Mobility And Temperature Dependence Of Resistivity- Current And Electricity

Slide 1: Mobility and Temperature Dependence of Resistivity

Resistivity ($\rho$) is a measure of how strongly a material opposes the flow of electric current.
Mobility ($\mu$) is a property of charge carriers in a material that determines their ability to move under the influence of an electric field.
The resistivity of a material is related to its mobility through the formula:
- $\rho = \frac{m}{ne^2\mu}$
- where $m$ is the mass of the charge carriers, $n$ is the number density of charge carriers, and $e$ is the charge of an electron.

Slide 2: Temperature Dependence of Resistivity

In most materials, the resistivity increases with an increase in temperature.
This can be explained by the scattering of charge carriers by lattice vibrations.
As the temperature increases, the lattice vibrations increase, leading to more scattering of charge carriers and a higher resistivity.
The temperature dependence of resistivity can be approximated by the equation:
- $\rho(T) = \rho_0(1 + \alpha(T-T_0))$
- where $\rho_0$ is the resistivity at a reference temperature $T_0$, and $\alpha$ is the temperature coefficient of resistivity.

Slide 3: Temperature Coefficient of Resistivity

The temperature coefficient of resistivity, $\alpha$, is a measure of how much the resistivity of a material changes with temperature.
It is given by the equation:
- $\alpha = \frac{1}{\rho_0}\frac{d\rho}{dT}$
Different materials have different temperature coefficients of resistivity.
For metallic conductors, $\alpha$ is typically positive, indicating an increase in resistivity with temperature.
For semiconductors, $\alpha$ can be positive or negative, depending on the type of semiconductor and the temperature range.

Slide 4: Mobility of Charge Carriers

The mobility of charge carriers in a material determines how easily they can move under the influence of an electric field.
It is given by the equation:
- $\mu = \frac{e\tau}{m}$
- where $e$ is the charge of an electron, $\tau$ is the average time between collisions, and $m$ is the mass of the charge carriers.
Materials with higher charge carrier mobilities have lower resistivities and are better conductors.
The mobility of charge carriers can be affected by impurities, defects, and temperature.

Slide 5: Relation between Mobility and Resistivity

The resistivity of a material is inversely proportional to the mobility of its charge carriers.
Higher mobility leads to lower resistivity, and vice versa.
This can be seen in the equation for resistivity:
- $\rho = \frac{m}{ne^2\mu}$
Materials with high mobility, such as metals, have low resistivities and are good conductors.
Materials with low mobility, such as insulators, have high resistivities and are poor conductors.

Slide 6: Example: Calculation of Resistivity

Let’s consider an example to calculate the resistivity of a material.
A wire of length 2 m and cross-sectional area 1 mm^2 has a resistance of 4 Ω.
Using the formula $R = \rho \frac{L}{A}$, we can rearrange it to find $\rho$:
- $\rho = \frac{RA}{L}$
Substituting the given values, we get $\rho = \frac{4 , \Omega \cdot 1 \times 10^{-6} , m^2}{2 , m}$.
Calculating, we find that the resistivity of the material is $\rho = 2 \times 10^{-6} , \Omega \cdot m$.

Slide 7: Example: Temperature Dependence of Resistivity

Consider a metal wire with an initial resistivity of $2 \times 10^{-8} , \Omega \cdot m$ at a reference temperature $T_0 = 293 , K$.
The temperature coefficient of resistivity for this wire is $\alpha = 0.003 , K^{-1}$.
To find the resistivity at a higher temperature $T$, we can use the equation $\rho(T) = \rho_0(1 + \alpha(T-T_0))$.
For example, at $T = 323 , K$, we find $\rho(323) = 2 \times 10^{-8}(1 + 0.003(323-293))$.
Calculating, we find that the resistivity at this temperature is $\rho(323) = 2.18 \times 10^{-8} , \Omega \cdot m$.

Slide 8: Temperature Coefficient of Resistivity for Different Materials

Different materials have different temperature coefficients of resistivity.
Graphite, for example, has a negative temperature coefficient of resistivity, meaning its resistivity decreases with increasing temperature.
Metals, on the other hand, generally have positive temperature coefficients of resistivity, meaning their resistivity increases with increasing temperature.
Semiconductor materials can have either positive or negative temperature coefficients of resistivity, depending on the specific material and temperature range.

Slide 9: Effect of Impurities and Defects on Mobility

Impurities and defects in a material can affect the mobility of its charge carriers.
Impurities such as doping agents in semiconductors can increase the number of charge carriers, leading to higher conductivity and lower resistivity.
Defects in the crystal lattice structure can scatter charge carriers, reducing their mobility and increasing resistivity.
The presence of impurities and defects can be controlled during the manufacturing process to optimize the conductivity and resistivity of materials.

Slide 10: Summary

Resistivity is a measure of how strongly a material opposes the flow of electric current.
Mobility is a property of charge carriers in a material that determines their ability to move under the influence of an electric field.
The resistivity of a material is related to its mobility through the formula $\rho = \frac{m}{ne^2\mu}$.
The resistivity of most materials increases with increasing temperature due to increased scattering of charge carriers by lattice vibrations.
The temperature coefficient of resistivity, $\alpha$, measures the change in resistivity with temperature and can be positive or negative depending on the material.

Mobility and Temperature Dependence of Resistivity

The mobility of charge carriers in a material determines their ability to move in response to an electric field.
It is influenced by factors such as impurities, defects, and temperature.
The resistivity of a material is inversely proportional to its mobility.
Higher mobility leads to lower resistivity, resulting in better electrical conductivity.
The resistivity of most materials increases with an increase in temperature.

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity, denoted by α, indicates how much the resistivity of a material changes with temperature.
It is defined as α = (1/ρ₀) * (dρ/dT), where ρ₀ is the resistivity at a reference temperature and dρ/dT is the rate of change of resistivity with temperature.
For metals, α is positive, indicating an increase in resistivity with temperature.
For semiconductors, α can be positive or negative, depending on whether they are p-type or n-type.
Insulators typically have a very small temperature coefficient of resistivity.

Effect of Temperature on Resistivity

As temperature increases, the lattice vibrations in a material also increase.
These vibrations can scatter charge carriers, impeding their motion and increasing resistivity.
In metals, the increase in resistivity is primarily due to increased scattering of electrons by lattice vibrations.
Semiconductors exhibit more complex temperature dependence, influenced by factors such as carrier mobility, carrier concentration, and energy bandgap.
Insulators generally have negligible changes in resistivity over a wide range of temperatures.

Calculation of Resistivity

The resistivity of a material can be calculated using the formula ρ = (RA)/L, where R is the resistance, A is the cross-sectional area, and L is the length.
This formula assumes the material has a uniform cross-section and obeys Ohm’s law.
Example: A wire with length 2 m and cross-sectional area 1 mm² has a resistance of 4 Ω. Using ρ = (RA)/L, we find ρ = (4 Ω * 1 mm²) / 2 m = 2 × 10⁻⁶ Ω·m.

Overview of Mobility

The mobility of charge carriers represents their ability to move through a material under the influence of an electric field.
It depends on factors like charge carrier concentration, scattering mechanisms, and the presence of impurities or defects.
Mobility is expressed in units of m²/V·s and is influenced by the presence of free electrons or holes in a material.
High mobility is desirable as it enables efficient current flow and low resistivity.

Role of Impurities on Mobility

The presence of impurities in a material can influence the mobility of charge carriers.
For example, in semiconductors, intentional doping with impurities imparts either excess electrons (n-type doping) or holes (p-type doping).
These additional charge carriers affect mobility and alter the electrical behavior of the material.
Impurities can also create scattering centers, restricting the motion of charge carriers and reducing mobility.

Temperature Dependency of Mobility

Temperature can impact the mobility of charge carriers in a material.
In some cases, as temperature increases, carrier mobility decreases due to increased scattering by lattice vibrations.
However, in certain systems, such as intrinsic semiconductors, temperature dependency might not be dominant.
The temperature effect on mobility is influenced by the intrinsic properties of the material, as well as the presence of impurities and defects.

Example: Temperature Coefficient of Resistivity

Consider a copper wire with an initial resistivity of 1.7 × 10⁻⁸ Ω·m at 20°C (293 K).
The experimentally determined temperature coefficient of resistivity for copper is 0.00427°C⁻¹.
To find the resistivity at 80°C (353 K), we can use the equation ρ(T) = ρ₀(1 + α(T - T₀)).
Substituting the values, ρ(353 K) = 1.7 × 10⁻⁸ Ω·m(1 + 0.00427°C⁻¹(353 K - 293 K)).
Calculating, we find that the resistivity at 80°C is approximately 2.28 × 10⁻⁸ Ω·m.

Example: Mobility Calculation

For a given material with charge carrier concentration n = 5 × 10²⁰ m⁻³ and resistivity ρ = 3 × 10⁻⁴ Ω·m, we can determine the mobility.
Using the formula ρ = (m)/(ne²μ), we can rearrange it to solve for mobility: μ = (m)/(ne²ρ).
Suppose the charge carriers have a mass m = 9.1 × 10⁻³¹ kg and an electron charge e = 1.6 × 10⁻¹⁹ C.
Substituting the values, we get μ = (9.1 × 10⁻³¹ kg) / (5 × 10²⁰ m⁻³ × (1.6 × 10⁻¹⁹ C)² × 3 × 10⁻⁴ Ω·m).
Calculating, we find that the mobility of the charge carriers is ≈ 3.02 m²/V·s.

Summary and Recap

Resistivity is a measure of how strongly a material resists the flow of electric current.
The mobility of charge carriers affects the resistivity of a material.
The resistivity of most materials increases with temperature due to increased lattice vibrations and subsequent scattering of charge carriers.
The temperature coefficient of resistivity indicates the change in resistivity with temperature.
Both impurities and defects can influence the mobility of charge carriers in a material.

Mobility and Temperature Dependence of Resistivity

In a material, the resistivity is inversely proportional to the mobility of its charge carriers.
Mobility is affected by factors such as impurities, defects, and temperature.
Higher mobility leads to lower resistivity, resulting in better conductivity.
The resistivity of most materials increases with temperature due to increased scattering of charge carriers.
The temperature coefficient of resistivity, α, indicates how resistivity changes with temperature.

Current and Electricity

Electric current is the flow of electric charge per unit time.
It is measured in Amperes (A) and is represented by the symbol “I”.
Current can be either direct current (DC) or alternating current (AC).
The current in a material is directly proportional to the applied voltage and inversely proportional to the resistance.
Ohm’s Law relates current, voltage, and resistance: V = IR.

Problem on Resistivity and Temperature

Problem: A copper wire has a resistivity of 1.7 × 10^(-8) Ω·m at 20°C and a temperature coefficient of resistivity of 0.00427 °C^(-1). Find its resistivity at 80°C.
Solution: Using the formula ρ(T) = ρ₀(1 + α(T - T₀)), we can substitute the given values: ρ(80°C) = (1.7 × 10^(-8) Ω·m)(1 + 0.00427 °C^(-1)(80°C - 20°C)).
Calculating, we find that the resistivity at 80°C is approximately 2.28 × 10^(-8) Ω·m.

Conductivity and Temperature Dependence

Conductivity (σ) is the inverse of resistivity and measures a material’s ability to conduct electric current.
σ = 1/ρ, where ρ is the resistivity of the material.
The conductivity of most materials increases with temperature due to increased charge carrier mobility.
Higher temperatures provide more energy to charge carriers, allowing them to overcome lattice vibrations and move more freely.
The temperature dependence of conductivity is characterized by the same temperature coefficient as resistivity but with the opposite sign.

Examples of Temperature Coefficients

Copper has a positive temperature coefficient of resistivity, meaning its resistivity increases with temperature.
Tungsten has a similar positive temperature coefficient and is commonly used in incandescent light bulbs.
Pure silicon has a negative temperature coefficient of resistivity, but when doped with specific impurities, it can exhibit positive or even zero temperature coefficient.
Carbon, in the form of graphite, has a negative temperature coefficient of resistivity, making it suitable for some high-temperature applications.

Calculation of Resistivity

To calculate the resistivity of a material, we need to know the resistance (R), length (L), and cross-sectional area (A) of the sample.
The formula for resistivity is ρ = (R × A) / L.
Example: A wire with a length of 2 meters and a cross-sectional area of 1 mm^2 has a resistance of 4 Ω. Using the formula, ρ = (4 Ω × 1 mm^2) / 2 m.
Calculating, we find that the resistivity of the material is 2 × 10^(-6) Ω·m.

Relation Between Mobility and Resistivity

The resistivity (ρ) of a material is inversely proportional to the mobility (μ) of its charge carriers.
ρ = m / (n × e² × μ), where m is the mass of the charge carriers, n is the charge carrier concentration, e is the charge of an electron, and μ is the mobility.
Higher mobility leads to lower resistivity, indicating better conductivity.
Materials with high mobility, such as metals, have low resistivities and are good conductors.
Materials with low mobility, like insulators, have high resistivities and are poor conductors.

Temperature Coefficient of Resistivity and Thermal Expansion

The temperature coefficient of resistivity (α) is related to the thermal expansion coefficient (β) of a material.
The relationship is given by the equation α = β - α’, where α’ is the thermal expansion coefficient of the crystal lattice.
The fact that α is usually positive indicates that the increase in resistivity with temperature is mainly due to increased lattice vibrations.
This relationship helps explain the general trend of resistivity increasing with temperature.

Applications of Temperature Dependence of Resistivity

The temperature dependence of resistivity has various practical applications.
Thermistors are temperature-sensitive resistors used in thermostats, temperature sensors, and thermal compensation circuits.
Overcurrent protection devices, like fuses and circuit breakers, take advantage of the temperature rise caused by increased resistance to prevent damage.
Temperature sensors and resistance thermometers rely on the correlation between resistance and temperature.
Integrated circuits use the temperature dependence of resistivity to monitor temperature and adjust performance.

Summary and Key Takeaways

The resistivity of a material depends on mobility, temperature coefficient, and other factors.
Mobility represents the ability of charge carriers to move in response to an electric field.
The resistivity of most materials increases with temperature due to increased scattering by lattice vibrations.
The temperature coefficient of resistivity measures the change in resistivity with temperature.
The relation between resistivity and mobility highlights the importance of charge carrier behavior in determining a material’s conductivity.