11.17 Variation of Resistance with Temperature | Electricity | Physics 11

The resistance of a material is a measure of its opposition to the flow of electric current. This property is not constant; it is significantly influenced by the material's temperature. Understanding this relationship is crucial for designing and operating electrical and electronic components in various thermal environments.

Graph showing resistance increasing with temperature for a metal.

Temperature Coefficient of Resistance

For many materials, especially metals, the change in resistance with temperature is approximately linear over a certain range. This relationship is described by the temperature coefficient of resistance ( $α$ ).

Definition: The temperature coefficient of resistance is the fractional change in resistance per unit change in temperature: $α = \frac{R _{T} - R _{0}}{R _{0} Δ T}$

The formula to calculate the resistance ( $R_{T}$ ) at a certain temperature ( $T$ ) is: $R_{T} = R_{0} [1 + α (T - T_{0})]$

Where:

$R_{T}$ is the resistance at the final temperature $T$ .
$R_{0}$ is the resistance at a reference temperature $T_{0}$ (often $0^{\circ} C$ or $2 0^{\circ} C$ ).
$α$ is the temperature coefficient of resistance.
SI unit of $α$ : per Kelvin ( $K^{- 1}$ ), also written as $^{\circ} C^{- 1}$ .

Behavior of Different Material Types

The effect of temperature on resistance varies greatly depending on the type of material.

Material Type	Resistance Behavior with Increasing Temperature	Reason
Metals (Conductors)	Increases (Positive $α$ )	Increased thermal vibrations of lattice atoms cause more frequent collisions with free electrons, impeding their flow.
Semiconductors	Decreases (Negative $α$ )	Higher temperature frees more charge carriers (electrons and holes), increasing conductivity.
Insulators	Very high; decreases slightly	Similar mechanism to semiconductors but requires much more energy to free charge carriers.
Electrolytes	Decreases	Higher temperature increases ion mobility in solution, allowing easier current flow.
Alloys	Increases very slightly	Designed to have a very low $α$ , making them stable for precision resistors.

Resistance and Resistivity

The resistance ( $R$ ) of a specific conductor depends on both its material and its physical dimensions.

$R = ρ \frac{L}{A}$

Where:

$L$ = length of the conductor (resistance is directly proportional to $L$ )
$A$ = cross-sectional area (resistance is inversely proportional to $A$ )
$ρ$ = resistivity of the material

Resistivity ( $ρ$ )

An intrinsic property of a material quantifying how strongly it resists current flow.
Unit: Ohm-metre ( $Ω \cdot m$ )
Resistivity itself varies with temperature, similar to resistance.

Conductivity ( $σ$ )

The reciprocal of resistivity: $σ = \frac{1}{ρ}$
Measures how well a material conducts electricity.
Unit: Siemens per metre (S/m)

Example Resistivity Values at 20°C:

Material	Resistivity ( $ρ$ ) ( $Ω \cdot m$ )	Classification
Copper	$1.7 \times 1 0^{- 8}$	Conductor
Silicon	$\approx 2.3 \times 1 0^{3}$	Semiconductor
Quartz (Fused)	$\approx 5 \times 1 0^{16}$	Insulator

Special Cases

Superconductors: Materials whose resistance drops to exactly zero below a critical temperature. In this state, they conduct electricity with no energy loss.

Thermistors: Semiconductor-based resistors with a large and predictable change in resistance with temperature, used in temperature sensors and control circuits.

Temperature Coefficient of Resistance

Definition: The temperature coefficient of resistance is the fractional change in resistance per unit change in temperature: $α = \frac{R _{T} - R _{0}}{R _{0} Δ T}$

The formula to calculate the resistance ( $R_{T}$ ) at a certain temperature ( $T$ ) is: $R_{T} = R_{0} [1 + α (T - T_{0})]$

Where:

$R_{T}$ is the resistance at the final temperature $T$ .
$R_{0}$ is the resistance at a reference temperature $T_{0}$ (often $0^{\circ} C$ or $2 0^{\circ} C$ ).
$α$ is the temperature coefficient of resistance.
SI unit of $α$ : per Kelvin ( $K^{- 1}$ ), also written as $^{\circ} C^{- 1}$ .

Behavior of Different Material Types

The effect of temperature on resistance varies greatly depending on the type of material.

Material Type	Resistance Behavior with Increasing Temperature	Reason
Metals (Conductors)	Increases (Positive $α$ )	Increased thermal vibrations of lattice atoms cause more frequent collisions with free electrons, impeding their flow.
Semiconductors	Decreases (Negative $α$ )	Higher temperature frees more charge carriers (electrons and holes), increasing conductivity.
Insulators	Very high; decreases slightly	Similar mechanism to semiconductors but requires much more energy to free charge carriers.
Electrolytes	Decreases	Higher temperature increases ion mobility in solution, allowing easier current flow.
Alloys	Increases very slightly	Designed to have a very low $α$ , making them stable for precision resistors.

Resistance and Resistivity

The resistance ( $R$ ) of a specific conductor depends on both its material and its physical dimensions.

$R = ρ \frac{L}{A}$

Where:

$L$ = length of the conductor (resistance is directly proportional to $L$ )
$A$ = cross-sectional area (resistance is inversely proportional to $A$ )
$ρ$ = resistivity of the material

Resistivity ( $ρ$ )

An intrinsic property of a material quantifying how strongly it resists current flow.
Unit: Ohm-metre ( $Ω \cdot m$ )
Resistivity itself varies with temperature, similar to resistance.

Conductivity ( $σ$ )

The reciprocal of resistivity: $σ = \frac{1}{ρ}$
Measures how well a material conducts electricity.
Unit: Siemens per metre (S/m)

Example Resistivity Values at 20°C:

Material	Resistivity ( $ρ$ ) ( $Ω \cdot m$ )	Classification
Copper	$1.7 \times 1 0^{- 8}$	Conductor
Silicon	$\approx 2.3 \times 1 0^{3}$	Semiconductor
Quartz (Fused)	$\approx 5 \times 1 0^{16}$	Insulator

Special Cases

Superconductors: Materials whose resistance drops to exactly zero below a critical temperature. In this state, they conduct electricity with no energy loss.

Thermistors: Semiconductor-based resistors with a large and predictable change in resistance with temperature, used in temperature sensors and control circuits.