Temperature Dependence of Resistivity: Formula and Graph
Physics · Current Electricity · NEET
The resistivity of a metal rises almost in a straight line with temperature, given by ρT = ρ0[1 + α(T − T0)], where ρ0 is the resistivity at reference temperature T0 and α is the temperature coefficient. For metals α is positive (ρ goes up when hot); for semiconductors and insulators α is negative (ρ goes down when hot). Memory hook: "Metals hate heat, semiconductors love it."
Resistivity vs temperature: metals rise on a near-straight line (positive α) from a small intercept ρ0, semiconductors fall (negative α), and alloys like Nichrome stay almost flat.
Your doubts, answered
Why does the resistivity of a metal go UP when it gets hot?
In a metal the number of free electrons stays almost the same. When temperature rises, the metal ions vibrate harder about their fixed positions. These stronger vibrations scatter the moving electrons more often, so the average time between collisions (relaxation time τ) becomes smaller. Since ρ = m/(ne²τ), a smaller τ means a larger ρ. So heat does not remove carriers in a metal; it just makes them collide more. That is why α is positive for metals like copper.
Why do semiconductors behave the opposite way (resistivity falls when hot)?
In a semiconductor the number of free carriers n is very small at room temperature because electrons are locked in bonds. Heating gives electrons enough energy to break free, so n grows very fast as temperature rises. This big increase in n makes ρ = m/(ne²τ) fall sharply, even though τ still drops a little. The gain in carriers wins, so resistivity decreases and α is negative. NCERT states insulators and semiconductors have resistivity that decreases with a rise in temperature.
Is the formula for resistivity ρ the same as the formula for resistance R?
Yes, both follow the same linear pattern over a small temperature range. Resistivity: ρT = ρ0[1 + α(T − T0)]. Resistance: RT = R0[1 + α(T − T0)]. This works because R = ρL/A, and for a modest temperature change the length L and area A change very little, so R and ρ scale together. In NEET you use the R-form for wire problems and the ρ-form for material/graph questions. The value of α is the same for both.
What does the graph of resistivity vs temperature look like for copper?
For copper (a metal) it is almost a straight line rising with a positive slope, starting from a small positive intercept ρ0 on the resistivity axis. It is not a line through the origin, because even near 0°C the metal still has some resistivity. At very low temperatures (far below 0°C) the real curve bends and flattens, so the straight line is only valid over a limited temperature range around T0.
Why is Nichrome used in heaters if its resistivity hardly changes with heat?
Alloys like Nichrome, Manganin and Constantan have a very weak (small) temperature coefficient, so their resistivity stays almost flat even when hot. This is exactly what you want in a heater or a standard resistor: the resistance should not swing around as the wire heats up. Nichrome also has high resistivity and does not oxidise easily at high temperature. So the near-constant resistivity is a feature, not a problem.
⚠️ The NEET trap ✗ Resistivity always increases with temperature, so hotter always means more resistivity. ✓ Only metals have positive α (ρ rises with heat). Semiconductors and insulators have negative α, so their resistivity DECREASES as temperature rises because more carriers are freed. 🧠 NTA loves the copper-vs-silicon flip. Read the material first: metal → α positive → ρ up; semiconductor/insulator → α negative → ρ down.
Real NEET questions
NEET 2020
Which of the following graphs represents the variation of resistivity (ρ) with temperature (T) for copper?
A · A. A steeply rising, concave-up curve
B · B. A decreasing curve
C · C. A straight line of positive slope starting from a small intercept ✓
D · D. A curve passing through the origin
Solution: Copper is a metal, so it follows ρT = ρ0[1 + α(T − T0)] with α positive. Expanding: ρT = ρ0 + ρ0·α·(T − T0), which is a straight line in T with positive slope and a small positive intercept ρ0 (the resistivity at the reference temperature). It does NOT pass through the origin because ρ0 is not zero. Semiconductors would give a decreasing curve, so options B and D are ruled out. Answer: C.
NEET 2020
The solids which have a negative temperature coefficient of resistance are:
A · A. Semiconductors only
B · B. Insulators and semiconductors ✓
C · C. Metals
D · D. Insulators only
Solution: Negative temperature coefficient means resistivity (and resistance) DECREASES as temperature rises. This happens when heating frees more charge carriers, i.e. n increases in ρ = m/(ne²τ). Both semiconductors and insulators behave this way (NCERT: their resistivity decreases with a rise in temperature). Metals instead have positive α because their carrier number is fixed and only scattering increases. So the correct set is 'insulators and semiconductors'. Answer: B.
NEET 2022
As the temperature increases, the electrical resistance:
A · A. increases for both conductors and semiconductors
B · B. decreases for both
C · C. increases for conductors but decreases for semiconductors ✓
D · D. decreases for conductors but increases for semiconductors
Solution: Conductors (metals): the free-electron count is fixed, but hotter ions vibrate more and scatter electrons more, so relaxation time τ falls and resistance rises (positive α). Semiconductors: heating breaks bonds and creates many more carriers, so n rises sharply and resistance falls (negative α). Therefore resistance increases for conductors but decreases for semiconductors. Answer: C.
Solved Current Electricity NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
What is the formula for temperature dependence of resistivity?
ρT = ρ0[1 + α(T − T0)], where ρT is resistivity at temperature T, ρ0 is resistivity at reference temperature T0, and α is the temperature coefficient of resistivity. It is valid over a limited temperature range where the graph is nearly a straight line.
What is the unit and dimension of α (temperature coefficient)?
Its unit is per degree Celsius (°C⁻¹) or per kelvin (K⁻¹), and its dimension is (Temperature)⁻¹, because α multiplies a temperature difference and the product must be dimensionless.
Is α positive or negative for metals?
For metals α is positive. Resistivity increases with temperature because increased ion vibrations scatter electrons more and reduce the relaxation time τ, while the number of carriers stays fixed.
Why does resistivity of a semiconductor decrease with temperature?
Heating a semiconductor frees many more charge carriers (n increases rapidly). Since ρ = m/(ne²τ), the large rise in n lowers ρ, giving a negative temperature coefficient.
Which materials have almost temperature-independent resistivity?
Alloys such as Nichrome, Manganin and Constantan show a very weak change of resistivity with temperature. This makes them ideal for heating elements and standard resistors.
