Mobility And Temperature Dependence Of Resistivity- Current And Electricity

Mobility and Temperature Dependence of Resistivity

Resistivity of a material determines its ability to resist the flow of electric current.
The resistivity of a material is dependent on various factors, including temperature.
In this lecture, we will discuss the concept of mobility and the temperature dependence of resistivity.

Electrical Mobility

Electrical mobility is a measure of how easily charges can move through a material under the influence of an electric field.
It is denoted by the symbol μ and is expressed in square meters per volt-second (m²/Vs).
The mobility of charges is affected by factors such as impurities, crystal structure, and temperature.

Drift Velocity

The drift velocity is the average velocity with which charges (electrons or holes) move in a material when subjected to an electric field.
It is denoted by the symbol v_d and is expressed in meters per second (m/s).
The drift velocity is related to the mobility of charges through the equation v_d = μE, where E is the electric field strength.

Temperature Dependence of Resistivity

The resistivity of a material typically increases with an increase in temperature.
This increase in resistivity is due to the increased scattering of charges by lattice vibrations, impurities, and other imperfections in the material.
The relationship between resistivity and temperature can be expressed using the equation ρ = ρ₀(1 + α(T - T₀)), where ρ₀ is the resistivity at a reference temperature T₀, α is the temperature coefficient of resistivity, and T is the absolute temperature.

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity (α) is a measure of how much the resistivity of a material changes with a change in temperature.
It is defined as the fractional change in resistivity per degree Celsius or Kelvin.
The temperature coefficient of resistivity can be positive, negative, or zero depending on the type of material.

Positive Temperature Coefficient

Materials with a positive temperature coefficient of resistivity exhibit an increase in resistivity with an increase in temperature.
Examples include most metals like copper, silver, and gold.
These materials show increased scattering of charges at higher temperatures due to increased lattice vibrations.

Negative Temperature Coefficient

Materials with a negative temperature coefficient of resistivity exhibit a decrease in resistivity with an increase in temperature.
Examples include semiconductors like silicon and germanium.
In semiconductors, the increased thermal energy at higher temperatures leads to more charge carriers, hence lowering the resistivity.

Zero Temperature Coefficient

Some materials have a zero temperature coefficient of resistivity, which means their resistivity remains constant over a wide temperature range.
One such material is a superconductor, which exhibits zero resistivity at or below a critical temperature.
Superconductors have unique properties that allow for the nearly perfect flow of electric current.

Application: Temperature Sensors

The temperature dependence of resistivity is commonly employed in the design of temperature sensors.
Resistance thermometers, or RTDs, use the temperature coefficient of resistivity to measure temperature accurately.
The change in resistivity with temperature can be calibrated to provide a precise and reliable temperature reading.

Summary

Mobility is a measure of how easily charges can move through a material under the influence of an electric field.
The mobility of charges influences the drift velocity and the overall resistivity of a material.
Resistivity typically increases with an increase in temperature due to increased scattering of charges.
The temperature coefficient of resistivity is a measure of the change in resistivity with a change in temperature.
Materials can have positive, negative, or zero temperature coefficients of resistivity.

Mobility and Temperature Dependence of Resistivity

Current and Electricity
Problem on Mobility

Mobility and Electric Field

Electrical mobility is the measure of how easily charges can move through a material under the influence of an electric field.
Mobility is given by the formula μ = v_d / E, where v_d is the drift velocity and E is the electric field strength.
It is easier for charges to move through a material with higher mobility.

Factors Affecting Mobility

Impurities: Impurities in a material can increase scattering of charges, reducing mobility.
Crystal Structure: The crystal structure of a material can influence the movement of charges.
Temperature: Temperature affects the mobility of charges, as discussed earlier.

Mobility and Conductivity

Conductivity is a measure of how well a material can conduct electric current.
It is defined as the reciprocal of resistivity, given by the formula σ = 1 / ρ.
Materials with higher mobility have higher conductivity, allowing for better flow of electric current.

Temperature Dependence of Resistivity

As discussed earlier, resistivity generally increases with temperature.
This is because higher temperatures lead to increased scattering of charges.
The relationship between resistivity and temperature is given by the equation ρ = ρ₀(1 + α(T - T₀)), where ρ₀ is the resistivity at a reference temperature T₀ and α is the temperature coefficient of resistivity.

Positive Temperature Coefficient (PTC) Materials

Materials with a positive temperature coefficient of resistivity (PTC) exhibit increased resistivity with increasing temperature.
Examples include most metals such as copper, silver, and gold.
At higher temperatures, lattice vibrations increase, leading to more scattering of charges, hence higher resistivity.

Negative Temperature Coefficient (NTC) Materials

Materials with a negative temperature coefficient of resistivity (NTC) show a decrease in resistivity with increasing temperature.
Examples include certain semiconductors like silicon and germanium.
In these materials, higher temperatures create more charge carriers, resulting in lower resistivity.

Zero Temperature Coefficient (ZTC) Materials

Materials with a zero temperature coefficient of resistivity (ZTC) have constant resistivity over a wide temperature range.
One example is a superconductor, which has zero resistivity at or below a critical temperature.
Superconductors offer almost perfect flow of electric current due to their unique properties.

Application: Temperature Sensors

The temperature dependence of resistivity is utilized in the design of temperature sensors.
Resistance thermometers, or RTDs, use the temperature coefficient of resistivity to measure temperature accurately.
The change in resistivity with temperature can be calibrated to provide precise and reliable temperature readings.

Summary

Mobility measures how easily charges move through a material under an electric field.
Mobility is influenced by factors like impurities, crystal structure, and temperature.
Resistivity generally increases with temperature due to increased scattering of charges.
Different materials exhibit positive, negative, or zero temperature coefficients of resistivity.
The temperature dependence of resistivity finds applications in temperature sensors.

Mobility and Temperature Dependence of Resistivity

Current and Electricity

Electric current is the flow of electric charge in a conductor.
It is caused by the movement of electrons or positive charge carriers.
The flow of electric charge requires the presence of an electric field.

Problem on Mobility

A copper wire of length 2m and cross-sectional area 2mm^2 has a resistance of 5Ω.
Calculate the mobility of charges in the copper wire if an electric field of 10V/m is applied.

Mobility and Electric Field

Electrical mobility (μ) is a measure of how easily charges move through a material under the influence of an electric field.
It is defined as the ratio of the drift velocity (v_d) to the electric field strength (E).
Mathematically, μ = v_d / E.

Example

Consider a material where the drift velocity is 0.5 m/s under an electric field of 1V/m.
Calculate the electrical mobility of the charges in the material.

Factors Affecting Mobility

Impurities: The presence of impurities in a material can lead to increased scattering of charges, reducing the mobility.
Crystal Structure: The arrangement and symmetry of atoms in the crystal lattice can affect the mobility of charges.
Temperature: The mobility of charges is also influenced by temperature, as discussed in subsequent slides.

Example

A semiconductor material has a higher concentration of impurities compared to a pure crystal.
Predict the effect on the mobility of charges in the semiconductor material.

Mobility and Conductivity

Conductivity (σ) is a measure of how well a material can conduct electric current.
It is the reciprocal of resistivity (ρ), given by the equation σ = 1 / ρ.
Materials with higher mobility exhibit higher conductivity.

Example

Consider two materials X and Y having equal resistivities.
If material X has a higher mobility than material Y, which material will have higher conductivity?

Temperature Dependence of Resistivity

Resistivity (ρ) is the inherent property of a material that determines its ability to resist the flow of electric current.
It typically increases with an increase in temperature.
This effect is due to increased scattering of charges by lattice vibrations and impurities.

Example

A wire made of a metal has a resistivity of 2Ω m at 300K.
If the resistivity at 500K is 2.5Ω m, calculate the temperature coefficient of resistivity (α).

Positive Temperature Coefficient (PTC) Materials

Materials with a positive temperature coefficient of resistivity (PTC) exhibit an increase in resistivity with increasing temperature.
Most metals, such as copper, silver, and gold, demonstrate PTC behavior.
At higher temperatures, lattice vibrations increase, leading to more scattering of charges and hence higher resistivity.

Example

A copper wire has a resistivity of 1.5Ω m at 300K.
What will be the resistivity of the wire at 400K, considering a positive temperature coefficient of 0.0035 K^-1?

Negative Temperature Coefficient (NTC) Materials

Materials with a negative temperature coefficient of resistivity (NTC) exhibit a decrease in resistivity with increasing temperature.
Certain semiconductors, like silicon and germanium, demonstrate NTC behavior.
At higher temperatures, more charge carriers are generated due to increased thermal energy, resulting in lower resistivity.

Example

A silicon semiconductor has a resistivity of 5Ω m at 300K.
If the resistivity decreases to 4Ω m at 500K, calculate the temperature coefficient of resistivity (α).

Zero Temperature Coefficient (ZTC) Materials

Materials with a zero temperature coefficient of resistivity (ZTC) have a constant resistivity over a wide temperature range.
One example of such a material is a superconductor, which exhibits zero resistivity at or below a critical temperature.
Superconductors possess unique properties that allow for almost perfect flow of electric current.

Example

A superconductor wire has a resistivity of zero at 15K.
If the wire is subjected to a temperature of 20K, what will be the resistivity of the wire?

Application: Temperature Sensors

The temperature dependence of resistivity finds applications in the design of temperature sensors.
Resistance thermometers, or RTDs, utilize the temperature coefficient of resistivity to accurately measure temperature.
The change in resistivity with temperature can be calibrated to provide precise and reliable temperature readings.

Example

An RTD made of platinum has a temperature coefficient of resistivity of 0.00392 K^-1.
If the resistance of the RTD at 25°C is 100Ω, calculate its resistance at 75°C.

Summary

Mobility measures how easily charges move through a material under an electric field.
Mobility depends on factors like impurities, crystal structure, and temperature.
Resistivity generally increases with temperature due to increased scattering of charges.
Materials can exhibit positive, negative, or zero temperature coefficients of resistivity.
The temperature dependence of resistivity finds applications in temperature sensors.