Drift Velocity And Resistance

The drift velocity of charge carriers is the average velocity with which they drift toward the positive terminal when a potential difference is applied across a conductor.
Resistance is the opposition offered by a substance to the flow of electric current.
There is a relation between the current density and drift velocity of charge carriers in a conductor.

Current Density (J)

Current density is defined as the amount of current flowing per unit area perpendicular to the direction of current.
Mathematically, current density (J) is given by:
- J = I / A
- Where I is the current flowing through the conductor and A is the cross-sectional area of the conductor.

Charge Carrier Density (n)

Charge carrier density, denoted by ’n’, is defined as the number of charge carriers per unit volume in a conductor.
It represents the concentration of charge carriers in the conductor.
Charge carrier density depends on the substance and its physical properties.

Electron Drift Velocity (vd)

Electron drift velocity, denoted by ‘vd’, is the average velocity at which the free electrons drift toward the positive terminal in a conductor.
It is directly proportional to the applied electric field and inversely proportional to the collision frequency of electrons.

Relation between Current Density and Drift Velocity

The relation between current density (J) and drift velocity (vd) can be given as:
- J: Current density
- n: Charge carrier density
- e: Charge of an electron
- vd: Drift velocity of electrons

Derivation of the Relation

Consider a conductor of cross-sectional area A, length l, and resistance R.

The current through the conductor is given by Ohm’s law: I = V / R, where V is the potential difference applied across the conductor.

Expressing the current density J in terms of current I and cross-sectional area A:
- J = I / A

Expressing the current I in terms of charge carrier density n, charge of an electron e, and drift velocity vd:
- I = n * e * A * vd
- (from the charge continuity equation: I = n * e * A * vd)

Substituting the value of I in the expression for current density:
- J = (n * e * A * vd) / A
- J = n * e * vd

More about the Relation

The relation J = n * e * vd shows that the current density in a conductor is directly proportional to the charge carrier density and the drift velocity of electrons.
It also implies that the current density can be increased by increasing the charge carrier density or the drift velocity.

Units of Current Density and Drift Velocity

The SI unit of current density (J) is Ampere per square meter (A/m²).
The SI unit of drift velocity (vd) is meter per second (m/s).

Example: Calculation of Current Density

Consider a copper wire of diameter 2 mm through which a current of 2 A is flowing. Calculate the current density.
Given:
- Current (I) = 2 A
- Diameter of wire (d) = 2 mm = 0.002 m
Solution:
- Radius (r) = d/2 = 0.001 m
- Cross-sectional area (A) = π * r² = 3.14 * (0.001)² = 3.14 * 0.001 = 0.00314 m²
- Current density (J) = I / A = 2 / 0.00314 = 636.94 A/m²

Summary

Drift velocity is the average velocity at which charge carriers drift in a conductor.
Current density is the amount of current flowing per unit area.
The relation between current density (J) and drift velocity (vd) is given by J = n * e * vd.
Current density is directly proportional to charge carrier density (n) and drift velocity (vd).
Units of current density and drift velocity are A/m² and m/s respectively.

Drift Velocity and Resistance - Relation between Current Density and Drift Velocity

The current density through a conductor is given by the equation: J = n * e * vd
This equation shows the direct relationship between current density, charge carrier density, and drift velocity.
The current density can be increased by increasing the charge carrier density or the drift velocity.

Applications

The relation between current density and drift velocity is used in various practical applications.
It helps in understanding the behavior of conductors and the flow of current through them.
The knowledge of current density is crucial for designing electrical circuits and analyzing their performance.

Example: Calculation of Drift Velocity

Consider a copper wire of length 2 m and resistance 5 Ω. If a potential difference of 10 V is applied across the wire, calculate the drift velocity of electrons.
Given:
- Length of wire (l) = 2 m
- Resistance (R) = 5 Ω
- Potential difference (V) = 10 V

Example (Continued)

Solution:
- Using Ohm’s law: V = I * R, we can find the current (I) flowing through the wire.
- From I = V / R, I = 10 / 5 = 2 A
- Now, using the relation J = n * e * vd, we can find the drift velocity (vd).
- Rearranging the equation: vd = J / (n * e)
- Since the values of charge carrier density (n) and charge (e) are not given, we cannot calculate the drift velocity in this example.

Factors Affecting Drift Velocity

The drift velocity of charge carriers in a conductor depends on several factors:
1. Electric Field: The stronger the electric field, the greater the drift velocity.
2. Charge Carrier Density: Higher charge carrier density leads to a higher drift velocity.
3. Temperature: Increasing temperature reduces the drift velocity due to increased collision frequency.
4. Material Properties: Different conductors have different collision frequencies, affecting the drift velocity.

Conductors vs. Insulators

Conductors and insulators have different characteristics when it comes to drift velocity.
Conductors have higher charge carrier densities and higher electrical conductivity, allowing more current to flow.
Insulators have lower charge carrier densities and lower conductivity, resulting in lower drift velocities and resistance to current flow.

Limitations of the Model

The relation between current density and drift velocity assumes a simplified model of conductors.
It does not take into account the complex behavior of charge carriers in real-world materials.
The model assumes that all charge carriers have the same drift velocity, which is an oversimplification.

Significance of Drift Velocity

Understanding drift velocity is crucial in various applications and fields.
It helps in the design and analysis of electrical circuits, transmission lines, and semiconducting devices.
The concept of drift velocity is essential for understanding electrical conduction in materials and the behavior of charge carriers.

Summary

The current density can be calculated using the equation J = n * e * vd.
The relationship between current density and drift velocity helps in understanding the flow of current in conductors.
Drift velocity depends on factors such as electric field strength, charge carrier density, and material properties.
Conductors have higher drift velocities compared to insulators.
The model of drift velocity has its limitations but provides a useful understanding of electrical conduction.

Questions

Calculate the drift velocity in a copper wire with a charge carrier density of 8.5 x 10^28 m^-3 and a current density of 500 A/m².
Explain the relationship between current density and drift velocity.
How does temperature affect the drift velocity of charge carriers in a conductor?
Compare the drift velocities in a conductor and an insulator.
Discuss the limitations of the drift velocity model.

Drift velocity and Resistance - Relation between Current Density and Drift Velocity

The current density in a conductor is given by the equation: J = n * e * vd
This equation shows the direct relationship between current density, charge carrier density, and drift velocity.
By increasing the charge carrier density or drift velocity, the current density can be increased.

Factors Affecting Drift Velocity

Electric Field: The strength of the electric field affects the drift velocity. A stronger electric field leads to a higher drift velocity.
Charge Carrier Density: A higher charge carrier density results in a higher drift velocity.
Temperature: Increasing temperature reduces the drift velocity because of increased collision frequency.
Material Properties: Different materials have different collision frequencies, which impacts the drift velocity.

Conductors vs. Insulators

Conductors have higher charge carrier densities and higher electrical conductivity, allowing more current to flow.
Insulators have lower charge carrier densities and lower conductivity, resulting in lower drift velocities and resistance to current flow.
The drift velocity in conductors is higher compared to insulators.

Limitations of the Model

The relation between current density and drift velocity assumes a simplified model of conductors.
It does not consider the complex behavior of charge carriers in real-world materials.
The model assumes that all charge carriers have the same drift velocity, which is an oversimplification.
In reality, the drift velocity varies for different charge carriers.

Significance of Drift Velocity

Understanding drift velocity is crucial in various applications and fields.
It helps in the design and analysis of electrical circuits, transmission lines, and semiconducting devices.
The concept of drift velocity is essential for understanding electrical conduction in materials and the behavior of charge carriers.
Without a proper understanding of drift velocity, it would be challenging to explain and predict the behavior of electrical currents.

Example: Calculation of Drift Velocity

Consider a copper wire of length 2 m and resistance 5 Ω. If a potential difference of 10 V is applied across the wire, calculate the drift velocity of electrons.
Given:
- Length of wire (l) = 2 m
- Resistance (R) = 5 Ω
- Potential difference (V) = 10 V

Example (Continued)

Solution:
- Using Ohm’s law: V = I * R, we can find the current (I) flowing through the wire.
- From I = V / R, I = 10 / 5 = 2 A
- Now, using the relation J = n * e * vd, we can find the drift velocity (vd).
- Rearranging the equation: vd = J / (n * e)
- Since the values of charge carrier density (n) and charge (e) are not given, we cannot calculate the drift velocity in this example.

Example: Calculation of Current Density

Consider a copper wire of diameter 2 mm through which a current of 2 A is flowing. Calculate the current density.
Given:
- Current (I) = 2 A
- Diameter of wire (d) = 2 mm = 0.002 m

Example (Continued)

Solution:
- Radius (r) = d/2 = 0.001 m
- Cross-sectional area (A) = π * r² = 3.14 * (0.001)² = 3.14 * 0.001 = 0.00314 m²
- Current density (J) = I / A = 2 / 0.00314 = 636.94 A/m²

Summary

The current density can be calculated using the equation J = n * e * vd.
The relationship between current density and drift velocity helps in understanding the flow of current in conductors.
Drift velocity depends on factors such as electric field strength, charge carrier density, and material properties.
Conductors have higher drift velocities compared to insulators.
The model of drift velocity has its limitations but provides a useful understanding of electrical conduction.