 × 10⁻⁷)

R = 3.36 × 10⁻⁸ / 1.963 × 10⁻⁷

R = 0.171 Ω

Exam Tip: Always convert diameter to radius, and convert mm to m, before calculating the area. A very common error is to forget to halve the diameter or to forget to square the radius. Also, remember to use the area formula for a circle: A = πr², not 2πr (which is the circumference).

Worked Example 2 — Finding Resistivity

A wire of length 1.50 m and cross-sectional area 3.0 × 10⁻⁷ m² has a resistance of 8.8 Ω. Calculate the resistivity of the material.

Solution:

ρ = RA/L = (8.8 × 3.0 × 10⁻⁷) / 1.50

ρ = 2.64 × 10⁻⁶ / 1.50

ρ = 1.76 × 10⁻⁶ Ω m

This value is close to the resistivity of nichrome, so the wire is likely made of nichrome (a nickel-chromium alloy used in heating elements).

Worked Example 3 — Comparing Two Wires

Wire A is made of copper (ρ = 1.68 × 10⁻⁸ Ω m), is 3.0 m long and has a diameter of 1.0 mm. Wire B is made of the same material, is 1.5 m long and has a diameter of 0.50 mm. Calculate the resistance of each wire and determine which has the greater resistance.

Solution:

Wire A: r = 0.50 × 10⁻³ m, A = π(0.50 × 10⁻³)² = 7.854 × 10⁻⁷ m² R_A = ρL/A = (1.68 × 10⁻⁸ × 3.0) / 7.854 × 10⁻⁷ = 5.04 × 10⁻⁸ / 7.854 × 10⁻⁷ = 0.0642 Ω

Wire B: r = 0.25 × 10⁻³ m, A = π(0.25 × 10⁻³)² = 1.963 × 10⁻⁷ m² R_B = ρL/A = (1.68 × 10⁻⁸ × 1.5) / 1.963 × 10⁻⁷ = 2.52 × 10⁻⁸ / 1.963 × 10⁻⁷ = 0.128 Ω

Wire B has the greater resistance (0.128 Ω vs 0.064 Ω), even though it is shorter. This is because halving the diameter reduces the area by a factor of 4, which quadruples the resistance per unit length. Halving the length only halves the resistance. The net effect is that Wire B has double the resistance of Wire A.

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Variation of Resistivity with Temperature

Metals

For metals, resistivity increases with temperature.