The resistance of a uniform conductor depends on its length, cross-sectional area, material and temperature. Resistivity separates the effect of the material from the conductor's dimensions.
Effect of length and area
Key equation
R∝ L
A longer conductor gives charge carriers a longer path and more opportunities for collisions, so its resistance is greater.
Key equation
R∝1A
A larger cross-sectional area provides more parallel paths for charge flow, so resistance is smaller.
Key equation
R=ρLA
R
resistance (Ω)
ρ
resistivity (Ω·m)
L
conductor length (m)
A
cross-sectional area (m²)
Key equation
ρ=RAL
Meaning and unit of resistivity
Resistivity is an intrinsic property that measures how strongly a material opposes electric current at a stated temperature. It does not change merely because a sample is made longer or thicker.
Key equation
[ρ]=Ωm^2m=Ωm
Quantity
Depends on
Resistance R
Material, length, cross-sectional area and temperature
Resistivity ρ
Nature of material and temperature
Conductivity
Conductivity measures how readily a material conducts and is the reciprocal of resistivity.
Key equation
σ=1ρ
σ
conductivity (S·m⁻¹)
Good conductors have low resistivity and high conductivity; good insulators have high resistivity and low conductivity.
Temperature dependence for metals
Over a moderate temperature range, the resistivity of a metal changes approximately linearly with temperature.
Key equation
ρT=ρ0[1+α(T-T0)]
ρT
resistivity at temperature T (Ω·m)
ρ0
resistivity at reference temperature T0 (Ω·m)
α
temperature coefficient of resistivity (K⁻¹ or °C⁻¹)
Key equation
Δ T=T-T0
For a conductor whose dimensions change negligibly, the same linear form can be written for resistance.
Key equation
RT=R0[1+α(T-T0)]
Metals usually have positive α. Higher temperature makes lattice ions vibrate more strongly, causing more electron collisions and increasing resistivity.
Semiconductors and insulators
Semiconductors generally have a negative temperature coefficient. Heating releases more mobile charge carriers, and this carrier increase usually causes their resistance to fall.
Material type
Typical temperature effect
Metal
ρ and R increase; α is positive
Semiconductor
ρ and R decrease; effective α is negative
Alloy such as manganin or constantan
Small temperature change; α is close to zero
Alloys with small temperature coefficients are useful for standard resistors because their resistance remains comparatively stable when temperature changes.
Worked example 1: finding resistivity
A wire has L=2.0 m, A=1.5×10⁻⁶ m² and R=0.020 Ω. Find its resistivity.
Key equation
ρ=RAL=(0.020)(1.5×10-6)2.0
Key equation
ρ=1.5×10-8\ Ωm
Worked example 2: temperature change
A material has ρ0=5.6×10⁻⁸ Ω·m and α=0.0045 K⁻¹. Find its resistivity after a 30 K rise.
Key equation
ρT=ρ0(1+αΔ T)
Key equation
ρT=(5.6×10-8)[1+(0.0045)(30)]
Key equation
ρT=6.356×10-8\ Ωm
Exam-ready method
Convert every length to metres and area to square metres.
Use R=ρL/A for a uniform conductor.
Distinguish resistance from resistivity in words.
For temperature questions, calculate ΔT=T−T0 first.
Use the correct sign of α for the material.
Write resistivity in Ω·m and temperature coefficient in K⁻¹ or °C⁻¹.
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Exam preparation
Exam Focus
Important points
R=ρL/A
Meaning and unit of resistivity
Conductivity
Temperature equation
Positive α for metals
Negative temperature behavior of semiconductors
Common mistakes
Confusing resistance with resistivity.
Writing Ωm instead of Ω·m.
Forgetting to convert area into m².
Using T instead of ΔT=T−T0.
Assuming all materials have positive temperature coefficient.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.
Resistivity and Temperature
Sticky Notes — Quick Revision
1.
Key Point 1
For a uniform conductor, R=ρL/A.
2.
Key Point 2
Resistivity is a material property with SI unit Ω·m.
3.
Key Point 3
For moderate temperature changes, ρT=ρ0[1+α(T−T0)].
4.
Key Point 4
Semiconductors normally show decreasing resistance as temperature rises.
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8 concept questions
Easy → Medium → Hard
Explanation after each answer
Exam Practice
Definition
Q1. Define resistivity and state its SI unit.
Derivation
Q2. State how resistance depends on length and cross-sectional area.
Reasoning
Q3. Why does metallic resistance increase with temperature?
Numerical
Q4.Find ρ for R=0.020 Ω, A=1.5×10⁻⁶ m² and L=2.0 m.
Comparison
Q5. Compare the temperature behavior of metals and semiconductors.
