Temperature dependence of resistance

Q: How do I handle reference temperature in calculations?

For CIE exams, the reference temperature for $R0$ 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.

CIE A-Level Physics· 10 min read

1. Physical Mechanism of Resistance Change★★☆☆☆⏱ 4 min

📘 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.

📐 Worked Example

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

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Step 1: Recall the properties of metallic conductors:
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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.
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Step 2: Describe the effect of temperature on the metal lattice:
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When temperature increases, positive copper ions gain kinetic energy and vibrate with larger amplitude around their fixed positions.
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Step 3: Link to increased resistance:
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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 Change★★☆☆☆⏱ 3 min

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📘 Definition

Temperature coefficient of resistance

\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 = 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.

📐 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}).

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

Exam tip:

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

3. Practical Applications★★★☆☆⏱ 3 min

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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.

📐 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})?

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Step 1: Calculate temperature change:
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$\Delta \theta = 35 - 25 = 10^\circ \text{C}$
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Step 2: Calculate new resistance:
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$R = 10 \left(1 + (-0.04)(10)\right) = 10 (1 - 0.4) = 6 \, \text{k}\Omega$
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Step 3: Calculate percentage change:
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$\text{\% change} = \frac{6 - 10}{10} \times 100\% = -40\%$
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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.

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.