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Chapter 6: Problem 7

Calculate the mobility of electrons in \(\mathrm{Cu}\). The resistivity of \(\mathrm{Cu}\) is \(1.72 \times\) \(10^{-8} \Omega \cdot \mathrm{m}\) at \(25^{\circ} \mathrm{C}\) and its density is \(8.9 \mathrm{~g} / \mathrm{cm}^{3}\). Assume each copper atom donates one valence electron to the conduction band.

Short Answer

Expert verified
The electron mobility in copper is \( 4.5 \times 10^{-3} \text{ m}^2/\text{V}\cdot \text{s}\).

Step by step solution

01

Determine the number of atoms per unit volume

First, we need to calculate the number of copper atoms per unit volume. The molar mass of copper is approximately 63.55 g/mol. Using the density of copper and Avogadro's number \( N_A = 6.022 \times 10^{23} \) atoms/mol, calculate the number of atoms per cubic meter.\[\text{Number of atoms per } m^3 = \left( \frac{8.9 \text{ g/cm}^3 \times 10^6 \text{ cm}^3/m^3}{63.55 \text{ g/mol}} \right) \times 6.022 \times 10^{23} \text{ atoms/mol}\approx 8.49 \times 10^{28} \text{ atoms/mol}.\]Thus, there are \(8.49 \times 10^{28}\) conduction electrons per cubic meter of copper.
02

Calculate electron charge and conductivity

We know one copper atom donates one valence electron, so the number of conduction electrons is the same as the number of atoms. Each electron has a charge of \( e = 1.6 \times 10^{-19} \text{ C} \). Next, calculate the conductivity \( \sigma \) using resistivity \( \rho \), given by\[\sigma = \frac{1}{\rho} = \frac{1}{1.72 \times 10^{-8} \Omega\cdot m} = 5.81 \times 10^7 \text{ S/m}.\]
03

Calculate mobility using the formula

Electron mobility \( \mu \) is related to conductivity \( \sigma \), charge \( e \), and the concentration of electrons \( n \) as follows: \( \sigma = n \cdot e \cdot \mu \). Substitute the known values:\[5.81 \times 10^7 \text{ S/m} = (8.49 \times 10^{28} \text{ electrons/m}^3) \cdot (1.6 \times 10^{-19} \text{ C}) \cdot \mu.\]Solving for \( \mu \), we have:\[\mu = \frac{5.81 \times 10^7}{8.49 \times 10^{28} \times 1.6 \times 10^{-19}}\approx 4.5 \times 10^{-3} \text{ m}^2/\text{V}\cdot \text{s}.\]
04

Verify units and calculate final result

Ensure the units for electron mobility \( \mu \) are indeed \( \text{m}^2/\text{V}\cdot\text{s} \). Since conductivity has units \( \text{S/m} = \text{A}/\text{V}\cdot\text{m} \) and electron concentration \( n \) has units \( \text{electrons/m}^3 \), when multiplying by electron charge \( e \), units simplify correctly, confirming the dimension for \( \mu \) is accurate. The calculated mobility is \( 4.5 \times 10^{-3} \, \text{m}^2/\text{V}\cdot\text{s} \).

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Key Concepts

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

Resistivity
Resistivity is a fundamental property that quantifies how strongly a material opposes the flow of electric current. It is expressed in ohm-meters (a d). A key aspect of understanding resistivity is recognizing that it is an intrinsic property of materials, meaning it does not change with the amount or shape of the material.

- With higher resistivity, electrical currents face more resistance, leading to more energy being dissipated as heat. - In the case of copper, known for its excellent electrical conductivity, the resistivity is quite low, specifically, copper has a resistivity of approximately 1.72 x 10^{-8} a d.

The relation between resistivity (c) and conductivity (3) is essential to remember: 3 = 1/c. This means that as resistivity decreases, conductivity increases, showcasing copper's efficiency in conducting electricity.
Conductivity
Conductivity is a measure of a material's ability to conduct an electric current. It is measured in siemens per meter (S/m) and is the reciprocal of resistivity.

- A high conductivity indicates that a material can easily allow the passage of electric current with minimal energy loss.
- Copper is often used in electrical applications due to its high conductivity of about 5.81 x 10^7 S/m.

The high conductivity of copper is a result of its atomic structure, which allows electrons to flow freely. Conductivity is crucial in electrical wiring and electronic circuits, and it directly affects how efficiently energy is transmitted.
Valence Electron
A valence electron is an outer-shell electron associated with an atom and capable of participating in the formation of a chemical bond.

- In metals like copper, valence electrons are not tightly bound to any atom and can move relatively freely. This freedom allows metals to conduct electric current efficiently. - Copper atoms donate one valence electron to the conduction band, creating a sea of free electrons that facilitate electrical conductivity. - The ease of movement of these valence electrons is what gives metals like copper their characteristic shiny appearance and high conductivity.

Understanding valence electrons is key to exploiting a metal's ability to conduct electricity, impacting both material science and electronics industries.
Copper Atoms
Copper atoms, with the atomic number 29, are renowned for their excellent electrical and thermal conductivity. These properties are attributed to the structure of copper atoms and their specific electronic configuration.

- Each copper atom contributes a single valence electron to the conduction band, fostering a rich environment of electrons ready for conduction. - Copper's molar mass is approximately 63.55 g/mol and its density is 8.9 g/cm^3, based on which the number of copper atoms in a material can be determined.

Copper's exceptional capacity to conduct electricity stems from its ability to donate one valence electron per atom, making it a staple in electrical engineering applications. The low resistivity and high conductivity of copper also ensure minimal energy loss, making it ideal for wiring and other conductive materials.

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Most popular questions from this chapter

Look up the refractive indices for fused silica and dense flint glass, and calculate the ratio of their reflectivities. Cite the source of your information.

Silicon has a density of \(2.40 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the concentration of the silicon atoms per cubic centimeter? (b) Phosphorus is added to the silicon to make it an \(n\)-type semiconductor with a conductivity of \(1 \mathrm{mho} / \mathrm{cm}\) and an electron mobility of \(1700 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}\). What is the concentration of the conduction electrons per cubic centimeter?

What will be the resistance of a copper wire \(0.08\) in. in diameter and \(100 \mathrm{ft}\) long if its resistivity is \(1.7 \mu \Omega \cdot \mathrm{cm}\) ?

A coil of wire \(0.1 \mathrm{~m}\) long and having 15 turns carries a current of \(1.0 \mathrm{~A}\). (a) Compute the magnetic induction if the coil is within a vacuum. (b) A bar of molybdenum is now placed in the coil, and the current adjusted to maintain the same magnetic induction as in part (a). Calculate the magnetization.
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