Resistivity and Materials

Overview

The resistance of a conductor depends on:

• the material used
• its length
• its cross-sectional area
• temperature

To compare materials fairly, we use resistivity, an intrinsic material property.

This page deepens ideas introduced in Current Electricity Fundamentals.

Related topics:

• I-V Characteristics
• DC Circuits
• Current Electricity Common Exam Traps

Definition

Resistance $R$

Resistance is the opposition a specific object offers to current flow.

It depends on:

• material
• length
• cross-sectional area
• temperature

Unit:

$$\Omega$$

Resistivity $\rho$

Resistivity is a property of the material itself (at a given temperature).

It allows comparison between materials independent of size and shape.

Unit:

$$\Omega \text{ m}$$

Resistivity is treated as a scalar quantity in this syllabus context.

Why It Matters

Resistivity links microscopic material behaviour to observable circuit resistance. It explains why different materials are chosen for wires, resistors, sensors, and heating elements.

Key Representations

Core Relationship

For a uniform conductor:

$$R=\rho \frac{l}{A}$$

Where:

• $R$ = resistance
• $\rho$ = resistivity
• $l$ = length
• $A$ = cross-sectional area

Geometry Effects

Effect of Length

$$R\propto l$$

If the conductor is longer:

• charge carriers travel further
• more collisions occur
• resistance increases

Effect of Cross-Sectional Area

$$R\propto \frac{1}{A}$$

If the conductor is thicker:

• more parallel conducting paths exist
• resistance decreases

Visual Comparison

Wire Type	Resistance
Long thin wire	High
Short thick wire	Low
Same size, higher $\rho$ material	Higher

Material Dependence

Different materials have different resistivities because of their atomic structure and available charge carriers.

Low Resistivity Materials

Good conductors:

• copper
• silver
• aluminium

Used for wiring.

Moderate Resistivity Materials

Useful for heating elements or resistors:

• nichrome
• manganin

High Resistivity Materials

Insulators:

• rubber
• glass
• plastic

Temperature Effects

Metals

For metallic conductors:

• resistance usually increases when temperature rises

Reason:

• stronger lattice vibration
• more collisions with electrons

So:

$$T\uparrow \Rightarrow R\uparrow$$

Semiconductors

For semiconductors:

• resistance usually decreases when temperature rises

Reason:

• more charge carriers become available

So:

$$T\uparrow \Rightarrow R\downarrow$$

See Thermistors and LDRs.

Microscopic Interpretation

Current depends on:

• number of mobile charge carriers
• ease of movement through lattice
• frequency of collisions

Metals

Many free electrons already exist. Temperature mainly increases scattering.

Semiconductors

Heating can greatly increase carrier density, reducing resistance.

Measuring Resistivity Experimentally

Apparatus

• wire sample
• metre rule
• micrometer screw gauge / calipers
• ammeter
• voltmeter
• power supply
• variable resistor

Method

Measure:

• length $l$
• diameter $d$
• current $I$
• voltage $V$

Then:

$$R=\frac{V}{I}$$

Cross-sectional area:

$$A=\frac{\pi d^2}{4}$$

Hence:

$$\rho =\frac{RA}{l}$$

Graph Method

From:

$$R=\rho \frac{l}{A}$$

If material and area remain constant:

$$R\propto l$$

So plot:

• vertical axis: $R$
• horizontal axis: $l$

Straight line through origin.

Gradient:

$$\frac{R}{l}=\frac{\rho }{A}$$

Thus:

$$\rho =(\text{gradient})A$$

Worked Examples

Example 1: Change of Length

A wire is stretched to twice its original length with constant volume.

Then:

• length doubles
• area halves

So:

$$R=\rho \frac{l}{A}$$

New resistance becomes:

$$4R$$

Example 2: Finding Resistivity

A wire has:

• $R=2.0\Omega$
• $l=5.0\text{ m}$
• $A=1.0\times 10^{-6}{\text{ m}}^2$

$$\rho =\frac{RA}{l}=\frac{(2.0)(1.0\times 10^{-6})}{5.0}=4.0\times 10^{-7}\Omega \text{ m}$$

Common Exam Pitfalls

1. Confusing Resistance with Resistivity

Resistance depends on dimensions. Resistivity is a material property.

2. Using Diameter as Area

Wrong:

$$A=d$$

Correct:

$$A=\frac{\pi d^2}{4}$$

3. Forgetting SI Units

Resistivity unit:

$$\Omega \text{ m}$$

4. Ignoring Temperature Changes

Heating during experiment may change resistance.

5. Wrong Proportional Reasoning

Doubling length alone doubles resistance only if area unchanged.

For a compact revision warning sheet, see:

Current Electricity Common Exam Traps

Quick Revision Summary

• Resistance depends on both material and geometry.
• Resistivity compares materials.
• Longer wire → larger resistance.
• Thicker wire → smaller resistance.
• Metals: resistance rises with temperature.
• Semiconductors: resistance often falls with temperature.

Summary Formula Table

Quantity	Formula
Resistance law	$R=\rho \frac{l}{A}$
Resistance from measurements	$R=\frac{V}{I}$
Area from diameter	$A=\frac{\pi d^2}{4}$
Resistivity rearranged	$\rho =\frac{RA}{l}$
Length dependence	$R\propto l$
Area dependence	$R\propto \frac{1}{A}$

Related Links

• Current Electricity Fundamentals
• I-V Characteristics
• Internal Resistance
• Electrical Power and Ratings
• Current Electricity Common Exam Traps
• DC Circuits
• Thermistors and LDRs

Links

• Related: Current Electricity Fundamentals
• Related: I-V Characteristics
• Related: DC Circuits
• Related: Current Electricity Common Exam Traps