# Temperature dependence of resistance

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u9-temperature-dependence-of-resistance/

This sub-topic explains how resistance of common materials (metals, thermistors) changes with temperature, the underlying physical mechanisms, and the mathematical relationship to calculate resistance at any temperature.

**Prerequisites:** [Resistance and resistivity](https://www.owlsprep.com/study/cie-9702-u9-resistance-and-resistivity/); [Ohm's law](https://www.owlsprep.com/study/cie-9702-u9-ohms-law/)

## Learning objectives

- Describe how resistance changes with temperature for metallic conductors and thermistors
- Recall and use the temperature dependence equation for resistance
- Explain the physical mechanisms behind resistance change with temperature
- Solve CIE exam problems on this topic

## Physical Mechanism of Resistance Change

**Electron scattering and resistance** — In any conductor, current is carried by free electrons moving through a lattice of positive ions. Resistance arises from collisions (scattering) of electrons with these ions, which impede electron motion and increase opposition to current.

*Example:* Changing temperature alters the rate of scattering and/or the number of free electrons, leading to a change in total resistance.

- **Metallic conductors**: Number of free charge carriers is constant for moderate temperature changes. Increased temperature makes ions vibrate more, increasing electron scattering, so resistance increases.
- **NTC thermistors**: Semiconductor devices. Increased temperature releases more free charge carriers, and this effect outweighs increased scattering, so resistance decreases.
- **PTC thermistors**: Less common variant where resistance increases sharply above a threshold temperature, used for switching circuits.

**Worked example:** Explain why the resistance of pure copper wire increases when temperature is raised.

1. Step 1: Recall the properties of metallic conductors:
2. Pure copper is a metallic conductor, so it has a fixed number of free charge carriers that does not change significantly with moderate temperature increases.
3. Step 2: Describe the effect of temperature on the metal lattice:
4. When temperature increases, positive copper ions gain kinetic energy and vibrate with larger amplitude around their fixed positions.
5. Step 3: Link to increased resistance:
6. Increased vibration leads to more frequent collisions between free electrons and ions. More scattering means greater opposition to current, so resistance increases.

> **Exam tip:** When asked to explain resistance change, always mention both charge carrier density and scattering for full marks in CIE exams.

## Mathematical Relationship for Resistance Change

**Temperature coefficient of resistance** — The fractional change in resistance per unit change in temperature, measured from a reference temperature.

*Notation:* \alpha

*Example:* Copper: \(+4.0 \times 10^{-3} \, ^\circ\text{C}^{-1}\), typical NTC thermistor: \(-5 \times 10^{-2} \, ^\circ\text{C}^{-1}\)

For small temperature changes, the relationship between resistance and temperature is approximately linear, given by:

$$R = R_0 \left( 1 + \alpha \Delta \theta \right)$$

Where \(R\) = resistance at new temperature, \(R_0\) = resistance at reference temperature, \(\alpha\) = temperature coefficient, and \(\Delta \theta = \theta - \theta_0\) is the change in temperature.

**Exam command terms**

- **Show that** — You must state the equation and show all substitution steps to reach the given answer *(Always write \(R = R_0(1 + \alpha \Delta \theta)\) before substituting values)*

- **Explain** — You must give physical reasoning, not just state if resistance increases or decreases

**Worked example:** A copper wire has resistance \(2.50 \, \Omega\) at \(20^\circ \text{C}\). \(\alpha = 4.0 \times 10^{-3} \, ^\circ \text{C}^{-1}\). Calculate resistance at \(100^\circ \text{C}\).

1. Step 1: Identify known values:
2. $$R_0 = 2.50 \, \Omega, \quad \alpha = 4.0 \times 10^{-3} \, ^\circ \text{C}^{-1}, \quad \Delta \theta = 100 - 20 = 80^\circ \text{C}$$
3. Step 2: Substitute into the equation:
4. $$R = 2.50 \left(1 + (4.0 \times 10^{-3})(80)\right)$$
5. Step 3: Calculate:
6. $$R = 2.50 (1 + 0.32) = 2.50 \times 1.32 = 3.30 \, \Omega$$

> **Exam tip:** Always check the sign of \(\alpha\): NTC thermistors have negative \(\alpha\), so resistance will decrease for temperature increases.

*Calculator:* allowed

## Practical Applications

The predictable temperature dependence of resistance is used in a wide range of practical temperature sensing and control devices:

- **Digital thermometers**: NTC thermistors have large resistance change per degree, making them very sensitive to small temperature changes.
- **Thermostats**: PTC thermistors are used to switch circuits on/off when a threshold temperature is reached.
- **Temperature compensation**: Predictable resistance change of metals is used to offset temperature-induced changes in precision circuits.

> **tip**
>
> Percentage change in resistance is simply \(\alpha \Delta \theta \times 100\%\), a useful shortcut for multiple choice questions.

**Worked example:** An NTC thermistor has resistance \(10 \, \text{k}\Omega\) at \(25^\circ \text{C}\), \(\alpha = -0.04 \, ^\circ \text{C}^{-1}\). What is the percentage change in resistance at \(35^\circ \text{C}\)?

1. Step 1: Calculate temperature change:
2. $$\Delta \theta = 35 - 25 = 10^\circ \text{C}$$
3. Step 2: Calculate new resistance:
4. $$R = 10 \left(1 + (-0.04)(10)\right) = 10 (1 - 0.4) = 6 \, \text{k}\Omega$$
5. Step 3: Calculate percentage change:
6. $$\text{\% change} = \frac{6 - 10}{10} \times 100\% = -40\%$$
7. Resistance decreases by 40%, which matches the behavior of an NTC thermistor.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting the negative sign of \(\alpha\) for NTC thermistors, calculating an increase in resistance instead of a decrease
  - Why it fails: Students default to positive \(\alpha\) from metallic examples and ignore the sign given in the question
  - Correct: Always check the material type: NTC thermistors always have a negative \(\alpha\), so use the given sign in your calculation
- **Wrong:** Explaining lower resistance in NTC thermistors by only saying 'less scattering at higher temperatures'
  - Why it fails: This ignores the dominant effect of increased charge carrier density, which is the main cause of resistance decrease
  - Correct: Always state that higher temperature releases more free charge carriers in semiconductors, which outweighs increased scattering, leading to lower resistance
- **Wrong:** Using \(0^\circ\text{C}\) as the reference temperature when \(R_0\) is given for a different temperature
  - Why it fails: Students assume the reference temperature is always freezing point, but it varies by problem
  - Correct: Always calculate \(\Delta \theta\) as new temperature minus the reference temperature given for \(R_0\)
- **Wrong:** Mixing up the direction of resistance change for metals and NTC thermistors
  - Why it fails: The mechanisms produce opposite net effects, and students often confuse the two
  - Correct: Use the mnemonic: *M*etals = *M*ore resistance when hot; *N*TC = *N*egative change (less resistance when hot)

## Cheatsheet

| Material Type | Sign of \(\alpha\) | Resistance at higher temperature | Physical Reason |
| --- | --- | --- | --- |
| Pure metal | Positive (+) | Increases | More electron scattering, constant charge carrier density |
| NTC thermistor | Negative (-) | Decreases | Large increase in free charge carriers outweighs scattering |
| PTC thermistor | Positive (+) | Increases sharply (above threshold) | Doped semiconductor for switching applications |

## What's next

Understanding the temperature dependence of resistance is critical for analyzing potential divider circuits that use thermistors for temperature sensing, a very common topic in CIE A-Level Physics exams. This concept also forms the foundation for more advanced topics like semiconductor physics, which you will cover later in your A-Level course. Mastery of the physical mechanisms and the mathematical relationship will help you tackle both multiple choice and structured questions on this topic. Below are related sub-topics you should review next to build complete mastery of current electricity.

- [Resistance and resistivity](https://www.owlsprep.com/study/cie-9702-u9-resistance-and-resistivity/)
- [D.C. circuits](https://www.owlsprep.com/study/cie-9702-u10-overview/)
- [Circuit symbols and diagrams](https://www.owlsprep.com/study/cie-9702-u10-circuit-symbols-and-diagrams/)

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