Temperature dependence of resistivity

How does resistivity change with temperature?

In plain words: heat a metal wire and its electrons collide more often, so it resists current more. In an intrinsic semiconductor’s thermal-generation regime heating also frees many more charge carriers, and that wins, so it conducts better when hot. Alloys such as nichrome hardly change at all.

Temperature coefficient of resistivity

For a metal, over a limited range of temperature (NCERT Eq. (3.26), page 90):

• ρ_T is the resistivity at temperature T and ρ₀ at a reference temperature T₀; α is the temperature coefficient of resistivity at the reference temperature T_₀, with dimension (temperature)⁻¹ (page 90).
• For metals, α is positive (page 90).
• The equation says the graph of ρ_T against T is a straight line. At temperatures much lower than 0 °C the real graph bends away from a straight line, so the equation is used only over a limited range around a chosen T₀ (NCERT page 90, Figure 3.8).
• Rearranged, α = (ρ_T − ρ₀)/[ρ₀(T − T₀)]: the fractional change in resistivity per degree (NCERT Summary point 9, page 103).
• NCERT’s worked examples apply the same form to the resistance of a wire, R₂ = R₁[1 + α(T₂ − T₁)], with α called the temperature coefficient of resistance (NCERT Example 3.3, page 91). This works because heating changes l/A far less than ρ. Metals expand by roughly 1 to 3 × 10⁻⁵ per °C, silver by 1.9 × 10⁻⁵ per °C (NCERT Class 11, Table 10.1, page 205; see linear expansion). For a silver wire, l grows by a fraction 1.9 × 10⁻⁵ per °C and A by twice that, so l/A falls by about 2 × 10⁻⁵ per °C, some 200 times less than silver’s α of 3.9 × 10⁻³ per °C (Example 3 below). For nichrome the factor is only 10 to 20, which is why NCERT Example 3.3 quotes the temperature coefficient of resistance itself (reasoned from NCERT Eq. (3.9)).

Metals, nichrome and semiconductors

• Nichrome (an alloy of nickel and chromium, with iron in some grades; NCERT, page 90, describes it as nickel, iron and chromium), manganin and constantan show a very weak dependence of resistivity on temperature. They are used in wire-wound standard resistors, whose values must hardly change with temperature (NCERT page 90, Figure 3.9).
• Intrinsic semiconductors in the thermal-generation regime: their resistivity decreases as the temperature rises (NCERT page 91, Figure 3.10).
• Why, from the drift picture: ρ = 1/σ = m/(ne²τ) (NCERT Eq. (3.27), page 91). NCERT gives a classical speed-and-collision explanation (page 91). In a quantum metal, the main temperature-dependent scattering comes from increased lattice vibrations; electron speeds near the Fermi surface do not simply scale as √T.
• In a metal, n hardly depends on temperature, so the fall in τ makes ρ rise (page 91).
• In the thermal-generation regime of insulators and semiconductors, n increases with temperature, and this more than makes up for the fall in τ, so ρ falls (page 91).

Solved examples

Common mistakes to avoid

Quick revision summary

• Over a limited range, ρ(T) = ρ_₀[1 + α(T − T_₀)]; α is the temperature coefficient of resistivity, in °C^⁻¹ or K^⁻¹, positive for metals.
• Far below 0 °C a metal’s ρ–T graph bends away from a straight line.
• Nichrome, manganin and constantan change very little, so they are used in standard resistors.
• Intrinsic semiconductors in the thermal-generation regime: ρ falls as T rises; doped samples need not follow this throughout their range.
• From ρ = m/(ne^²τ): τ falls as T rises; in metals n is nearly constant, so ρ rises; in semiconductors n rises faster, so ρ falls.

Questions students ask

Why does the resistivity of alloys such as manganin and constantan hardly change with temperature?

An alloy’s resistivity changes little with temperature because most of it does not depend on temperature: beyond the NCERT text, the irregular mix of different atoms scatters electrons strongly at every temperature, and in the approximation called Matthiessen’s rule, this fixed part and the temperature-dependent part add (A. Matthiessen and C. Vogt, Philosophical Transactions of the Royal Society, 1864). In a pure metal near room temperature the growing part is nearly all of it. So nichrome’s coefficient, 1.70 × 10^⁻⁴ °C^⁻¹ (NCERT Example 3.3, page 91), is about a twentieth of copper’s, 3.93 × 10^⁻³ °C^⁻¹ (NBS Copper Wire Tables, 1966), and manganin and constantan suit standard resistors (NCERT page 90).

What is the temperature coefficient of resistance of copper?

The temperature coefficient of resistance of copper is about 3.93 × 10^⁻³ °C^⁻¹ at 20 °C, the value set for standard annealed copper wire and measured between points fixed on a wire free to expand (beyond the NCERT text, US National Bureau of Standards, Copper Wire Tables, Handbook 100, 1966). So a copper wire’s resistance rises by about 0.4 % for each degree: a copper coil of 12.0 Ω at 20 °C has about 12.0 × (1 + 3.93 × 10^⁻³ × 45) ≈ 14.1 Ω at 65 °C.

What is superconductivity, and how does it differ from an ordinary metal’s behaviour near absolute zero?

Superconductivity is the complete loss of electrical resistance that some materials show below a critical temperature, provided the magnetic field and current remain below their critical limits: beyond the NCERT text, Heike Kamerlingh Onnes discovered it in mercury in 1911, a few kelvin above absolute zero (US Department of Energy, ‘DOE Explains…Superconductivity’). An ordinary metal such as copper behaves differently: cooled towards absolute zero, its resistivity levels off at a small residual value set by impurities and defects, the part Matthiessen’s rule (1864) treats as independent of temperature. A superconductor’s resistance instead drops suddenly to zero, a jump no temperature coefficient describes, and a steady current then flows without loss as heat.

Is temperature dependence of resistance in the JEE Main and NEET syllabus?

Yes, temperature dependence of resistance is in both syllabuses, in Unit 12, Current Electricity: the JEE Main syllabus names “temperature dependence of resistance”, and the NEET (UG) 2026 syllabus has the same words in its Unit 12. NCERT Class 12 Physics Part I treats it as Section 3.8, Temperature Dependence of Resistivity (pages 90–92), with NCERT Examples 3.3 and 3.4 and Exercises 3.3, 3.5 and 3.6. The syllabuses say resistance and NCERT says resistivity, but NCERT’s worked examples apply the same formula to the resistance of a wire (NCERT Example 3.3, page 91).

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. What are the dimensions of the temperature coefficient of resistivity?

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   (Temperature)⁻¹ (NCERT page 90).

2. Q2. Why are manganin and constantan used for standard resistors?

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   Their resistivity changes very little with temperature, so the resistance values stay nearly constant (NCERT page 90).

3. Q3. A wire’s resistance is 20 Ω at 20 °C, and α referred to 20 °C is 4.0 × 10⁻³ °C⁻¹. Find its resistance at 70 °C.

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   R = 20[1 + 4.0 × 10⁻³ × 50] = 20 × 1.2 = 24 Ω (NCERT Eq. (3.26), page 90).