# Resistance and resistivity > CIE A-Level Physics · 9702 > Source: https://www.owlsprep.com/study/cie-9702-u9-resistance-and-resistivity/ This subtopic explores resistance and resistivity, their relationship, temperature dependence, and experimental methods to measure resistivity. These concepts are core to all circuit problems in both multiple choice and structured questions. **Prerequisites:** [Electric current and potential difference](https://www.owlsprep.com/study/cie-9702-u9-electric-current-potential-difference/) ## Learning objectives - Define resistance and resistivity and state their correct units - Use the formula $R = \rho L / A$ to solve calculation problems - Explain how temperature affects resistance for metals and semiconductors - Describe the experiment to determine resistivity of a metal wire ## Resistance and Ohm's Law **Resistance** — The ratio of the potential difference across a component to the current flowing through it, given by: $R = \frac{V}{I}$. Measured in ohms ($\Omega$). *Notation:* R *Example:* A 5 $\Omega$ resistor with 10 V across it carries a current of 2 A. > **info** > > Ohm's law states that for an ohmic conductor at constant temperature, $V \propto I$, so resistance is constant. Non-ohmic conductors (filament lamps, diodes) do not obey Ohm's law, so their resistance changes with current. **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. $$R = \frac{12}{0.25} = 48 \; \Omega$$ ## Resistivity and the $R = \rho L / A$ formula **Resistivity** — 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: *Notation:* $\rho$ *Example:* Copper has low resistivity ($\approx 1.7 \times 10^{-8} \Omega \cdot \text{m}$), 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 $1.7 \times 10^{-8} \Omega \cdot \text{m}$. 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. $$R = \frac{\rho L}{A} = \frac{(1.7 \times 10^{-8})(2.0)}{1.96 \times 10^{-7}} \approx 0.17 \; \Omega$$ **Check your understanding** Check your understanding of the difference between resistance and resistivity: 1. 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 *Answer:* Resistivity is constant for a material at constant temperature *Why:* Correct! Resistivity is an intrinsic property, so it does not depend on the dimensions of the wire, only on the material and temperature. ## Temperature dependence of resistance 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. > **info** > > NTC thermistors are used as temperature sensors in circuits because of their predictable change in resistance with temperature. ## Experiment to determine resistivity of a wire CIE frequently asks practical questions about measuring resistivity of a metal wire. The standard method is: 1. Measure the total length $L$ of the wire using a meter rule. 2. Measure the diameter $d$ at 3 different points along the wire using a micrometer screw gauge, calculate the average diameter. 3. Connect the wire in series with a battery, ammeter and variable resistor, add a voltmeter in parallel across the wire. 4. Vary the length of the wire, record $V$ and $I$ for each length, calculate $R = V/I$ for each length. 5. Plot a graph of $R$ against $L$. Gradient = $\rho / A$, so $\rho = \text{gradient} \times A$. > **Exam tip:** Always state that you need to average the diameter measurement to reduce random error from the micrometer. ## Common pitfalls - **Wrong:** Forgetting to convert diameter/area units to meters before calculation - Why it fails: Diameter is usually measured in mm, leaving it in mm gives an answer 10⁶ times incorrect - Correct: Convert all length measurements to meters before calculating area and substituting into the resistivity formula - **Wrong:** Confusing resistance and resistivity, claiming copper has very low resistance - Why it fails: Copper has low resistivity (material property), but a long thin copper wire can have high resistance - Correct: Remember resistivity is intrinsic to the material, resistance depends on both material and dimensions - **Wrong:** Stating all materials have increasing resistance with increasing temperature - Why it fails: This only applies to metals, not semiconductors - Correct: Distinguish: metals → resistance increases with T; semiconductors/NTC thermistors → resistance decreases with T - **Wrong:** Squaring diameter instead of radius when calculating cross-sectional area - Why it fails: Common arithmetic error when working from diameter measurements - Correct: Always divide diameter by 2 to get radius before calculating $A = \pi r^2$ - **Wrong:** Claiming ohmic conductors have constant resistance regardless of temperature - Why it fails: Ohm's law only holds at constant temperature; resistivity changes with temperature for all materials - Correct: Remember resistance of ohmic conductors only stays constant when temperature is constant ## Cheatsheet | Quantity/Type | Key Relationship | Unit | Key Property | | --- | --- | --- | --- | | Resistance | $R = \frac{V}{I}$ | $\Omega$ | Depends on material, dimensions, T | | Resistivity | $R = \frac{\rho L}{A}$ | $\Omega \cdot \text{m}$ | Intrinsic material property, depends only on material and T | | Metals | $\rho \propto T$ | - | Resistance increases with temperature | | NTC Thermistors | $\rho \propto 1/T$ | - | Resistance decreases with temperature | ## 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. - [I-V Characteristics](https://www.owlsprep.com/study/cie-9702-u9-i-v-characteristics/) - [Temperature dependence of resistance](https://www.owlsprep.com/study/cie-9702-u9-temperature-dependence-of-resistance/) - [D.C. circuits](https://www.owlsprep.com/study/cie-9702-u10-overview/) --- From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-9702-u9-resistance-and-resistivity/