The Physics
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Opus in profectus

Blackbody Radiation

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Problems

practice

  1. A laser used in a fiber optic communication system operates at a wavelength of 635 nm, has a power output of 1 mW, and can transmit data at a rate of 2.5 gigabits per second. Compute the following quantities…
    1. the frequency of the electromagnetic radiation
    2. the color of the electromagnetic radiation
    3. the duration of a single pulse
    4. the length of a single pulse
    5. the number of wavelengths in a single pulse
    6. the energy of a single pulse
    7. the energy of a single photon
    8. the number of photons in a single pulse
  2. Write something.
  3. Write something.
  4. Combine the speed of light (c), gravitational constant (G), and the reduced Planck constant (ℏ) so as to generate the Planck units of…
    1. length
    2. mass
    3. time
    Add boltzmann constant (k) to the list and generate the Planck unit of…
    1. temperature
    Combine your results and generate the Planck units of…
    1. acceleration
    2. force
    3. momentum
    4. energy
    5. power
    6. pressure
    7. density
    8. angular frequency
    Include the constant from Coulomb's law (1/4πε0) and generate the Planck units of…
    1. charge
    2. current
    3. voltage
    4. resistance
    5. magnetic flux

conceptual

  1. Pigments are used to give color to paint (on a house or on an artist's canvas), fibers (in clothing, carpeting, or tapestries), and glaze (on ceramic pottery or mosaic tiles).
    1. Why does red pigment appear red? Why does blue pigment appear blue?
    2. How does the energy of a photon of red light compare to the energy of a photon of blue light?
    3. Which pigment is more likely to fade first when exposed to sunlight, red or blue? Justify your answer.
  2. Three related questions.
    1. Why are ultraviolet, x-rays, and gamma rays always regarded as harmful while infrared, microwaves, and radio waves are generally regarded as benign (by sensible people, anyway).
    2. In what sense can visible light be thought of as "just right" as an energy source for life (and other forms of chemistry) on Earth?
    3. Under what circumstances could exposure to the generally benign electromagnetic waves listed in part a. be considered harmful?

numerical

  1. A microwave oven emits 700 W of radiation at a frequency of 2.45 GHz. How many photons does it generate per second?
  2. The lasers used in many lidar systems are typically one of two infrared wavelengths: 1550 nm or 905 nm.
    1. How many photons of 1550 nm IR does it take to equal 1 nJ of energy?
    2. How many photons of 905 nm IR does it take to equal 1 nJ of energy?
  3. The filament of a 60 W incandescent light bulb has a temperature of 3300 K.
    1. Assume the filament is an ideal blackbody and determine the peak wavelength of its spectrum.
    2. Assume the peak wavelength is equal to the mean wavelength of the visible photons emitted and determine the number of photons emitted per second.
  4. The world's most powerful laser is the Petawatt at Lawrence Livermore National Laboratory in California. At full energy of about 680 joules, the shots delivered by the Petawatt last a mere 440 femtoseconds. The resulting rapid dump of energy when applied to a small space is sufficient to knock neutrons loose from medium sized nuclei like gold or to split heavier nuclei like uranium in two. When applied to light nuclei like hydrogen, the power of the Petawatt could be used to initiate nuclear fusion (the process that powers the Sun) in reactors on Earth.
    1. What is the power of the Petawatt beam?
    2. What is the energy per photon assuming a wavelength of 1057 nm?
    3. How many photons are delivered per shot?
  5. Complete the following table for photons of visible light.
    characteristic longest wavelength shortest wavelength
    wavelength (m) 700 nm 400 nm
    frequency (Hz)
    energy per photon (J)
    energy per photon (eV)
    color
  6. Complete the following table for different types of electromagnetic radiation.
    energy per
    photon (eV)    		 energy per
    photon (J)    		 frequency
    (Hz)    		 wavelength
    (m)    		 type of
    radiation
    100 eV
    103 eV
    106 eV
  7. The Balmer series is a set of wavelengths emitted by excited hydrogen atoms. Complete the following table for the four visible wavelengths in the Balmer series identified by the Greek letters α (alpha), β (beta), γ (gamma), and δ (delta).
    characteristic α β γ δ
    wavelength (nm) 656 nm 486 nm 434 nm 410 nm
    frequency (Hz)
    energy per photon (J)
    energy per photon (eV)
    color
  8. Photosynthesis is the process used by plants, algae, and some bacteria to convert light energy into chemical energy. A counterintuitive feature of photosynthesis is that the primary pigments involved absorb light on either side of the central part of the spectrum (typically red and blue photons) and reflect light near the center of the spectrum (green photons). Some pigments from three diverse photosynthetic systems are given in the table below. Each has two peak wavelengths around which photons are most often absorbed and used to do work. Determine the corresponding frequencies, photon energies, and color (if visible) or spectral band (infrared or ultraviolet).
    line graph line graph line graph
    Absorbance peaks of some photosynthetic pigments Chl a/b: Chlorophyll a and b are found in green plants. LH2: The light harvesting complex of purple bacteria. BChl a/b: Bacteriochlorophyll a and b are found in green sulfur bacteria. Source: Arp, et al. 2020
    pigment peak λ (nm) f (THz) E (10−19 J) E (eV) color
    Chl a shorter 428
    Chl a longer 660
    Chl b shorter 440
    Chl b longer 652
    LH2 shorter 801
    LH2 longer 857
    BChl c shorter 431
    BChl c longer 740
    BChl e shorter 461
    BChl e longer 655
  9. Aerial mapping using light detection and ranging (lidar), also known as airborne laser scanning (ALS), is a remote sensing technology for mapping the Earth's surface that is especially useful in areas with heavy vegetation cover. Multispectral lidar systems use lasers with different wavelengths to determine vegetation type (forest, grassland, cropland, etc.) as well as altitude. With the right combination of wavelengths, it is even possible to subtract the vegetation from an image making it possible to "see" terrain hidden beneath a forest canopy. Specifications for the lasers in one multispectral lidar system are given in the table below. Complete the table using the information given and your knowledge of physics.
    Laser specifications of a multispectral lidar mapping system Adapted from: Fernandez-Diaz, et al. 2016.
    property channel 1 channel 2 channel 3
    wavelength (nm) 1550 1064 532
    look angle (°) 3.5 forward nadir 7.0 forward
    repetition rate (kHz) 50–300 50–300 50–300
    beam divergence (°) 0.02 0.02 0.06
    pulse energy (μJ) 35 15 30
    pulse duration (ns) 2.7 3.5 3.7
    pulse power (kW)
    frequency (THz)
    energy per photon (10−19 J)
    energy per photon (eV)
    photons per pulse (1012)
    color (or IR or UV)

    Magnify

  10. Read the following information-rich paragraph about orbiting lidar systems used to map the moon, then complete the table below it.

    The first topographic mapping of the Moon from orbit was performed by Apollo 15 in 1970, which was then extended by the Apollo 16 and Apollo 17 missions. The lasers were flash-lamp pumped based and mechanically Q-switched with a ruby laser emitting at 694.23 nm with a pulse energy of 200 nJ and a pulse width of 10 ns. In 1994, the Clementine mission was launched, which mapped the surface of the Moon over a period of more than two months. It carried on board a compact, lightweight, diode-pumped Nd:Cr:YAG laser, emitting pulses of 180 mJ energy and 10 ns width at 1064 nm. The Q-switching was performed by a Lithium Niobate (LiNbO3) crystal-based Pockels cell. SELENE, also known as Kaguya, was a Japanese mission launched in 2007 that orbited the Moon for more than one and a half years and surveyed it with LIDAR technology using an Nd:YAG laser with 100 mJ output power at 1064 nm, a pulse width of 15 ns and a repetition rate of 1 Hz. The Q-switch was also performed actively by a LiNbO3-based Pockels cell. China also created a lunar exploration program called Chang'E. The first mission was launched in 2007 and the lunar surface was mapped between November 2007 and July 2008. The laser specifications were similar to those of previous missions, with an output wavelength of 1064 nm emitted from a diode-pumped Nd:YAG crystal actively Q-switched with a potassium dideuterium phosphate (KD*P) crystal. The energy emitted was 150 mJ per pulse and a pulse width of less than 7 ns at repetition rate of 1 Hz. A similar laser was used for the Chang'E-2 mission launched in 2010, but with a repetition rate of 5 Hz. The resolution of the measurements was increased to achieve a 5 m vertical accuracy. In 2008, India launched its first lunar probe called Chandrayaan, equipped with the Lunar Laser Ranging Instrument (LLRI) that mapped the surface topography using a Q-switched Nd:YAG laser at 1064 nm with a pulse energy of 100 mJ and a pulse width of 15 ns. The active Q-switching was also performed by an LiNbO3 crystal. The NASA Lunar Orbiter Laser Altimeter (LOLA) was also launched in 2008, one of its objectives being the validation of technologies for following missions to the Moon. It uses a DPSSL with an Nd:Cr:YAG slab with passive Q-switch and a cross-Porro resonator configuration working at 1064 nm, 28 Hz repetition rate and about 3 mJ at 5 ns for the both mounted redundant lasers.

    Guilhot and Ribes-Pleguezuelo, 2019.

    Lidar systems used on lunar missions
    Apollo Clementine Selene Chang'E Chandrayaan LOLA
    nation US US Japan China India US
    initial year 1970 1994 2007 2007 2008 2008
    laser type ruby Nd:Cr:YAG Nd:YAG Nd:YAG Nd:YAG Nd:Cr:YAG
    wavelength (nm)
    pulse energy (mJ)
    pulse duration (ns)
    pulse power (kW)
    frequency (THz)
    energy per photon (10−19 J)
    energy per photon (eV)
    photons per pulse (1015)
    color (or IR or UV)
  11. At sufficiently high energies, all particles will scatter off the photons that make up the cosmic microwave background. Cosmic rays with energies greater than about 5 × 1019 eV cannot travel very far before they are scattered and lose energy. This is known as the Greisen-Zatsepin-Kuzmin limit (or GZK limit). Cosmic rays with energies greater than this are not impossible, just exceedingly rare. The most famous ultrahigh energy cosmic ray is probably the Oh-My-God particle detected at the Fly's Eye camera in Utah in 1991. It was reported to have an energy of 3.2 × 1020 eV.
    1. Determine the speed of a 45.5 g golf ball that would have a kinetic energy equivalent to…
      1. a cosmic ray at the GZK limit
      2. the Oh-My-God particle
    2. Determine the speed of a 145 g baseball that would have a kinetic energy equivalent to…
      1. a cosmic ray at the GZK limit
      2. the Oh-My-God particle

statistical

  1. led.txt
    Four physics students measured the voltage drop across a batch of light emitting diodes (LEDs) connected to a simple circuit. They used a spectrometer to measure the peak wavelength of the light emitted from each LED.
    1. Calculate the frequency (f) corresponding to the peak wavelength for each LED in hertz (Hz).
    2. Calculate the energy loss (E) of the electrons flowing through the LEDs in joules (J).
    3. Construct a graph of energy vs. frequency. Add a best fit straight line to the graph.
    4. Determine the value of Planck's constant (h) in joule seconds (J s).
  2. led.txt
    Four physics students measured the voltage drop across a batch of light emitting diodes (LEDs) connected to a simple circuit. They used a spectrometer to measure the peak wavelength of the light emitted from each LED.
    1. Calculate the inverse peak wavelength (λ−1) for each LED in inverse nanometers (nm−1).
    2. Calculate the energy loss (E) of the electrons flowing through the LEDs in electronvolts (eV).
    3. Construct a graph of energy vs. inverse wavelength. Add a best fit straight line to the graph.
    4. Determine the value of Planck's constant times the speed of light (hc) in electronvolt nanometers (eV nm).