91Ó°ÊÓ

An electron with kinetic energy \(K_{e}\) travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a \(z\) axis. The electron's motion is subject only to the force due to the field. (a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitude \(\mu=K_{e} / B\) and that it is in the direction opposite that of \(\vec{B}\). What are the (b) magnitude and (c) direction of the magnetic dipole moment of a positive ion with kinetic energy \(K_{i}\) under the same circumstances? (d) An ionized gas consists of \(5.3 \times 10^{21}\) electrons/m \(^{3}\) and the same number density of ions. Take the average electron kinetic energy to be \(6.2 \times 10^{-20} \mathrm{~J}\) and the average ion kinetic energy to be \(7.6 \times 10^{-21} \mathrm{~J}\). Calculate the magnetization of the gas when it is in a magnetic field of \(1.2 \mathrm{~T}\).

Step by step solution

01

Understand the relation for magnetic dipole moment

The magnetic dipole moment (μ) of a charged particle moving in a circular path within a magnetic field is related to its kinetic energy and the applied magnetic field. The relationship can be derived from the charge-to-mass ratio and kinetic energy relations.

02

Derive expression for magnetic moment of electron (a)

The centripetal force for electron motion is provided by the Lorentz force: \[ F = qvB = \frac{mv^2}{r} \] Using kinetic energy \( K_e = \frac{1}{2}mv^2 \), solve for \( v \) and \( r \); then the orbital magnetic dipole moment is \[ \mu = \frac{qv}{2r} \].Thus, \( \mu = \frac{K_e}{B} \) because the current loop area is \( \pi r^2 \) and using circular motion conditions.

03

Conclude magnetic dipole direction for electron (a)

The magnetic dipole moment direction follows the right-hand rule for motion of negative charges. The electron moving clockwise induces a dipole moment opposite to \( \vec{B} \).

04

Calculate magnetic dipole moment for positive ion (b)

For a positive ion, the derived formula holds: \[ \mu = \frac{K_i}{B} \] This formula similarly represents the magnetic dipole moment of a positive ion due to its orbital motion.

05

Determine direction of dipole moment for ion (c)

A positive ion moving in the same circular path generates a magnetic dipole moment in the direction parallel to \( \vec{B} \), determined by the right-hand rule applied to positive charges.

06

Calculate electron and ion contributions to magnetization (d)

Magnetization \( M \) is the sum of dipole moments per unit volume. Calculate:1) \( \mu_{\text{electron}} = \frac{K_e}{B} \) for electrons, 2) \( \mu_{\text{ion}} = \frac{K_i}{B} \) for ions. Thus, using total density, \[ M = n(\mu_{\text{electron}} - \mu_{\text{ion}}) \], where \( n = 5.3 \times 10^{21} \text{ m}^{-3} \). Substitute given values \( K_e = 6.2 \times 10^{-20} \text{ J} \), \( K_i = 7.6 \times 10^{-21} \text{ J} \), and \( B = 1.2 \text{ T} \).

07

Perform final magnetization calculation

Compute\[ \mu_{\text{electron}} = \frac{6.2 \times 10^{-20} \text{ J}}{1.2 \text{ T}}, \quad \mu_{\text{ion}} = \frac{7.6 \times 10^{-21} \text{ J}}{1.2 \text{ T}} \]Find total magnetization\[ M = 5.3 \times 10^{21} \times (\mu_{\text{electron}} - \mu_{\text{ion}}) \].Substitute values to obtain \[ M \].

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

kinetic energy

Understanding kinetic energy is essential when analyzing motion in a magnetic field. Kinetic energy is the energy possessed by an object due to its motion. For an electron, its kinetic energy can be expressed as \[ K = \frac{1}{2}mv^2 \]where \( m \) is the mass and \( v \) is the velocity of the electron. In a magnetic field, this energy determines how the electron moves along a circular path. Calculating kinetic energy is crucial for deriving the magnetic dipole moment of particles like electrons and ions.

magnetic field

A magnetic field is an invisible field that exerts a force on particles like electrons and ions, causing them to move in specific ways. Magnetic fields are vectors, meaning they have both a magnitude and a direction. Understanding the behavior of a charged particle in a magnetic field relies on knowing its relation:

• The direction of the magnetic field vector \( \vec{B} \) is critical as it influences the path of the charged particle.
• Charged particles moving in magnetic fields experience a force perpendicular to both their velocity and the magnetic field.

The interaction between charged particles and magnetic fields is at the heart of phenomena like magnetization.

electron motion

The motion of an electron within a magnetic field is fascinating. Electrons follow a circular path due to the force exerted by the magnetic field. This phenomenon can be explained by the Lorentz force:\[ F = qvB = \frac{mv^2}{r} \]where \( q \) is the charge, \( v \) is the velocity, \( B \) is the magnetic field strength, and \( r \) is the radius of the circle. The electron contributes to the formation of a magnetic dipole moment, a concept that can be derived using its orbital motion.

right-hand rule

The right-hand rule is a useful mnemonic for understanding the direction of forces and moments in magnetic fields. It helps in predicting the direction of the magnetic dipole moment generated by moving charges.

• For negative charges like electrons, point your thumb in the direction of the velocity (the direction the electron travels).
• Your fingers should curl in the direction of the magnetic field lines.

For electrons, this rule shows that the magnetic dipole moment is opposite to the field direction. Conversely, for positive ions, the dipole moment follows the direction of the magnetic field.

positive ion

Positive ions moving in a magnetic field have their own set of rules, similar to electrons but opposite in direction. When a positive ion circulates in a magnetic field, its magnetic dipole moment aligns parallel to the magnetic field vector. This distinction influences the overall magnetization of a material:

• For positive ions, the magnetic dipole moment \( \mu \) is parallel to the magnetic field \( \vec{B} \).
• Derived using the same principles as for electrons but with opposite orientation.

The behavior of positive ions, when mixed with electrons in a material, affects how strongly the material responds to magnetic fields.

magnetization

Magnetization refers to the vector field representing the density of magnetic dipole moments in a magnetic material. It is calculated as the net dipole moment per unit volume. Given a mix of electrons and positive ions, like in an ionized gas, their contributions are considered individually. The magnetization \( M \) can be expressed as:\[ M = n(\mu_{\text{electron}} - \mu_{\text{ion}}) \]where \( n \) is the number density of particles. Here are key points to remember:

• Magnetization is influenced by both the number density of particles and their individual dipole moments.
• The difference between electron and ion dipole moments is crucial to determining the net magnetization.

Understanding magnetization provides insight into how materials like gases respond to external magnetic fields.