Resistivity: (i) resistivity of a material; the equation R = ρL/A

The idea

Resistivity ρ is a property of the material itself, not of a particular specimen, so it lets you compare copper with nichrome regardless of shape. The defining equation R = ρL/A rearranges to ρ = RA/L. Here R is resistance in ohms (Ω), L is the length in metres (m) and A is the cross-sectional area in square metres (m²). Because ρ = RA/L, its unit is Ω m. A long thin wire has more resistance than a short thick one of the same material. Doubling L doubles R; doubling A halves R. Typical values: copper about 1.7 × 10⁻⁸ Ω m, nichrome about 1.1 × 10⁻⁶ Ω m. In a multiple-choice question you may be asked to identify the unit, to predict how R changes when L or A changes, or to calculate ρ from R, L and A.

What earns the marks

• Resistivity is defined by R = ρL/A, rearranged as ρ = RA/L.
• ρ is a property of the material; R also depends on the dimensions L and A.
• The unit of ρ is the ohm metre (Ω m), obtained from Ω × m² / m.
• Doubling the length L doubles the resistance R for a fixed material and area.
• Doubling the cross-sectional area A halves the resistance R for a fixed material and length.
• A larger ρ means a poorer conductor; copper has a much smaller ρ than nichrome.

Common mistakes and how to put them right

• Thinking resistivity depends on the length or thickness of the wire.

  Put it right: ρ is a material property; only R depends on L and A.

• Using the unit Ω m⁻¹ for resistivity.

  Put it right: ρ = RA/L gives Ω × m² / m = Ω m.

• Substituting A in mm² or cm² without converting to m².

  Put it right: Convert areas to m² before calculating, since 1 mm² = 1 × 10⁻⁶ m².

• Confusing resistance R with resistivity ρ in a rearrangement.

  Put it right: Write ρ = RA/L and check each symbol before substituting.

Exam tips

• Write the equation as ρ = RA/L before substituting any numbers, so the subject is clear.
• Check units of L and A first; convert mm² to m² by multiplying by 1 × 10⁻⁶.
• For proportional-reasoning options, change one variable at a time and state the effect on R.
• Sanity-check the order of magnitude: metals have ρ of order 10⁻⁸ to 10⁻⁶ Ω m.