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Chapter 18: Problem 9

Briefly tell what is meant by the drift velocity and mobility of a free electron.

Short Answer

Expert verified
Answer: The relationship between drift velocity and mobility of a free electron can be described by the equation \(v_d = \mu E\), where \(v_d\) is the drift velocity, \(\mu\) is the mobility, and \(E\) is the electric field. Drift velocity is directly proportional to the mobility of an electron, which means that a higher mobility indicates that the electron can move more easily in response to the electric field, resulting in a higher drift velocity. The mobility of an electron depends on factors such as the nature of the medium, temperature, and impurity levels present in the material.

Step by step solution

01

Defining Drift Velocity of a Free Electron

Drift velocity is the average velocity of free charge carriers, such as electrons, when they are subjected to an external electric field inside a conducting medium like a metal. The drift velocity depends on the strength of the applied electric field, the charge and the mass of the electron, and the interaction between the electron and the lattice of the conducting medium. The drift velocity can be calculated using the formula: \(v_d = \frac{I}{nAe}\) where \(v_d\) is the drift velocity, \(I\) is the current, \(n\) is the number density of charge carriers (electrons), \(A\) is the cross-sectional area of the conductor, and \(e\) is the elementary charge.
02

Defining Mobility of a Free Electron

Mobility is a property that characterizes how easily electrons can move in response to an electric field. Mathematically, it's defined as the ratio of the drift velocity of the charged particle to the strength of the electric field acting on it. The mobility of an electron can be represented as: \(\mu = \frac{v_d}{E}\) where \(\mu\) is the mobility, \(v_d\) is the drift velocity, and \(E\) is the electric field.
03

Relationship between Drift Velocity and Mobility

There is a direct relationship between drift velocity and mobility. As introduced earlier, mobility can be expressed as the ratio of drift velocity to the electric field: \(\mu = \frac{v_d}{E}\) By rearranging the formula and solving for drift velocity, we can find that: \(v_d = \mu E\) This equation shows that the drift velocity is directly proportional to the mobility of a free electron in an electric field. Higher mobility indicates that the electron can move more easily in response to the electric field, resulting in a higher drift velocity. The mobility of an electron depends on factors such as the nature of the medium (conductor, insulator or semiconductor), temperature, and impurity levels present in the material.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Mobility
Electron mobility is a fundamental concept that describes how easily an electron can travel through a conducting medium under the influence of an electric field. In simpler terms, it measures the electron's ability to move through a material when a force, such as an electric field, is applied. This property is crucial for determining the efficiency of materials in conducting electricity.
  • Mathematical Definition: Electron mobility is denoted by \( \mu \) and is mathematically expressed as: \[ \mu = \frac{v_d}{E} \]where \( v_d \) is the drift velocity and \( E \) is the electric field.
  • Factors Affecting Mobility: Mobility can vary depending on several factors such as the type of material (metal, semiconductor), temperature, and impurities present.
Knowing the electron mobility helps engineers design better electronic devices by selecting materials that allow electrons to move swiftly, thereby improving performance.
Electric Field
An electric field is a region around a charged particle where the particle exerts a force on other charged particles. Imagine it like an invisible aura that pushes or pulls other charges in its vicinity. Electric fields are fundamental to understanding how electric forces act at a distance.
  • Creation of Electric Fields: Electric fields are created by charged objects. If you've ever rubbed a balloon on your hair and stuck it to a wall, you've experienced the effects of static electricity and the charged field it creates.
  • Representation and Effects: They are often represented by lines pointing in the direction of the force they exert on a positive charge. The strength of the electric field is measured in volts per meter (V/m).
Electric fields are essential for calculating force interactions and understanding how electrons move in a circuit, which directly ties into concepts like drift velocity and electron mobility.
Conducting Medium
A conducting medium is any material that allows the free movement of charge carriers, such as electrons. These materials can easily transport electric current because they contain charge carriers that move freely when an electrical force is applied. Metals like copper and aluminum are typical examples.
  • Characteristics of Conductors: Conductors have low resistance to electricity and high electron mobility, making them ideal for electrical wiring and components.
  • Interaction with Electric Fields: When an electric field is applied, electrons move through these materials, creating an electric current.
The choice of conducting medium plays a vital role in electrical engineering because it impacts the efficiency and reliability of electronic devices.
Charge Carriers
Charge carriers are the particles responsible for carrying electric charge in a medium. In conductors, these are typically electrons, but in certain conditions like semiconductors, they can also be holes (positive charge carriers).
  • Types of Charge Carriers: Depending on the material, charge carriers can be either free electrons or holes. In metals, electrons are the primary charge carriers.
  • Role in Conducting Electricity: The movement of charge carriers under an electric field's influence results in a flow of electric current.
Understanding charge carriers helps explain how electricity moves in different materials, providing insights into designing more efficient electronic components and understanding concepts like drift velocity and electron mobility.

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Most popular questions from this chapter

Tin bronze has a composition of \(92 \mathrm{wt} \% \mathrm{Cu}\) and \(8 \mathrm{wt} \% \mathrm{Sn}\), and consists of two phases at room temperature: an \(\alpha\) phase, which is copper containing a very small amount of tin in solid solution, and an \(\epsilon\) phase, which consists of approximately 37 wt \(\%\) Sn. Compute the room temperature conductivity of this alloy given the following data: \begin{tabular}{ccc} \hline \multicolumn{2}{c}{ Electrical } \\ Phase & Resistivity \((\Omega \cdot m)\) & Density \(\left(g / c m^{5}\right)\) \\\ \hline\(\alpha\) & \(1.88 \times 10^{-8}\) & \(8.94\) \\ \(\epsilon\) & \(5.32 \times 10^{-7}\) & \(8.25\) \\ \hline \end{tabular}

Demonstrate that the two Ohm's law expressions, Equations \(18.1\) and \(18.5\), are equivalent.

Define the following terms as they pertain to semiconducting materials: intrinsic, extrinsic, compound, elemental. Now provide an example of each.

At room temperature the electrical conductivity and the electron mobility for copper are \(6.0 \times 10^{7}(\Omega \cdot \mathrm{m})^{-1}\) and \(0.0030 \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\), respectively. (a) Compute the number of free electrons per cubic meter for copper at room temperature. (b) What is the number of free electrons per copper atom? Assume a density of \(8.9 \mathrm{~g} / \mathrm{cm}^{3}\)

(a) Calculate the drift velocity of electrons in germanium at room temperature and when the magnitude of the electric field is \(1000 \mathrm{~V} / \mathrm{m}\). (b) Under these circumstances, how long does it take an electron to traverse a \(25-\mathrm{mm}\) (1-in.) length of crystal?
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