Textbook Question
The spectrum of the sodium atom is detected in the light from a distant galaxy. If the -nm line is redshifted to nm, at what speed is the galaxy receding from the earth?
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Ch 44: Particle Physics and Cosmology
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 44, Problem 32
In an experiment done in a laboratory on the earth, the wavelength of light emitted by a hydrogen atom in the to transition is nm. In the light emitted by the quasar 3C273 (see Problem ), this spectral line is redshifted to nm. Assume the redshift is described by Eq. () and use the Hubble law to calculate the distance in light-years of this quasar from the earth.
Verified step by step guidance1
Step 1: Understand the redshift phenomenon. Redshift occurs when the wavelength of light emitted by an object is stretched due to the object moving away from the observer. The redshift (z) is defined as \( z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} \), where \( \lambda_{observed} \) is the observed wavelength and \( \lambda_{emitted} \) is the emitted wavelength.
Step 2: Calculate the redshift \( z \) using the given wavelengths. Substitute \( \lambda_{observed} = 563.9 \, \text{nm} \) and \( \lambda_{emitted} = 486.1 \, \text{nm} \) into the formula \( z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} \).
Step 3: Use the Hubble law to relate the redshift to the distance of the quasar. The Hubble law states \( v = H_0 \cdot d \), where \( v \) is the recession velocity, \( H_0 \) is the Hubble constant, and \( d \) is the distance. The recession velocity \( v \) can be approximated as \( v \approx c \cdot z \), where \( c \) is the speed of light.
Step 4: Substitute \( v = c \cdot z \) into the Hubble law to find the distance \( d \). Rearrange the equation to \( d = \frac{v}{H_0} \), and substitute \( v \approx c \cdot z \) to get \( d = \frac{c \cdot z}{H_0} \). Use the known values for \( c \) (speed of light) and \( H_0 \) (Hubble constant, typically \( 70 \, \text{km/s/Mpc} \) or another given value).
Step 5: Convert the distance \( d \) from megaparsecs (Mpc) to light-years. Use the conversion factor \( 1 \, \text{Mpc} = 3.26 \times 10^6 \, \text{light-years} \). Multiply the distance in Mpc by this factor to express the final distance in light-years.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Wavelength and Energy Levels
In atomic physics, the wavelength of light emitted during electron transitions between energy levels is crucial. For hydrogen, when an electron moves from a higher energy level (n=4) to a lower one (n=2), it emits a photon with a specific wavelength, which can be calculated using the Rydberg formula. This relationship between energy levels and emitted wavelengths is foundational for understanding atomic spectra.
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Redshift
Redshift occurs when light from an object is stretched to longer wavelengths as it moves away from an observer, often due to the expansion of the universe. It is quantified by the formula z = (λ_observed - λ_emitted) / λ_emitted, where λ represents the wavelengths. In cosmology, redshift is a key indicator of the velocity and distance of celestial objects, such as quasars.
Hubble's Law
Hubble's Law states that the recessional velocity of a galaxy (v) is directly proportional to its distance (d) from Earth, expressed as v = H₀ * d, where H₀ is the Hubble constant. This relationship allows astronomers to estimate distances to faraway galaxies and quasars based on their redshift, providing insights into the structure and expansion of the universe.
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