Resistivity and Materials

Branch note: This page deepens one part of Current Electricity Fundamentals.

Overview

This note explains why conductors made from different materials or with different dimensions can have different resistance.

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

Core Ideas

• Resistance belongs to a specific object; resistivity belongs to the material.
• For a uniform conductor, $R=\rho l/A$.
• Longer conductors have larger resistance, while conductors with larger cross-sectional area have smaller resistance.
• Resistivity depends on material and temperature.

Exam Relevance

Resistivity questions often combine measurements of length, diameter, voltage, and current. The common risk is forgetting that cross-sectional area depends on diameter squared.

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 made of one material, with uniform cross-section and fixed temperature,

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

Where:

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

Figure: For the same material and temperature, compare length while holding $A$ constant, and compare area while holding $l$ constant. A longer uniform conductor has greater resistance; a larger cross-sectional area provides more parallel conducting paths and gives smaller resistance.

Geometry Effects

If the conductor is longer,

$$R\propto l$$

because adding length adds more scattering region in series, so a larger p.d. is required to maintain the same current.

If the conductor is thicker,

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

because there are more parallel paths for current flow.

For a circular wire,

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

The square matters: doubling diameter makes $A$ four times larger and, with $\rho$ and $l$ unchanged, makes $R$ four times smaller.

If a wire is stretched without changing its volume, $Al$ is constant. Hence $A\propto 1/l$ and, if $\rho$ remains unchanged,

$$R=\rho \frac{l}{A}\propto l^2.$$

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

Practical enrichment

The measurement method below supports practical reasoning but goes beyond the minimum conceptual statement of $R=\rho l/A$.

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

A useful graph method is to measure $R$ for several lengths at constant temperature. A graph of $R$ against $l$ has gradient $\rho /A$, so $\rho =A\times \text{gradient}$.

Worked Example

A wire has length $1.5\mathrm{m}$, diameter $0.40\mathrm{mm}$, and resistance $0.20\Omega$.

$$A=\frac{\pi (0.40\times 10^{-3}{)}^2}{4}=1.26\times 10^{-7}{\mathrm{m}}^2$$ $$\rho =\frac{RA}{l}=\frac{(0.20)(1.26\times 10^{-7})}{1.5}=1.7\times 10^{-8}\Omega \mathrm{m}.$$

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