Linear Temperature Dependence – First Encounter
You've probably noticed that many things change when you heat them up. A metal rail expands on a hot day. The resistance of a wire increases when current makes it hot. The pressure in a sealed tyre rises after a long drive.
The simplest way this happens is linear temperature dependence — the property changes in direct proportion to the change in temperature. Double the temperature rise, double the change in the property. No surprises, no sudden jumps.
The Intuition
Imagine a rubber band. If you pull it gently, it stretches a little. Pull twice as hard, it stretches twice as much — that's a linear relationship between force and stretch.
Now replace "force" with "temperature change" and "stretch" with "some physical quantity" (length, resistance, pressure, volume). That's linear temperature dependence: equal increments of temperature produce equal increments of the quantity.
This is the first approximation for most materials over a limited temperature range. It's never perfectly true forever, but it's remarkably accurate for small temperature changes.
The Precise Statement
If a physical quantity Q depends linearly on temperature T, then:
Q(T)=Q0+α(T−T0)
Where:
- Q0 is the value at some reference temperature T0 (often 0∘C or 25∘C)
- α is the temperature coefficient — the rate of change per degree
- T is the current temperature
The change ΔQ=Q−Q0 is directly proportional to the change ΔT=T−T0:
ΔQ=αΔT
Q(T)=Q0[1+β(T−T0)]
where β=α/Q0 is the fractional temperature coefficient (units: ∘C−1 or K−1)
Real Examples You'll Meet in Exams
| Quantity | Symbol | Typical behaviour | Common β value |
|---|
| Length of a metal rod | L | Expands on heating | ≈1.2×10−5∘C−1 (steel) |
| Resistance of a copper wire | R | Increases with temperature | ≈3.9×10−3∘C−1 |
| Volume of an ideal gas (constant pressure) | V | Increases linearly with T (in Kelvin) | 1/273.15∘C−1 |
| Pressure of an ideal gas (constant volume) | P | Increases linearly with T (in Kelvin) | 1/273.15∘C−1 |
For gases, the linear law works only when temperature is measured in Kelvin, not Celsius. The formula becomes V=V0(1+273.15T) where T is in °C — but this is just a disguised version of V∝T (Kelvin).
Why This Matters
Linear temperature dependence is the foundation of:
- Thermometers (mercury in glass, resistance thermometers, thermocouples)
- Thermal expansion calculations in bridges and railway tracks
- Temperature compensation in electronic circuits
- Charles's Law and Gay-Lussac's Law for ideal gases
The key insight: when you see a straight-line graph of a physical quantity against temperature, you're looking at linear temperature dependence. The slope of that line is α, and the intercept at T=0 (or T=T0) gives you Q0.
Linear temperature dependence is not a law of nature — it's an approximation that works well for small temperature ranges. For large temperature changes, higher-order terms (T2, T3, ...) become important, and the relationship becomes non-linear.
Students preparing for boards often pair a search for "Linear Temperature Dependence class 11 physics" with "NCERT Physics syllabus" — Linear Temperature Dependence is a syllabus-aligned topic under Thermal Properties of Matter in NCERT Class 11 Physics, making it a natural fit for both board exams and JEE/NEET practice sets. Working through the worked examples above alongside the official NCERT Physics textbook is the most reliable way to turn this understanding into exam-ready recall.