Resistance and resistivity

CIE A-Level Physics· 40 min read

1. Resistance and Ohm's Law★★☆☆☆⏱ 10 min

📘 Definition

Resistance

R

The ratio of the potential difference across a component to the current flowing through it, given by: $R = \frac{V}{I}$ . Measured in ohms ( $Ω$ ).

Example:

A 5 $Ω$ resistor with 10 V across it carries a current of 2 A.

📐 Worked Example

A filament lamp carries a current of 0.25 A when 12 V is applied across it. Calculate its resistance.

1
Recall the definition of resistance:
2
$R = \frac{V}{I}$
3
Substitute the given values $V = 12$ V, $I = 0.25$ A:
4
$R = \frac{12}{0.25} = 48 \; \Omega$

2. Resistivity and the $R = \rho L / A$ formula★★★☆☆⏱ 15 min

📘 Definition

Resistivity

$ρ$

An intrinsic material property that describes opposition to current flow. It is independent of the dimensions of the material, only depends on material type and temperature. The relationship between resistance and resistivity is:

Example:

Copper has low resistivity ( $\approx 1.7 \times 1 0^{- 8} Ω \cdot m$ ), making it ideal for electrical wiring.

R = \frac{\rho L}{A}

📐 Worked Example

A copper wire of length 2.0 m, diameter 0.50 mm has resistivity $1.7 \times 1 0^{- 8} Ω \cdot m$ . Calculate its resistance.

1
Convert diameter to meters and calculate cross-sectional area $A$ :
2
$d = 0.50 \; \text{mm} = 0.50 \times 10^{-3} \; \text{m}, \; r = 0.25 \times 10^{-3} \; \text{m} \\ A = \pi r^2 = \pi (0.25 \times 10^{-3})^2 \approx 1.96 \times 10^{-7} \; \text{m}^2$
3
Substitute into the resistivity formula:
4
$R = \frac{\rho L}{A} = \frac{(1.7 \times 10^{-8})(2.0)}{1.96 \times 10^{-7}} \approx 0.17 \; \Omega$

✓ Quick check

Check your understanding of the difference between resistance and resistivity:

Which of the following statements is true?
- Resistance is an intrinsic material property
- Resistivity depends on the length of the wire
- Resistivity is constant for a material at constant temperature
- Resistance does not depend on temperature
Reveal answer
2 —
Correct! Resistivity is an intrinsic property, so it does not depend on the dimensions of the wire, only on the material and temperature.

3. Temperature dependence of resistance★★★☆☆⏱ 12 min

Resistivity changes with temperature, and the effect is different for metals and semiconductors:

Metals: Increasing temperature increases ion vibration, leading to more frequent collisions between free electrons and ions. This increases resistivity, so resistance increases with temperature.
Semiconductors (e.g. NTC thermistors): Increasing temperature releases more free charge carriers. This effect outweighs increased collisions, so resistivity and resistance decrease with increasing temperature.

📐 Worked Example

An NTC thermistor is connected to a constant voltage battery. Explain what happens to the current as temperature increases.

1
NTC stands for Negative Temperature Coefficient, meaning the thermistor's resistance decreases as temperature increases.
2
From Ohm's law, current is given by $I = \frac{V}{R}$ , and $V$ is constant here.
3
As $R$ decreases, the value of the current $I$ increases.

4. Experiment to determine resistivity of a wire★★★★☆⏱ 15 min

CIE frequently asks practical questions about measuring resistivity of a metal wire. The standard method is:

Measure the total length $L$ of the wire using a meter rule.
Measure the diameter $d$ at 3 different points along the wire using a micrometer screw gauge, calculate the average diameter.
Connect the wire in series with a battery, ammeter and variable resistor, add a voltmeter in parallel across the wire.
Vary the length of the wire, record $V$ and $I$ for each length, calculate $R = V / I$ for each length.
Plot a graph of $R$ against $L$ . Gradient = $ρ / A$ , so $ρ = gradient \times A$ .

Exam tip:

Always state that you need to average the diameter measurement to reduce random error from the micrometer.

5. Common Pitfalls

Wrong move:

Forgetting to convert diameter/area units to meters before calculation

Why:

Diameter is usually measured in mm, leaving it in mm gives an answer 10⁶ times incorrect

Correct move:

Convert all length measurements to meters before calculating area and substituting into the resistivity formula

Wrong move:

Confusing resistance and resistivity, claiming copper has very low resistance

Why:

Copper has low resistivity (material property), but a long thin copper wire can have high resistance

Correct move:

Remember resistivity is intrinsic to the material, resistance depends on both material and dimensions

Wrong move:

Stating all materials have increasing resistance with increasing temperature

Why:

This only applies to metals, not semiconductors

Correct move:

Distinguish: metals → resistance increases with T; semiconductors/NTC thermistors → resistance decreases with T

Wrong move:

Squaring diameter instead of radius when calculating cross-sectional area

Why:

Common arithmetic error when working from diameter measurements

Correct move:

Always divide diameter by 2 to get radius before calculating $A = π r^{2}$

Wrong move:

Claiming ohmic conductors have constant resistance regardless of temperature

Why:

Ohm's law only holds at constant temperature; resistivity changes with temperature for all materials

Correct move:

Remember resistance of ohmic conductors only stays constant when temperature is constant

6. Quick Reference Cheatsheet

Quantity/Type	Key Relationship	Unit	Key Property
Resistance	$R = \frac{V}{I}$	$Ω$	Depends on material, dimensions, T
Resistivity	$R = \frac{ρ L}{A}$	$Ω \cdot m$	Intrinsic material property, depends only on material and T
Metals	$ρ \propto T$		Resistance increases with temperature
NTC Thermistors	$ρ \propto 1/ T$		Resistance decreases with temperature

When this came up on past exams

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

2023 · P2
Resistivity calculation problem
2022 · P1
Temperature dependence of resistance
2021 · P1
Resistance of wire calculation

Going deeper

syllabusCIE 9702 Current of electricity syllabus

What's Next

Resistance and resistivity are the foundation for all circuit analysis in A-Level Physics. The calculation skills you learned here will be used repeatedly when solving problems for series and parallel circuits, potential dividers, and practical circuit questions. Understanding temperature dependence of resistance is also key to explaining the behaviour of common components like thermistors and light-dependent resistors, which are frequent exam topics. Mastery of this subtopic will make all subsequent electricity topics much easier to understand.