understand that the resistance of a thermistor decreases as the temperature increases (it will be assumed that thermistors have a negative temperature coefficient)

Resistance, Resistivity and Temperature‑Dependent Behaviour (Cambridge IGCSE/A‑Level Physics 9702)

This note covers the full syllabus content for 9.2 Potential difference & power (as context) and 9.3 Resistance, resistivity & temperature dependence. It focuses on NTC thermistors (negative‑temperature‑coefficient) while also recognising PTC devices (positive‑temperature‑coefficient).

9.2 Potential Difference & Power – link to resistance

• Ohm’s law: \(V = I R\)  (Voltage = Current × Resistance)
• Electrical power in a resistor:

  • \(P = V I\)
  • Using Ohm’s law,
     \(P = I^{2} R\) or \(P = \dfrac{V^{2}}{R}\)

Example: A 10 Ω heater carries 2 A.

\(P = I^{2}R = (2\ \text{A})^{2}\times10\ \Omega = 40\ \text{W}\). The heater converts 40 J of electrical energy into heat each second – a direct illustration of why resistance matters in heating elements.

9.3 Resistance, Resistivity & Temperature Dependence

9.3 a – Definitions (required by the syllabus)

• Resistance (R) – opposition to the flow of charge; unit Ω.
• Resistivity (ρ) – intrinsic property of a material; unit Ω·m.
• Conductivity (σ) – reciprocal of resistivity, σ = 1/ρ; unit S·m⁻¹.
• Temperature coefficient of resistance (α) – fractional change in resistance per kelvin; positive for most metals, negative for many semiconductors.
• Thermistor – a semiconductor resistor whose resistance varies strongly with temperature.

  • NTC (Negative‑temperature‑coefficient): R falls as T rises.
  • PTC (Positive‑temperature‑coefficient): R rises as T rises (recognised only by sign).

9.3 b – Geometry and the basic resistance formula

For a uniform conductor of length L (m) and cross‑sectional area A (m²):

R = ρ · L ⁄ A

• Longer → larger R.
• Wider → smaller R.
• Higher‑ρ material → larger R for the same geometry.

Unit reminder: use metres for L and square metres for A so that R is in ohms.

9.3 c – Typical resistivity values (Ω·m)

9.3 d – Temperature coefficient for metals (positive α)

Resistivity of most metals varies approximately linearly with temperature:

ρ = ρ₀ [1 + α (T − T₀)]

Substituting into R = ρL/A gives the familiar resistance expression:

R = R₀ [1 + α (T − T₀)]

• R₀ – resistance at reference temperature T₀ (usually 20 °C or 293 K).
• α – positive temperature‑coefficient (≈ 3.9 × 10⁻³ K⁻¹ for copper).

Typical α values (order of magnitude)

9.3 e – Semiconductors – NTC Thermistors

In a semiconductor the number of charge carriers rises sharply with temperature, so resistivity – and therefore resistance – falls. The syllabus adopts the empirical β‑model:

R = R₀ · e^{ β (1/T − 1/T₀) }

• R₀ – resistance at reference temperature T₀ (usually 25 °C = 298 K).
• β – material‑specific constant (typical 3000–5000 K).
• T – absolute temperature in kelvin.

Because \(1/T\) decreases as \(T\) increases, the exponent becomes more negative and \(R\) drops exponentially.

Qualitative explanation (syllabus point 9.3 f)

1. At low temperature few electrons have enough energy to cross the band gap → low carrier concentration → high resistance.
2. Heating supplies energy, promoting more electrons into the conduction band → carrier concentration rises sharply.
3. More carriers allow a larger current for the same applied voltage, so the measured resistance falls.

Worked example (NTC thermistor)

Given \(R₀ = 10\ \text{kΩ}\) at \(T₀ = 298\ \text{K}\), \(\beta = 3500\ \text{K}\). Find \(R\) at \(T = 350\ \text{K}\).

R = R₀·e^{β(1/T – 1/T₀)}
= 10 000 Ω·e^{3500(1/350 – 1/298)}
1/350 – 1/298 = 0.002857 – 0.003356 = –0.000499 K⁻¹
Exponent = 3500 × (–0.000499) = –1.7465
R = 10 000 Ω·e^{–1.7465} ≈ 10 000 Ω·0.174 ≈ 1.74 kΩ

The resistance has fallen to about 17 % of its 25 °C value, illustrating the strong NTC effect.

9.3 g – PTC Thermistors (recognition of sign only)

A PTC thermistor has a positive temperature coefficient: its resistance increases as temperature rises. In most exam questions it is sufficient to state the sign and give one everyday example (e.g., self‑resetting heater protection).

Practical Use in Circuits (syllabus point 9.3 h)

• Thermistors are commonly placed in a voltage divider:
  

  \(V{\text{out}} = V{\text{supply}}\dfrac{R{\text{thermistor}}}{R{\text{fixed}}+R_{\text{thermistor}}}\). The output voltage varies with temperature.

• Because the change is exponential, a small temperature shift can produce a large change in resistance – ideal for precise temperature sensing.
• When designing, check the datasheet for the β‑value and the temperature range over which the exponential model is guaranteed.

Key Take‑aways (mapped to syllabus codes)

• 9.2 – Resistance links directly to \(V = IR\) and \(P = I^{2}R\); power dissipation in resistors is the basis of heating elements.
• 9.3 a – Definitions of R, ρ, σ, α and thermistor types.
• 9.3 b – Geometry formula \(R = ρL/A\).
• 9.3 c – Typical resistivity values for common materials.
• 9.3 d – Linear temperature dependence for metals; α values table provided.
• 9.3 e–f – Exponential NTC behaviour, β‑model, and qualitative carrier‑generation explanation.
• 9.3 g – Recognition of PTC sign.
• 9.3 h – Practical circuit use (voltage divider, importance of β‑value).

Remember to keep units consistent (L in m, A in m², ρ in Ω·m, T in K) and to quote the reference temperature when using any temperature‑coefficient formula.