Topic 11.411 min read
Resistivity and Temperature
For a uniform conductor, R=ρL/A.
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Resistance describes one particular conductor, while resistivity describes the material from which it is made. A long thin wire has more resistance than a short thick wire of the same material.
- R
- resistance (Ω)
- ρ
- resistivity (Ω·m)
- L
- length (m)
- A
- cross-sectional area (m²)
For most metals, resistance and resistivity increase when temperature rises. For semiconductors, they usually decrease as temperature rises.
Recommended for Class 11
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
A longer conductor gives charge carriers a longer path and more opportunities for collisions, so its resistance is greater.
A larger cross-sectional area provides more parallel paths for charge flow, so resistance is smaller.
- R
- resistance (Ω)
- ρ
- resistivity (Ω·m)
- L
- conductor length (m)
- A
- cross-sectional area (m²)
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.
| 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.
- 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.
- 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)
For a conductor whose dimensions change negligibly, the same linear form can be written for resistance.
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.
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.
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.
Exam Focus
Exam Focus
Important points
- R=ρL/A
- Meaning and unit of resistivity
- Conductivity
- Temperature equation
- Positive α for metals
- Negative temperature behavior of semiconductors
Important formulas
- R=ρLA
- ρ=RAL
- σ=1ρ
- ρT=ρ0[1+α(T-T0)]
Common exam traps
- Convert mm² to m².
- Use temperature difference, not final temperature alone.
Long question structure
Dimensional factors → define resistivity → unit → temperature equation → microscopic explanation → material comparison → applications
Common Mistakes
Common Mistakes
- Mistake
Confusing resistance with resistivity.
- Mistake
Writing Ωm instead of Ω·m.
- Mistake
Forgetting to convert area into m².
- Mistake
Using T instead of ΔT=T−T0.
- Mistake
Assuming all materials have positive temperature coefficient.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
For a uniform conductor, R=ρL/A.
Resistance is directly proportional to length and inversely proportional to cross-sectional area.
Resistivity is a material property with SI unit Ω·m.
Conductivity is σ=1ρ.
Resistivity is the intrinsic opposition of a material to current; R=ρL/A.
R=ρLA,ρ=RAL,ρT=ρ0[1+α(T-T0)]
- Definition
- Resistivity is the intrinsic opposition of a material to current; R=ρL/A.
- Formula
- R=ρLA,ρ=RAL,ρT=ρ0[1+α(T-T0)]
- Example
- For R=0.020 Ω, A=1.5×10⁻⁶ m² and L=2.0 m, ρ=1.5×10⁻⁸ Ω·m.
- Critical warning
- Area must be in m² and resistivity in Ω·m.
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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.