Resistivity and Materials

Overview

This note supports the Topic 13 hub by explaining why two conductors with the same voltage and current can still behave differently.

The core idea is that resistance belongs to a specific object, while resistivity belongs to the material.

Related topics:

• I-V Characteristics
• 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 material property at a given temperature.

It allows fair comparison between materials independent of size and shape.

Unit:

$$\Omega \text{ m}$$

Why It Matters

Resistivity links microscopic material structure to macroscopic circuit resistance.

It explains why different materials are used for:

• wires
• resistors
• heating elements
• sensors

Key Representations

For a uniform conductor,

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

Where:

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

Geometry Effects

If the conductor is longer,

$$R\propto l$$

because charge carriers travel farther and undergo more collisions.

If the conductor is thicker,

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

because there are more parallel paths for current flow.

Quick 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

These are used for wiring.

Moderate Resistivity Materials

Useful for resistors and heating elements:

• nichrome
• manganin

High Resistivity Materials

Insulators:

• rubber
• glass
• plastic

Temperature Effects

Metals

For metallic conductors,

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

The lattice vibrations become stronger, so electrons experience more collisions.

Semiconductors

For semiconductors,

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

More charge carriers become available as temperature rises.

This is the material link behind thermistors in I-V Characteristics.

Measuring Resistivity

To measure a wire’s resistivity:

• measure length $l$
• measure diameter and calculate cross-sectional area $A$
• measure voltage $V$ and current $I$
• calculate resistance using $R=V/I$
• then use $R=\rho l/A$ to find $\rho$

Good Practice

• take diameter measurements carefully; the area depends on the square of the diameter
• use a short current pulse if heating would noticeably change the resistance
• keep the wire temperature as steady as possible

Quick Checks

• Resistance is for the object.
• Resistivity is for the material.
• Longer wire means larger resistance.
• Thicker wire means smaller resistance.
• Hot metal usually has larger resistance; hot semiconductor usually has smaller resistance.

Links

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