Domanda del libro di testo
INT A beam of electrons is incident upon a gas of hydrogen atoms. Through what potential difference must the electrons be accelerated to have this speed?
111
views
Ch 38: Quantization
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 38, Problema 72b
Consider an electron undergoing cyclotron motion in a magnetic field. According to Bohr, the electron’s angular momentum must be quantized in units of ℏ. Compute the first four allowed radii in a 1.0 T magnetic field.
Guida verificata passo dopo passo1
Step 1: Begin by understanding the quantization condition for angular momentum. According to Bohr's quantization rule, the angular momentum of the electron is given by \( L = n \hbar \), where \( n \) is a positive integer (quantum number), and \( \hbar \) is the reduced Planck's constant.
Step 2: Relate the angular momentum \( L \) to the cyclotron motion of the electron. The angular momentum for circular motion is \( L = m_e v r \), where \( m_e \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the circular orbit.
Step 3: Use the force balance for cyclotron motion. The centripetal force is provided by the Lorentz force due to the magnetic field \( B \): \( m_e v^2 / r = e v B \), where \( e \) is the charge of the electron. Simplify this to \( v = e B r / m_e \).
Step 4: Substitute \( v \) from the force balance equation into the angular momentum expression \( L = m_e v r \). This gives \( L = m_e (e B r / m_e) r = e B r^2 \). Equating this to Bohr's quantization condition \( L = n \hbar \), we find \( e B r^2 = n \hbar \). Solve for \( r \): \( r = \sqrt{(n \hbar) / (e B)} \).
Step 5: Compute the first four allowed radii by substituting \( n = 1, 2, 3, 4 \), \( \hbar \approx 1.054 \times 10^{-34} \, \mathrm{J \cdot s} \), \( e \approx 1.602 \times 10^{-19} \, \mathrm{C} \), \( B = 1.0 \, \mathrm{T} \), and \( m_e \approx 9.109 \times 10^{-31} \, \mathrm{kg} \) into the formula \( r = \sqrt{(n \hbar) / (e B)} \).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Concetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Cyclotron Motion
Cyclotron motion refers to the circular motion of a charged particle, such as an electron, in a magnetic field. The magnetic force acts as a centripetal force, causing the particle to move in a circular path. The frequency of this motion, known as the cyclotron frequency, is directly proportional to the strength of the magnetic field and the charge-to-mass ratio of the particle.
Video consigliato:
Percorso guidato
09:57
Motional EMF
Quantization of Angular Momentum
In quantum mechanics, the angular momentum of a particle is quantized, meaning it can only take on discrete values. For an electron in a magnetic field, Bohr's model states that the angular momentum must be an integer multiple of ℏ (reduced Planck's constant). This quantization leads to specific allowed energy levels and corresponding radii for the electron's orbit.
Video consigliato:
Percorso guidato
06:18Intro to Angular Momentum
Magnetic Field Strength
Magnetic field strength, measured in teslas (T), is a measure of the magnetic force experienced by a charged particle. In this context, a 1.0 T magnetic field indicates a strong magnetic influence on the electron's motion. The strength of the magnetic field directly affects the radius of the electron's cyclotron motion, as stronger fields result in smaller radii for the allowed orbits.
Video consigliato:
Percorso guidato
05:30Magnetic Fields and Magnetic Dipoles
Pratica correlata
Domanda del libro di testo
Consider an electron undergoing cyclotron motion in a magnetic field. According to Bohr, the electron’s angular momentum must be quantized in units of ℏ. Find an expression for the allowed energy levels En in terms of ℏ and the cyclotron frequency fcyc.
126
views
Domanda del libro di testo
In the atom interferometer experiment of Figure 38.13, laser-cooling techniques were used to cool a dilute vapor of sodium atoms to a temperature of 0.0010 K=1.0 mK. The ultracold atoms passed through a series of collimating apertures to form the atomic beam you see entering the figure from the left. The standing light waves were created from a laser beam with a wavelength of 590 nm. Because interference is observed between the two paths, each individual atom is apparently present at both point B and point C. Describe, in your own words, what this experiment tells you about the nature of matter.
1442
views
Domanda del libro di testo
The electrons in a cathode-ray tube are accelerated through a 250 V potential difference and then shot through a 33-nm-diameter circular aperture. What is the diameter of the bright spot on an electron detector 1.5 m behind the aperture?
78
views