B.5.3 Resistance and Resistivity

Resistance and Resistivity

What is Resistance?

Definition

Resistance

Resistance is a measure of how much a material opposes the flow of electric current.

It determines how easily electrons can move through a conductor.

Note

The unit of resistance is the ohm (Ω).

The Formula for Resistance

Resistance is defined by the equation:

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

where:

$R$ is the resistance in ohms (Ω)
$V$ is the potential difference across the conductor in volts (V)
$I$ is the current flowing through the conductor in amperes (A)

Example

If a potential difference of 12 V is applied across a resistor and a current of 2 A flows through it, the resistance is:

$R = \frac{12 V}{2 A} = 6 Ω$

Ohm’s Law and Resistance

Definition

Ohm's law

Ohm’s Law states that the current through a conductor is directly proportional to the potential difference across it, provided the temperature remains constant.

For ohmic materials (those that obey Ohm’s Law), the resistance remains constant regardless of the current or voltage.

Example

A graph of current (I) versus voltage (V) for an ohmic material is a straight line through the origin, indicating constant resistance.

Graph for an ohmic and non-ohmic material.

Note

Not all materials obey Ohm’s Law.
For non-ohmic materials, such as filament lamps or diodes, the resistance changes with current or voltage.

Tip

Resistance in a wire originates from the collisions between free electrons and the metal atoms in the conductor.
As electrons move through the wire, they interact with the vibrating lattice of atoms, losing energy in the form of heat.
The higher the temperature or the longer and thinner the wire, the greater the resistance.

Factors Affecting Resistance

Resistance depends on several factors:

Material:
- Different materials have different abilities to conduct electricity.
- Metals like copper have low resistance, while insulators like rubber have high resistance.
Length:
- Resistance is directly proportional to the length of the conductor.
- A longer wire has more resistance.
Cross-Sectional Area:
- Resistance is inversely proportional to the cross-sectional area.
- A thicker wire has less resistance.
Temperature:
- For most conductors, resistance increases with temperature because increased atomic vibrations hinder electron flow.

Tip

When solving problems involving resistance, always check if the material is ohmic or non-ohmic, as this affects how resistance behaves.

Resistivity: A Material Property

What is Resistivity?

Definition

Resistivity

Resistivity is a fundamental property of a material that quantifies how strongly it resists the flow of electric current.

It is denoted by the symbol $ρ$ (rho) and is measured in ohm-meters (Ω·m).

Note

Resistivity is independent of the shape or size of the material and depends only on its intrinsic properties and temperature.

In contrast, that is resistance which changes in response to a change in the dimension of the wire.

The Formula for Resistivity

The relationship between resistance and resistivity is given by:

$ρ = \frac{R \cdot A}{L}$

where:

$ρ$ is the resistivity in ohm-meters (Ω·m)
$R$ is the resistance in ohms (Ω)
$A$ is the cross-sectional area of the conductor in square meters (m²)
$L$ is the length of the conductor in meters (m)

Example

A copper wire with a length of 2 m, a cross-sectional area of $1 \times 10^{- 6} m^{2}$ , and a resistance of 0.034 Ω has a resistivity of:

$ρ = \frac{0.034 Ω \cdot 1 \times 10^{- 6} m^{2}}{2 m}$
$= 1.7 \times 10^{- 8} Ω \cdot m$

Analogy

Think of resistivity as the "friction" electrons experience as they move through a material. Just as rough surfaces create more friction for a sliding object, materials with high resistivity create more opposition to electron flow.

Illustrating resistance and resistivity.

Connecting Resistance and Resistivity

The relation between resistance and resistivity can be observed from the formula given earlier:

$R = ρ \cdot \frac{L}{A}$

Example

A wire with a length of 5 m, a cross-sectional area of $2 \times 10^{- 6} m^{2}$ , and a resistivity of $1.5 \times 10^{- 8}$ Ω·m has a resistance of:

$R = 1.5 \times 10^{- 8} Ω \cdot m \cdot \frac{5 m}{2 \times 10^{- 6} m^{2}} = 0.0375 Ω$

Practical Applications

Understanding resistance and resistivity is crucial in designing electrical systems:

Wires:
- Low-resistivity materials like copper are used for electrical wiring to minimize energy loss.
Resistors:
- High-resistivity materials are used to make resistors, which control current in circuits.
Heating Elements:
- Materials with high resistivity, such as nichrome, are used in heating elements because they convert electrical energy into heat efficiently.

Reflection

Theory of Knowledge

How does the concept of resistivity connect to sustainability?
Consider how choosing materials with optimal resistivity can reduce energy loss in power transmission, contributing to more efficient energy use.

Self review

What is the resistance of a wire with a potential difference of 10 V across it and a current of 2 A flowing through it?
A wire has a length of 3 m, a cross-sectional area of $1 \times 10^{- 6}$ m², and a resistivity of $2 \times 10^{- 8}$ Ω·m. What is its resistance?
How does increasing the length of a wire affect its resistance? What about increasing its cross-sectional area?

Summary

Resistance is the opposition to the flow of electric current, defined as $R = \frac{V}{I}$ .
Resistivity is a material property that determines resistance, expressed as $ρ = \frac{R \cdot A}{L}$ .
Resistance depends on the material, length, cross-sectional area, and temperature of the conductor.
Understanding these concepts is essential for designing efficient electrical systems.

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