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3.5 - DRIFT OF ELECTRONS AND THE ORIGIN OF RESISTIVITY

Interactive Audio Lesson

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Introduction to Electron Movement

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Teacher
Teacher

Today, we're going to discuss how electrons behave in a conductor. Can anyone tell me what happens to electrons when there’s no electric field present?

Student 1
Student 1

They just move randomly, right?

Teacher
Teacher

Exactly! Without an electric field, electrons have random thermal motion, resulting in no net drift velocity. We can remember this by thinking of them dancing in all directions. What's the average velocity in this case?

Student 2
Student 2

It would be zero!

Teacher
Teacher

Correct! It’s like a group of people moving around, where some go left and some go right, but overall no one is moving from their starting point. Now, let’s see what happens when we apply an electric field.

Effects of Electric Field

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Teacher
Teacher

When we apply an electric field, what do you think happens to the electrons?

Student 3
Student 3

They get pushed in the direction of the field?

Teacher
Teacher

Exactly, they accelerate in the direction opposite to the field. This results in a phenomenon we call drift. Discuss with your neighbor, what do you think drift velocity means?

Student 4
Student 4

Does it measure how fast the electrons are moving overall?

Teacher
Teacher

Yes! Drift velocity is the average velocity of electrons due to the electric field. We denote it as v_d and can express it mathematically as v_d = -eE/mτ. Understanding this helps us relate charge movement with electric field strength.

Collisions and Relaxation Time

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Teacher
Teacher

Now, let’s think about the collisions electrons experience in a conductor. Why do these collisions matter?

Student 1
Student 1

They can slow the electrons down?

Teacher
Teacher

Great answer! These collisions can deflect the electrons and slow down their drift. The average time between collisions is called relaxation time τ. Does anyone want to explain how we can calculate drift velocity using this?

Student 2
Student 2

We use that formula you just mentioned, right? v_d is related to τ?

Teacher
Teacher

Exactly! The more frequent the collisions, the shorter the relaxation time and therefore the lower the drift velocity.

Understanding Resistivity

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Teacher
Teacher

Now that we understand drift velocity better, let's connect it to resistivity. Who can tell me what resistivity tells us about a material?

Student 3
Student 3

It shows how much the material resists the flow of electric current.

Teacher
Teacher

Exactly! Resistivity ρ is defined as the resistance of a unit length of material. The formula we discussed earlier shows that resistivity is influenced by the number of carriers and how long they can move before colliding. It's represented as ρ = m/n e² τ. Can someone explain what each symbol represents?

Student 4
Student 4

m is the mass, n is the number density of electrons, and e is their charge!

Teacher
Teacher

Very well summarized! This creates a foundation for how materials can be categorized as conductors, semiconductors, or insulators.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section discusses the drift of electrons in conductors under an electric field and the resulting concepts of resistivity and conductivity.

Standard

The section explores how electrons collide with fixed ions in a conductor and how their average drift velocity is established under the influence of an electric field. It introduces key concepts such as mobility, charge density, and the relationship between drift velocity and electric field, laying the foundation for understanding resistivity and conductivity in materials.

Detailed

Drift of Electrons and the Origin of Resistivity

The drift of electrons in a conductor under an applied electric field is crucial for understanding electrical conduction. When no electric field is present, electrons move randomly due to thermal motion, resulting in an average velocity of zero. However, when an electric field is applied, electrons are accelerated, leading to a net drift in the direction opposite to the field. The factors affecting this motion include the charge and mass of the electron, as well as the frequency and nature of collisions with fixed positive ions in the metal lattice.

Key Points Covered:

  • Collisions and Random Motion: In the absence of an electric field, the average velocity of electrons remains zero due to random thermal energy.
  • Electric Field Impact: When an electric field is applied, electrons are subjected to constant acceleration, progressing towards a net average drift velocity.
  • Relaxation Time: This average time between collisions affects the drift velocity, as more frequent collisions reduce the electrons' average speed.
  • Drift Velocity Calculation: The average drift velocity (v_d) can be expressed as

v_d = - rac{e E}{m} au
where
- e = charge of the electron
- E = applied electric field
- m = mass of the electron
- τ = relaxation time
- Resistivity: The resistivity ρ is defined through relationships involving the number density of free electrons, their charge, and the relaxation time, leading to the formula:

ρ = rac{m}{n e^2 τ},
where n is the electron density
(a measure of the number of conduction electrons per unit volume).

Understanding drift velocity, charge density, and resistivity not only explains basic electrical conduction in metals but also forms the basis for advanced topics in material science and engineering.

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Audio Book

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Collisions and Average Velocity of Electrons

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As remarked before, an electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Thus, if there are N electrons and the velocity of the ith electron (i = 1, 2, 3, ... N ) at a given time is v, then

$$ \frac{1}{N} \sum_{i=1}^{N} v_i = 0 $$

Detailed Explanation

Electrons in a conductor are constantly colliding with fixed ions, which scatter their motion in random directions. After such a collision, even though the speed of the electron remains the same, the direction may change unpredictably. As a result, when you calculate the average velocity of a large number of electrons, their directions cancel out, leading to an average velocity of zero. This means that, although individual electrons are moving, their collective motion does not result in a net flow without the influence of an external electric field.

Examples & Analogies

Think of a crowded room filled with people moving around randomly. If you were to average the direction in which everyone is walking, you would find that there's no overall movement towards any door; people are just bumping into each other and changing directions without contributing to a mass exit.

Effect of Electric Field on Electron Movement

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Consider now the situation when an electric field is present. Electrons will be accelerated due to this field by

$$ a = \frac{-eE}{m} $$ where -e is the charge and m is the mass of an electron.

Consider again the ith electron at a given time t. This electron would have had its last collision some time before t, and let \( t_i \) be the time elapsed after its last collision. Thus, the average velocity of the electrons at time \( t \) is the average of all the velocities. The average of \( v_i \) is zero [since immediately after any collision, the direction of the velocity of an electron is completely random]. Let us denote by \( \tau \), the average time between successive collisions.

Detailed Explanation

When an electric field is applied to a conductor, due to the electric force acting on electrons, they experience acceleration in the direction of the field. The force is given by the formula above, where the charge of the electron and its mass determines how quickly it responds. While electrons do accelerate, their motion is interrupted by frequent collisions with the fixed ions in the conductor. This collision process averages out the acceleration over time, leading to a steady average drift velocity rather than continuous acceleration.

Examples & Analogies

Imagine you are driving a car on a straight road. Your speed increases as you press the accelerator, similar to how electrons accelerate due to an electric field. However, every time you hit a pothole (representing collisions), you are jolted and lose some of your speed. Even though you are pushing the pedal to accelerate, the frequent hits from potholes keep you from going faster than a certain speed; similarly, the random collisions keep electrons from continually accelerating.

Drift Velocity of Electrons

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Thus, averaging gives us for the average velocity \( v_d \):

$$ v_{d} = \frac{-eE\tau}{m} $$

Detailed Explanation

The drift velocity, represented by \( v_d \), is the average velocity of electrons in a conductor when an electric field is applied. It can be derived from considering the accumulated effects of electron acceleration over the average time between collisions. The negative sign indicates that electrons move in the opposite direction of the electric field, as they are negatively charged. The expression shows that the drift velocity is directly proportional to the strength of the electric field and the average relaxation time between collisions.

Examples & Analogies

Think about water flowing through a pipe. While the pressure from a pump pushes the water along, the water molecules may bump into each other and the sides of the pipe. Similar to how more pressure leads to higher flow rates, a stronger electric field causes electrons to drift faster. The 'drift velocity' is like the average speed of water flowing despite its turbulent path.

Current Density and Drift

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Because of the drift, there will be net transport of charges across any area perpendicular to E. Consider a planar area A, located inside the conductor such that the normal to the area is parallel to E. Then the total charge transported across this area A in time \( \Delta t \) is given by:

$$ I = -neA v_{d} $$ where n is the number of free electrons per unit volume in the metal.

Detailed Explanation

This equation shows how drift velocity leads to a measurable current. The drift of electrons results in a net flow of charge across any cross-sectional area that is perpendicular to their motion. 'n' represents the density of charge carriers (electrons), and 'A' is the area through which they are moving. The product of these values with the drift velocity gives the total current, which is a measure of how much charge is passing through that area over time.

Examples & Analogies

Imagine a group of people trying to exit a theater through a single door. The number of people trying to leave (analogous to 'n') and the width of the door (the area 'A') both influence how many can exit per second (the current). If they all move towards the exit at the same steady pace (like the drift velocity), you can easily calculate how many people will exit each second.

Relationship to Ohm's Law

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From the relations for current density \( j \) and drift velocity, we can derive Ohm's Law.
As found, \( j = \sigma E \), where \( \sigma \) is the conductivity of the material. Therefore, combining all these relations leads us to the conclusion that

$$ \sigma = \frac{ne^2}{m\tau}. $$

Detailed Explanation

This final relationship connects all the previous concepts into Ohm's Law. From understanding how electron drift produces a measurable current density, we can see how the performance of conductors can be characterized by conductivity, which incorporates the charge density and the average time between collisions. The result shows that higher charge density or lower collision time leads to better conductivity, explaining why different materials have varying electrical properties.

Examples & Analogies

Consider a crowded mall during a sale. If there's a good flow of shoppers (analogous to current density), the rate at which shoppers can make purchases (conductivity) depends on how many entrance/exit points are available and how quickly people move through the aisles (representing charge density and collision time). If more exits are available and crowd control is effective, shopping can proceed smoothly—as conductivity improves in materials with favorable properties.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Drift Velocity: This velocity represents the average speed of charge carriers in a conductor under the influence of an electric field.

  • Relaxation Time: The average time between collisions of electrons, affecting their drift speed.

  • Resistivity: A measure of a material's opposition to electric current flow, dependent on carrier density and collision time.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Using the drift velocity equation, we can estimate how long it takes for a charge carrier to traverse a certain distance in a conductor.

  • In metals, the resistivity is low due to high electron density, while in insulators, it is high due to low electron density.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Electrons free, in fields they flee; drift they must, in conductors trust.

📖 Fascinating Stories

  • Imagine a bustling city where pedestrians represent electrons, freely wandering until an electric barrier directs them to flow in one direction, simulating the drift velocity.

🧠 Other Memory Gems

  • MINE - Movement, Influence of Electric field, Number of Electrons, Energy (The factors for drift velocity).

🎯 Super Acronyms

RECORD - Resistivity, Electrons, Charges, Overcoming Random Drift.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Drift Velocity (v_d)

    Definition:

    The average velocity of charge carriers, such as electrons, in a conductor due to an applied electric field.

  • Term: Relaxation Time (τ)

    Definition:

    The average time interval between collisions of electrons with ions in a conductor.

  • Term: Resistivity (ρ)

    Definition:

    A property of a material that quantifies its resistance to electric conduction, influenced by electron density and relaxation time.

  • Term: Mobility (µ)

    Definition:

    A measure of how quickly electrons can move through a material in response to an electric field.

  • Term: Electric Field (E)

    Definition:

    A vector field that represents the force exerted on a unit charge at any point in space.