Study Guide

Temperature dependence of resistance

CIE A-Level PhysicsΒ· 10 min read

1. Physical Mechanism of Resistance Changeβ˜…β˜…β˜†β˜†β˜†β± 4 min

πŸ“˜ Definition

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.

πŸ“ Worked Example

Explain why the resistance of pure copper wire increases when temperature is raised.

  1. 1

    Step 1: Recall the properties of metallic conductors:

  2. 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. 3

    Step 2: Describe the effect of temperature on the metal lattice:

  4. 4

    When temperature increases, positive copper ions gain kinetic energy and vibrate with larger amplitude around their fixed positions.

  5. 5

    Step 3: Link to increased resistance:

  6. 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.

2. Mathematical Relationship for Resistance Changeβ˜…β˜…β˜†β˜†β˜†β± 3 min

βœ“ Calculator OK

πŸ“˜ Definition

Temperature coefficient of resistance

Ξ±\alpha

The fractional change in resistance per unit change in temperature, measured from a reference temperature.

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=R0(1+αΔθ)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.

πŸ“ 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. 1

    Step 1: Identify known values:

  2. 2
    R0=2.50 Ω,Ξ±=4.0Γ—10βˆ’3β€‰βˆ˜Cβˆ’1,Δθ=100βˆ’20=80∘CR_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. 3

    Step 2: Substitute into the equation:

  4. 4
    R=2.50(1+(4.0Γ—10βˆ’3)(80))R = 2.50 \left(1 + (4.0 \times 10^{-3})(80)\right)
  5. 5

    Step 3: Calculate:

  6. 6
    R=2.50(1+0.32)=2.50Γ—1.32=3.30 ΩR = 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.

3. Practical Applicationsβ˜…β˜…β˜…β˜†β˜†β± 3 min

βœ“ Calculator OK

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.

πŸ“ 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. 1

    Step 1: Calculate temperature change:

  2. 2
    Δθ=35βˆ’25=10∘C\Delta \theta = 35 - 25 = 10^\circ \text{C}
  3. 3

    Step 2: Calculate new resistance:

  4. 4
    R=10(1+(βˆ’0.04)(10))=10(1βˆ’0.4)=6 kΞ©R = 10 \left(1 + (-0.04)(10)\right) = 10 (1 - 0.4) = 6 \, \text{k}\Omega
  5. 5

    Step 3: Calculate percentage change:

  6. 6
    % change=6βˆ’1010Γ—100%=βˆ’40%\text{\% change} = \frac{6 - 10}{10} \times 100\% = -40\%
  7. 7

    Resistance decreases by 40%, which matches the behavior of an NTC thermistor.

4. Common Pitfalls

Wrong move:

Forgetting the negative sign of (\alpha) for NTC thermistors, calculating an increase in resistance instead of a decrease

Why:

Students default to positive (\alpha) from metallic examples and ignore the sign given in the question

Correct move:

Always check the material type: NTC thermistors always have a negative (\alpha), so use the given sign in your calculation

Wrong move:

Explaining lower resistance in NTC thermistors by only saying 'less scattering at higher temperatures'

Why:

This ignores the dominant effect of increased charge carrier density, which is the main cause of resistance decrease

Correct move:

Always state that higher temperature releases more free charge carriers in semiconductors, which outweighs increased scattering, leading to lower resistance

Wrong move:

Using (0^\circ\text{C}) as the reference temperature when (R_0) is given for a different temperature

Why:

Students assume the reference temperature is always freezing point, but it varies by problem

Correct move:

Always calculate (\Delta \theta) as new temperature minus the reference temperature given for (R_0)

Wrong move:

Mixing up the direction of resistance change for metals and NTC thermistors

Why:

The mechanisms produce opposite net effects, and students often confuse the two

Correct move:

Use the mnemonic: Metals = More resistance when hot; NTC = Negative change (less resistance when hot)

5. Quick Reference 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

6. Frequently Asked

How do I handle reference temperature in calculations?

For CIE exams, the reference temperature for (R_0) is always stated in the question, typically (0^\circ\text{C}) or room temperature (20^\circ\text{C}). If it is not explicitly stated, assume it is the starting temperature from which the change is measured.

When this came up on past exams

AI-estimated based on syllabus patterns β€” cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2022 Β· 1

    Resistance change of metallic wire

  • 2023 Β· 2

    Thermistor in potential divider

  • 2021 Β· 1

    Temperature coefficient calculation

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.