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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Explain the expansion of the universe in terms of a Hubble graph and cosmological redshift
  • Describe the analogy between cosmological expansion and an expanding balloon
  • Use Hubble’s law to make predictions about the measured speed of distant galaxies

We have been discussing elementary particles, which are some of the smallest things we can study. Now we are going to examine what we know about the universe, which is the biggest thing we can study. The link between these two topics is high energy: The study of particle interactions requires very high energies, and the highest energies we know about existed during the early evolution of the universe. Some physicists think that the unified force theories we described in the preceding section may actually have governed the behavior of the universe in its earliest moments.

Hubble’s Law

In 1929, Edwin Hubble published one of the most important discoveries in modern astronomy. Hubble discovered that (1) galaxies appear to move away from Earth and (2) the velocity of recession (v) is proportional to the distance (d) of the galaxy from Earth. Both v and d can be determined using stellar light spectra. A best fit to the sample illustrative data is given in Figure 11.18. (Hubble’s original plot had a considerable scatter but a general trend was still evident.)

Graph of velocity v in km per s versus distance d in Mpc. A line from the origin forms an angle of roughly 45 degrees with the x axis. Many dots close to the line are highlighted.
Figure 11.18 This graph of red shift versus distance for galaxies shows a linear relationship, with larger red shifts at greater distances, implying an expanding universe. The slope gives an approximate value for the expansion rate. (credit: John Cub)

The trend in the data suggests the simple proportional relationship:

v=H0d,v=H0d,
(11.12)

where H0=70km/s/MpcH0=70km/s/Mpc is known as Hubble’s constant. (Note: 1 Mpc is one megaparsec or one million parsecs, where one parsec is 3.26 light-years.) This relationship, called Hubble’s law, states that distant stars and galaxies recede away from us at a speed of 70 km/s for every one megaparsec of distance from us. Hubble’s constant corresponds to the slope of the line in Figure 11.18. Hubble’s constant is a bit of a misnomer, because it varies with time. The value given here is only its value today.

Interactive

Watch this video to learn more about the history of Hubble’s constant.

Hubble’s law describes an average behavior of all but the closest galaxies. For example, a galaxy 100 Mpc away (as determined by its size and brightness) typically moves away from us at a speed of

v=((70kms)/Mpc)(100Mpc)=7000km/s.v=((70kms)/Mpc)(100Mpc)=7000km/s.

This speed may vary due to interactions with neighboring galaxies. Conversely, if a galaxy is found to be moving away from us at speed of 100,000 km/s based on its red shift, it is at a distance

d=v/H0=(10,000km/s)/((70kms)/Mpc)=143Mpc.d=v/H0=(10,000km/s)/((70kms)/Mpc)=143Mpc.

This last calculation is approximate because it assumes the expansion rate was the same 5 billion years ago as it is now.

Big Bang Model

Scientists who study the origin, evolution, and ultimate fate of the universe (cosmology) believe that the universe began in an explosion, called the Big Bang, approximately 13.7 billion years ago. This explosion was not an explosion of particles through space, like fireworks, but a rapid expansion of space itself. The distances and velocities of the outward-going stars and galaxies permit us to estimate when all matter in the universe was once together—at the beginning of time.

Scientists often explain the Big Bang expansion using an inflated-balloon model (Figure 11.19). Dots marked on the surface of the balloon represent galaxies, and the balloon skin represents four-dimensional space-time (Relativity). As the balloon is inflated, every dot “sees” the other dots moving away. This model yields two insights. First, the expansion is observed by all observers in the universe, no matter where they are located. The “center of expansion” does not exist, so Earth does not reside at the “privileged” center of the expansion (see Exercise 11.24).

Figure a shows a balloon connected to a cylinder for inflation. The balloon is marked with a grid, and a few dots on the grid are highlighted. Figure b shows the same balloon, now inflated. The highlighted dots are further apart from each other.
Figure 11.19 An analogy to the expanding universe: The dots move away from each other as the balloon expands; compare (a) to (b) after expansion.

Second, as mentioned already, the Big Bang expansion is due to the expansion of space, not the increased separation of galaxies in ordinary (static) three-dimensional space. This cosmological expansion affects all things: dust, stars, planets, and even light. Thus, the wavelength of light (λ)(λ) emitted by distant galaxies is “stretched” out. This makes the light appear “redder” (lower energy) to the observer—a phenomenon called cosmological redshift. Cosmological redshift is measurable only for galaxies farther away than 50 million light-years.

Example 11.8

Calculating Speeds and Galactic Distances A galaxy is observed to have a redshift:

z=λobsλemitλemit=4.5.z=λobsλemitλemit=4.5.

This value indicates a galaxy moving close to the speed of light. Using the relativistic redshift formula (given in Relativity), determine (a) How fast is the galaxy receding with respect to Earth? (b) How far away is the galaxy?

Strategy We need to use the relativistic Doppler formula to determine speed from redshift and then use Hubble’s law to find the distance from the speed.

Solution

  1. According to the relativistic redshift formula:
    z=1+β1β−1,z=1+β1β−1,

    where β=v/c.β=v/c. Substituting the value for z and solving for ββ, we get β=0.93.β=0.93. This value implies that the speed of the galaxy is 2.8×108m/s2.8×108m/s.
  2. Using Hubble’s law, we can find the distance to the galaxy if we know its
    recession velocity:
    d=vH0=2.8×108m/s73.8×103m/sper Mpc=3.8×103Mpc.d=vH0=2.8×108m/s73.8×103m/sper Mpc=3.8×103Mpc.

Significance Distant galaxies appear to move very rapidly away from Earth. The redshift of starlight from these galaxies can be used to determine the precise speed of recession, over 90%90% of the speed of light in this case. This motion is not due to the motion of galaxy through space but by the expansion of space itself.

Check Your Understanding 11.8

The light of a galaxy that moves away from us is “redshifted.” What occurs to the light of a galaxy that moves toward us?

Interactive

View this video to learn more about the cosmological expansion.

Structure and Dynamics of the Universe

At large scales, the universe is believed to be both isotropic and homogeneous. The universe is believed to isotropic because it appears to be the same in all directions, and homogeneous because it appears to be the same in all places. A universe that is isotropic and homogeneous is said to be smooth. The assumption of a smooth universe is supported by the Automated Plate Measurement Galaxy Survey conducted in the 1980s and 1900s (Figure 11.20). However, even before these data were collected, the assumption of a smooth universe was used by theorists to simplify models of the expansion of the universe. This assumption of a smooth universe is sometimes called the cosmological principle.

Picture shows an oval shape with a black background. Many galaxies are seen inside it.
Figure 11.20 The Automated Plate Measurement (APM) Galaxy Survey. Over 2 million galaxies are depicted in a region 100 degrees across centered toward the Milky Way’s south pole. (credit: 2MASS/T. H. Jarrett, J. Carpenter, & R. Hurt)

The fate of this expanding and smooth universe is an open question. According to the general theory of relativity, an important way to characterize the state of the universe is through the space-time metric:

ds2=c2dt2a(t)2dΣ2,ds2=c2dt2a(t)2dΣ2,
(11.13)

where c is the speed of light, a is a scale factor (a function of time), and dΣdΣ is the length element of the space. In spherical coordinates (r,θ,ϕ)r,θ,ϕ), this length element can be written

dΣ2=dr21kr2+r2(dθ2+sin2θdφ2),dΣ2=dr21kr2+r2(dθ2+sin2θdφ2),
(11.14)

where k is a constant with units of inverse area that describes the curvature of space. This constant distinguishes between open, closed, and flat universes:

  • k=0k=0 (flat universe)
  • k>0k>0 (closed universe, such as a sphere)
  • k<0k<0 (open universe, such as a hyperbola)

In terms of the scale factor a, this metric also distinguishes between static, expanding, and shrinking universes:

  • a=1a=1 (static universe)
  • da/dt>0da/dt>0 (expanding universe)
  • da/dt<0da/dt<0 (shrinking universe)

The scale factor a and the curvature k are determined from Einstein’s general theory of relativity. If we treat the universe as a gas of galaxies of density ρρ and pressure p, and assume k=0k=0 (a flat universe), than the scale factor a is given by

d2adt2=4πG3(ρ+3p)a,d2adt2=4πG3(ρ+3p)a,
(11.15)

where G is the universal gravitational constant. (For ordinary matter, we expect the quantity ρ+3pρ+3p to be greater than zero.) If the scale factor is positive (a>0a>0), the value of the scale factor “decelerates” (d2a/dt2<0d2a/dt2<0), and the expansion of the universe slows down over time. If the numerator is less than zero (somehow, the pressure of the universe is negative), the value of the scale factor “accelerates,” and the expansion of the universe speeds up over time. According to recent cosmological data, the universe appears to be expanding. Many scientists explain the current state of the universe in terms of a very rapid expansion in the early universe. This expansion is called inflation.

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