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Physics Workshop Class

This is a separate 1-credit course that requires registering with the registrar's office

Workshop sections

Course # Section Day, Time Place Workshop leader Email
PH 299-001 62985 Th 12:00-13:15 NH 385 Ameena Alattar aka6
PH 299-003 62982 Th 14:00-15:50 NH 391 Bee Bui bui
PH 299-002 62980 Fr 10:15-12:05 NH 389 Patrick Bradley pat5
PH 299-006 62983 Fr 14:00-15:50 CH 254 Eliza Slater elizaslater @ gmail

Workshop syllabus

ph299_syllabus_14sp.pdf: available for download as a PDF file

Workshop 9 (May 29,30)

Quantum Physics

Equations and Relations WS.9

  1. Wien's displacement law
    • $f_\text{peak}=$ $\big(5.88\times 10^{10}\,{\text s}^{-1}\cdot{\text K}^{-1}\big)T$
  2. Total energy at a frequency $f$ of a black-body system (Planck's hypothesis):
    • $E_n=$ $nhf\;$, where $\;n=0,\,1,\,2,\,3,\,...$ and $\;h=6.62606957\times 10^{-34}\,{\text J}\!\cdot\!{\text s}\,$ is the Planck constant
  3. Photon energy
    • $E=hf$
  4. Cutoff frequency in photoelectric effect
    • $f_0=\frac{W_0}{h}$
  5. Maximum kinetic energy in photoelectric effect
    • $K_\text{max}=hf-W_0$
  6. Momentum of a photon
    • $p=$ $\frac{hf}{c}$ $=\frac{h}{\lambda}$
  7. Compton effect
    • $\Delta\lambda=$ $\lambda'-\lambda=$ $\frac{h}{m_ec}\big(1-\cos\theta\big)$
  8. de Broglie wavelength
    • $\lambda=$ $\frac{h}{p}$
  9. Energy
    • $E=\frac{p^2}{2m}$
  10. Crystal diffraction
    • $2d\sin\theta$ $=m\lambda\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;m=1,\, 2,\, 3,...$
  11. Heisenberg uncertainty principle
    • $\Delta p_y\Delta y\ge\frac{h}{2\pi}$
    • $\Delta E\Delta t\ge\frac{h}{2\pi}$

Problem 9.1

A photon of wavelength $2.0\times 10^{-11}\,$m strikes a free electron of mass $m_e$ that is initially at rest. After the collision, the photon is shifted in wavelength by an amount $\;\Delta\lambda=\frac{2h}{m_ec}\,$, and reversed in direction.

  • (a) Determine the kinetic energy of the electron after the collision (in joules and eV).
  • (b) Determine the momentum of the incident photon.
  • (c) Is the photon wavelength increased or decreased by the interaction? Explain.
  • (d) Determine the magnitude of the momentum acquired by the electron.

Solution:

  • (a) $\Delta\lambda=4.85 \times 10^{-12}\,$m
    • Initial photon energy $9.932\times 10^{-15}\,$J
    • Final photon energy $7.994\times 10^{-15}\,$J
    • The difference goes to the electron: $1.938\times 10^{-15}\,$J $=12.1\,$keV
  • (b) $3.313\times 10^{-23}\frac{ {\text{kg}}\cdot{\text m}}{\text s}$
  • (c) Increased, because some energy got transferred to the electron
  • (d) $5.98\times 10^{-23}\frac{ {\text{kg}}\cdot{\text m}}{\text s}$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.2

A photon of energy 240 keV is scattered by a free electron. If the recoil electron has a kinetic energy of 190 keV, what is the wavelength of the scattered photon?

Solution:

  • The energy of the scattered photon is $240-190=50\,$keV
  • The corresponding wavelength is $\lambda_\text{sc}=2.48\times 10^{-11}\,$m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.3

An incident photon of wavelength 0.0100 nm is Compton scattered; the scattered photon has a wavelength of 0.0124 nm. What is the change in kinetic energy of the electron that scattered the photon?

Solution:

  • $1.986\times 10^{-14}\,{\text J}$ $-1.602\times 10^{-14}\,{\text J}$ $=0.384\,$J

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 9.1

  1. The Compton shift is the same for x-rays and for visible light. Why is it that the Compton shift for x-rays can be measured readily but that for visible light cannot?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.4

Two different monochromatic light sources, one yellow (580 nm) and one violet (425 nm), are used in a photoelectric effect experiment. The metal surface has a photoelectric threshold frequency of $6.20\times 10^{14}\,$Hz.

  • (a) Are both sources able to eject photoelectrons from the metal? Explain.
  • (b) How much energy is required to eject an electron from the metal?

Solution:

  • (a) The violet will eject electrons, but the yellow will not (see b)
  • (b) $4.2\times 10^{-19}\,{\text J}$ $=2.6\,$eV (corresponding to 484-nm light)

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 9.2-9.4

  1. In a photoelectric effect experiment, how is the stopping potential determined? What does the stopping potential tell us about the electrons emitted from the metal surface?
  2. Of the following statements about the photoelectric effect, which are true and which are false?
    1. The greater the frequency of the incident light, the greater the stopping potential.
    2. The greater the intensity of the incident light, the greater the cutoff frequency.
    3. The greater the work function of the target material, the greater the stopping potential.
    4. The greater the work function of the target material, the greater the cutoff frequency.
    5. The greater the frequency of the incident light, the greater the maximum kinetic energy of the ejected electrons.
    6. The greater the energy of the photons, the smaller the stopping potential.
  3. A darkroom used for developing black-and-white film can be dimly lit by a red light bulb without ruining the film. Why is a red light bulb used rather than white or blue or some other color?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.5

What are the de Broglie wavelengths of electrons with the following values of kinetic energy?

  • (a) 1.0 eV
  • (b) 1.0 keV

Solution:

  • (a) $1.23\times 10^{-9}\,$m
  • (b) $3.88\times 10^{-11}\,$m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.6

What is the ratio of the wavelength of a 0.100-keV photon to the wavelength of a 0.100-keV electron?

Solution:

  • $\frac{1.2\,\times\,10^{-8}\,{\text m} }{1.23\,\times\,10^{-10}\,{\text m}}$ $\approx 98$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.7

What is the de Broglie wavelength of a basketball of mass 0.50 kg when it is moving at 10 m/s? Why don't we see diffraction effects when a basketball passes through the circular aperture of the hoop?

Solution:

  • $1.33\times 10^{-34}\,$m. Diffraction angles are really small.

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.8

An electron passes through a slit of width $1.0\times 10^{-8}\,$m. What is the uncertainty in the electron’s momentum component in the direction parallel to the slit?

Solution:

  • parallel to the slit, $\approx 0$
  • perpendicular to the slit, $1.05\times 10^{-26}\frac{ {\text{kg}}\cdot{\text m}}{\text s}$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.9

If the momentum of the basketball (see problem above) has a fractional uncertainty of $\frac{\Delta p}{p} = 10^{-6}$, what is the uncertainty in its position?

Solution:

  • $2.1\times 10^{-29}\,$m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 9.5-9.6

  1. The uncertainty principle does not allow us to think of the electron in an atom as following a well-defined trajectory. Why, then, are we able to define trajectories for golf balls, comets, and the like?
    • [Hint: How are the uncertainties in momentum and velocity related?]
  2. An electron and a proton have the same kinetic energy. Which has the greater de Broglie wavelength?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 9.10

Halogen light bulbs can have higher filament temperatures than regular incandescent bulbs. A standard light bulb operates at about 2900 K while a halogen bulbs might be 3500 K hot.

  • (a) What is the peak frequency for both bulbs?
  • (b) The human eye is most sensitive in the green (~550 nm) part of the visible spectrum. Which bulb produces a peak frequency closer to the frequency of green light?
  • (c) At what temperature would a light bulb need to be to have peak frequency that corresponds to the frequency were the eye is most sensitive?

Solution:

  • (a) 1000 nm for standard, 828 nm for halogen
  • (b) Halogen
  • (c) 5270 K

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 9.7-9.10

  1. An incandescent light bulb is connected to a dimmer switch. When the bulb operates at full power, it appears white, but as it is dimmed it looks more and more red. Explain.
  2. Some stars are reddish in color, others bluish, and others yellowish-white (like the Sun). How is the color related to the surface temperature of the star? What color are the hottest stars? What color are the coolest?
  3. How can we demonstrate the existence of matter waves?
  4. Of the electromagnetic waves generated in a microwave oven, and in your dentist’s x-ray machine, which has
    1. the greater wavelength,
    2. the greater frequency, and
    3. the greater photon energy?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Workshop 8 (May 22,23)

Relativity

Equations and Relations WS.8

  1. Time dilation
    • $\Delta t=$ $\frac{\Delta t_0}{\sqrt{1-v^2/c^2}}$
  2. Length contraction
    • $L=$ $L_0 \sqrt{1-\frac{v^2}{c^2}}$
    • $v=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}$
  3. Relativistic momentum
    • $p=\frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}$
  4. Relativistic energy
    • $E=$ $\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}$ $=mc^2$
  5. Rest energy
    • $E_0=m_0c^2$
  6. Relativistic kinetic energy
    • $K=$ $\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0c^2$
  7. Schwarzschild radius
    • $R=\frac{2GM}{c^2}$
    • where $G=6.67384\times 10^{-11}\frac{ {\text N}\cdot{\text m}^2}{ {\text{kg}}^2}$ is the gravitational constant

Problem 8.1

A spaceship moves at a constant velocity of $0.40c$ relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be 1.0 m long.

  • (a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer?
  • (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?

Solution:

  • (a) 1 m
  • (b) 0.917 m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.2

A rectangular plate of glass, measured at rest, has sides 30.0 cm and 60.0 cm.

  • (a) As measured in a reference frame moving parallel to the 60.0-cm edge at speed $0.25c$ with respect to the glass, what are the lengths of the sides?
  • (b) How fast would a reference frame have to move in the same direction that the plate of glass viewed in that frame is square?

Solution:

  • (a) $30.0\,{\text{cm}}\times 58.1\,{\text{cm}}$
  • (b) $0.866\,c$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.3

A spaceship travels at constant velocity from Earth to a point 510 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is $0.99c$. A passenger is 20 yr old when departing from Earth in the year 2000.

  • (a) How old is the passenger when the ship reaches its destination, as measured by the ship's clock?
  • (b) If the spaceship sends radio-signal reports back to Earth every 12 hours (by their clocks), at what interval are the reports sent to Earth, according to Earth clock?
  • (c) If the ship sends a radio signal back to Earth as soon as it reaches its destination, in what year, by Earth's calendar, does the signal reach Earth?

Solution:

  • (a) 92 years and 8 months
  • (b) 85.1 hours (time delay between “send” events according to Earth clock)
  • (c) The year 3025.

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 8.1-8.2

  1. A sprinter crosses the start line (event 1) and runs at constant velocity until he crosses the finish line (event 2). In what reference frame would an observer measure the proper time interval between these two events? In what reference frame would an observer measure the proper length of the track from start line to finish line?
  2. If the speed of light would be infinitely large, would you observe the effects of length contraction and time dilation?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.4

Electron A is moving west with speed $0.6c$ relative to the lab. Electron B is also moving west with speed $0.8c$ relative to the lab. What is the speed of electron B in a frame of reference which electron A is at rest?

Solution:

  • $0.385\,c$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.5

An observer on earth notices two spaceships approach at speeds of $0.75c$ and $0.5c$ respectively. What is the relative speed between the spaceships as measured by a passenger of one of the spaceships?

Solution:

  • $0.4\,c$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 8.3

  • You are moving at a speed $0.1c$ relative to Tom who shines a light toward you. At what speed do you see the light passing you by?

Solution:

  • c

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.6

A constant force is applied to a particle initially at rest. Sketch qualitative graphs of the particle's speed, momentum, and acceleration as functions of time. Assume that the force acts long enough so the particle achieves relativistic speeds.

Solution:

  • momentum grows linearly with time t
  • velocity plot looks like this
  • acceleration plot is here

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.7

An electron is accelerated from rest through a potential difference of $1 \times 10^7\,$V (also consider $1 \times 10^8\,$V).

  • (a) What is the relativistic kinetic energy of the electron?
  • (b) What is its speed?
  • (c) What relativistic kinetic energy and speed would a proton have after being accelerated by the same potential difference?
  • (d) What is the relativistic momentum of the proton and the electron?
  • (e) What potential difference do you need to accelerate an electron to $0.85c$?

Solution:

  • (a) same, $E_\text{kin}=$ $1 \times 10^7\,{\text{eV}}=$ $1.6\times 10^{-12}\,{\text J}\;\;$ (or $1 \times 10^8\,{\text{eV}}=$ $1.6\times 10^{-11}\,$J)
  • (b) $0.9988\,c\;\;$ (or, $0.99999\,c$)
  • (c) Same $E_\text{kin}$, but slower speed: 0.145 c (or, 0.428 c)
  • (d) $p_p=7.35\times 10^{-20}\,\frac{ {\text{kg}}\cdot{\text m}}{\text s}$, $p_e=5.6\times 10^{-21} \,\frac{ {\text{kg}}\cdot{\text m}}{\text s}\;\;$
    $\;\;\;\;\;$ (or, $p_p= 2.38\times 10^{-19}\,\frac{ {\text{kg}}\cdot{\text m}}{\text s}$, $p_e= 5.37\times 10^{-20} \,\frac{ {\text{kg}}\cdot{\text m}}{\text s}$)
  • (e) 459 kV

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 8.4

Why is it harder to accelerate a proton to a speed close to the speed of light, than is to accelerate an electron to the same speed?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.8

A nuclear power plant generates $10\times 10^9\,$W of power. Assuming 100% efficiency, by how much does the mass of the fuel change in one day to produce this much energy?

Solution:

  • 9.6 g

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 8.9

Find the radii to which the sun and the earth must be compressed for them to become black holes.

Solution:

  • For the Sun, 2964 m
  • For the Earth, 0.0088 m (about 1 cm)

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 8.5-8.6

  1. You are in a special compartment on a train that admits no light, sound, or vibration. Is there any way you can tell whether the train is at rest or moving at constant nonzero velocity? Explain.
  2. You are enclosed in a box with six opaque walls. Are there any experiments you can perform inside the box to prove that you are
    1. moving with constant linear velocity
    2. accelerating
    3. rotating with constant angular velocity?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Workshop 7 (May 15,16)

Interference and Diffraction

Solutions WS.7

Equations and Relations WS.7

  1. Double Slit interference:
    1. Bright fringes
      • $d\sin\theta$ $=m\lambda\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;m=0,\pm 1,\pm 2,\pm 3,...$
    2. Dark fringes
      • $d\sin\theta$ $=\left(m-\frac{1}{2}\right)\lambda\;\;\;\;\;$ $m=\pm 1,\pm 2,\pm 3,...$
  2. Air Wedge
    1. Bright fringes
      • $\frac{1}{2}+\frac{2d}{\lambda}$ $=m\;\;\;\;\;$ $\;\;\;\;\;\;$ $\;\;m=1,2,3,...$
    2. Dark fringes
      • $\frac{1}{2}+\frac{2d}{\lambda}$ $=m+\frac{1}{2}$ $\,\;\;\;\;$ $\;\;m=1,2,3,...$
  3. Single-Slit Diffraction, condition for dark fringes
    • $W\sin\theta=$ $m\lambda\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;m=\pm 1,\pm 2,\pm 3,...$
  4. First dark fringe for a circular aperture:
    • $\sin\theta$ $=1.22\frac{\lambda}{D}$
  5. Rayleigh's Criterion:
    • $\sin\theta_\text{min}$ $=1.22\frac{\lambda}{D}$
  6. Diffraction grating, principal maxima
    • $d\sin\theta=$ $m\lambda\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;m=0,\pm 1,\pm 2,\pm 3,...$

—-

Problem 7.1

The interference pattern shown in the figure below is produced by light with a wavelength of 450 nm passing through two slits with a separation of 50 $\mu$m. After passing through the slits, the light forms a pattern of bright and dark spots on a screen located 1.25 m from the slits. What is the distance between the two vertical, dashed lines in the figure below?

Solution:

  1. the distance is 5.1 cm

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 7.1-7.2

  1. Does the spacing between fringes in a two-slit interference pattern increase, decrease, or stay the same if
    1. the slit separation is increased
    2. the light intensity is increased
    3. the color of the light is switched from red to blue
    4. the whole apparatus is submerged in cooking sherry?
  2. If the slits are illuminated with white light, then at any side maximum, does the blue component or the red component peak closer to the central maximum?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.2

In a single-slit diffraction experiment, the width of the slit is 1.90 $\mu$m. If a beam of light of wavelength 632 nm forms a diffraction pattern,

  • (a) what is the angle associated with the first minimum above the central bright fringe?
  • (b) what is the angle associated with the third minimum above the central bright fringe?
  • (c) at what distance above the central bright fringe are those minima if the pattern is viewed on a screen 2 m from the slit.
  • (d) how many dark fringes will be produced on either side of the central bright fringe?

Solution:

  • (a) 19.4$^\circ$
  • (b) 86.3$^\circ$
  • (c) 0.7 m and 30.7 m for the minima in (a) and (b) respectively
  • (d) 3 on each side, 6 total

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 7.3

Why can you easily hear sound around a corner, while you cannot see around the same corner?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.3

If diffraction were the only limitation, what would be the maximum distance at which the headlights of a car could be resolved (seen as two separate sources) by the naked human eye? – The diameter of the pupil of the eye is about 7 mm when dark-adapted. Make reasonable estimates for the distance between the headlights and for the wavelength.

Solution:

  1. about 10 miles (or 10 km), depending on the assumptions

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.4

The Hubble space telescope has an aperture with a diameter of 2.4 m. How close together can two asteroids at $5\times 10^{10}\,$m distance be if they are still seen as two objects (assume $\lambda=500\,$nm)?

Solution:

  1. They are ~12 km apart

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 7.4-7.6

  1. The resolving power of some microscopes is increased by illuminating the object with ultraviolet light. Explain.
  2. Telescopes used in astronomy have large lenses (or mirrors). One reason is to let a lot of light in – important for seeing faint astronomical bodies. Can you think of another reason why it is an advantage to make these telescopes so large?
  3. At night, many people see rings (called entoptic halos) surrounding bright outdoor lamps in otherwise dark surroundings. The rings are the first of the side maxima in diffraction patterns produced by structures that are thought to be within the cornea (or possibly the lens) of the observer's eye. (The central patterns of such maxima overlap the image of the lamp.)
    1. Would a particular ring become smaller or larger if the lamp were switched from blue to red light?
    2. If a lamp emits white light, is blue or red on the outside edge of the ring?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.5

A film of soapy water in air is held vertically and viewed in reflected light. The film has index of refraction $n = 1.36$.

  • (a) Explain why the film appears black at the top.
  • (b) The light reflected perpendicular to the film at a certain point is missing the wavelengths 504 nm and 630.0 nm. No wavelengths between these two are missing. What is the thickness of the film at that point?
  • (c) What other visible wavelengths are missing, if any?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.6

A sodium light ($\lambda = 589.3\,$nm) is used to view a soap film to make it look black when directed perpendicular to the film. What is the minimum thickness of the soap film if the index of refraction of soap solution is 1.36?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.7

A nonreflective coating of magnesium fluoride ($n=1.38$) is applied to a camera lens ($n=1.5$). If one wants to prevent light at a wavelength of 560 nm to reflect from the lens, what minimum thickness does the coating need to be?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 7.7

The figure shows two rays of light encountering interfaces, where they reflect and refract. Which of the resulting waves are shifted in phase at the interface?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 7.8

A diffraction grading produces a bright fringe at an angle of 14$^\circ$ for 400 nm. For another wavelength the same order (m) fringe is at an angle of 27$^\circ$.

  • (a) What is the unknown wavelength?
  • (b) If $m=8$, what is the separation between the slits on the grating?
  • (c) At what angle would the bright fringes occur if the slit separation is doubled?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 7.8-7.15

  1. In a diffraction grating, how does the spacing of the lines affect the separation of the fringes in the interference pattern?
  2. If white light were incident upon a diffraction grating, instead of monochromatic light, what would the resulting interference pattern look like?
  3. When white light falls on a prism, it is dispersed, forming a spectrum of colors, with the red component receiving the least deviation. Compare this spectrum with that produced by a diffraction grating.
  4. AM radio waves have wavelengths hundreds of meters long. Can AM radio waves be used as radar to detect aircraft? Explain.
  5. Visible light has wavelengths from about $4.0\times 10^{-7}\,$m to about $7.5\times 10^{-7}\,$m. Can visible light be used to detect individual atoms? Explain.
  6. Photography using what type of light (i.e. infrared, ultraviolet, visible, x-ray, gamma rays, etc.) is needed to image individual atoms?
  7. Radio waves and light waves are both electromagnetic waves, yet radio waves can be received behind tall buildings. Explain why this is possible when light cannot reach these areas.
  8. What effect places a lower limit on the size of an object that can be clearly seen with the best optical microscope?

Solutions:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Workshop 6 (May 8,9)

Lens Systems

Solutions WS.6

Equations and Relations WS.6

  1. Index of refraction
    • $v=\frac{c}{n}$
  2. Snell's Law
    • $n_1\sin\theta_1=n_2\sin\theta_2$
  3. Thin Lens Equation
    • $\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$
      • $d_o$ is the object distance
      • $d_i$ is the image distance
  4. Magnification
    • $m=-\frac{d_i}{d_o}=\frac{h_i}{h_o}$
  5. ${\text{f-number}}$ $=\frac{\text{focal length}}{\text{diameter of aperture}}$ $=\frac{f}{D}$
  6. refractive power = $\frac{1}{f}$
  7. Magnifying glass - Magnification:
    • $M=\frac{N}{f}\;\;\;$ (image at infinity)
    • $M=1+\frac{N}{f}\;$ (image at near point, N)
  8. Compound microscope
    • $M_\text{total}$ $=-\frac{sN}{f_\text{objective}f_\text{eyepiece}}$
  9. Telescope
    • $M_\text{total}$ $=-\frac{f_\text{objective}}{f_\text{eyepiece}}$
    • $L=$ $f_\text{objective}+f_\text{eyepiece}$

Problem 6.1

A diverging lens ($f=-5\,$cm) is located 25 cm to the right of a converging lens ($f=10\,$cm). A beetle (length = 2 cm) is 30 cm to the left of the converging lens.

  1. Relative to the diverging lens, where is the image of the beetle?
  2. What is the magnification of the image?
  3. What size is the image of the beetle?
  4. Is the image real or virtual?
  5. Is it upright or inverted
  6. Sketch the ray diagram for the two-lens system.

Solution:

  1. $-3.3$ cm
  2. $-\frac{1}{6}=-0.167$
  3. $0.333\,$ cm
  4. virtual
  5. inverted

Questions:

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Problem 6.2

The near point of a farsighted person is 65 cm. What power contact lens he must use to correct this problem and be able to read a book at a normal near point of 25 cm?

Solution:

  1. +2.46 dpt

Questions:

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    • [Workshop leaders, post your answers here]

Problem 6.3

A stamp $1.5\,{\text{cm}}\times 2\,{\text{cm}}$ is viewed through a magnifying glass with $f=10\,$cm. What is the size of the stamp if the observer’s eye is relaxed and has a near point distance of 25 cm?

Solution:

  1. $3.75\,{\text{cm}}\times 5\,$cm

Questions:

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    • [Workshop leaders, post your answers here]

Problem 6.4

Jenna is farsighted; the nearest object she can see clearly without corrective lenses is 2.0 m away. It is 1.8 cm from the lens of her eye to the retina.

  1. Sketch a ray diagram to show (qualitatively) what happens when she tries to look at something closer than 2 m, without corrective lenses.
  2. What should the focal length of her contact lenses be so that she can see objects clearly as close as 20 cm from her eye?

Solution:

  1. see diagram to the right
  2. 4.5 dpt

Questions:

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    • [Workshop leaders, post your answers here]

Question 6.1

  1. Much of the bending of light rays necessary for human vision occurs at the cornea (at the air-eye interface). The cornea has an index of refraction somewhat greater than that of water.
    1. When your eye is submerged in a swimming pool, is the bending of light rays at the cornea greater than, less than, or the same as in air?
    2. The Central American fish Anableps can see simultaneously above and below water because it swims with its eyes partially extending above the water surface. To provide clear sight in both media, is the radius of curvature of the submerged portion of the cornea greater than, less than, or equal to that of the exposed portion?
  1. Solution:

Questions:

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    • [Workshop leaders, post your answers here]

Problem 6.5

A nearsighted woman cannot clearly see objects more than 2.0 m away. The distance from the lens of the eye to the retina is 1.8 cm, and the eye's power of accommodation is 4.0 D (the inverse focal length of the cornea-lens system increases by a maximum of 4.0 dpt over its relaxed inverse focal length when accommodating for nearby objects). Assume the corrective lenses are 2 cm from the eyes.

  1. As an amateur optometrist, what corrective eyeglass lenses would you prescribe to enable her to clearly see distant objects?
  2. Find the nearest object she can see clearly with and without her glasses.

Solution:

  1. $-0.5\,$dpt
  2. Here good accuracy (4-digits) is required:
    • find the focal length of the eye when relaxed (“far point”):
      • $\frac{1}{f_\text{far}}=\frac{1}{2\,{\text m}}+\frac{1}{0.018\,{\text m}}$ $=56.06\,$dpt
    • find the near-point focal distance of the eye:
      • $\frac{1}{f_\text{near}}$ $=\frac{1}{f_\text{far}}+4\,{\text{dpt}}$ $=60.06\,$dpt
    • without glasses, solve
      • $\frac{1}{f_\text{near}}$ $=\frac{1}{d_o}+\frac{1}{0.018\,{\text m}}$
      • $d_0=0.222\,{\text m}$ $=22.2\,$cm
    • with glasses, the optical power of the eye+glasses at the near point is 59.56 dpt (0.5 dpt less than naked eye). Solve
      • $\frac{1}{f_\text{near}}$ $=\frac{1}{d_o}+\frac{1}{0.018\,{\text m}}$
        • $d_0=0.250\,{\text m}$ $=25.0\,$cm
      • More accurately, we should treat eye+grasses as two separate thin lenses, and do a detailed calculation:
        • The glass maps the object at an unknown distance x onto a virtual image 22.2 cm away from the eye (20.2 cm from the glasses)
        • solve $\frac{1}{f_\text{glasses}}=$ $\frac{1}{-200\,{\text{cm}}}=\frac{1}{x}+\frac{1}{-20.2\,{\text{cm}}}$
        • $x=22.47\,$cm from the glasses
        • $d_0=24.47\,$cm from the eye, pretty close to our “single lens” estimate of 25 cm.

Questions:

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Problem 6.6

A telescope of an amateur astronomer has an angular magnification of $–200$. The eyepiece has a focal length of 5 mm.

  1. What is the focal length of the objective lens?
  2. How long is the astronomer's telescope?

Solution:

  1. $M_\text{total}=\frac{\theta'}{\theta}$ $=-\frac{f_\text{ojective}}{f_\text{eyepiece}}$ $-200$
    • $f_\text{ojective}=-M_\text{total}\,f_\text{eyepiece}$ $=1000\,{\text{mm}}$ $=1\,$m
  2. $L=f_\text{ojective}+f_\text{eyepiece}$ $=1.005\,$m

Questions:

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    • [Workshop leaders, post your answers here]

Question 6.2

In a microscope, the objective lens has a short focal length, whereas in a telescope, the objective lens has a long focal length. Explain the reason for the difference in focal lengths.


Problem 6.7

A photographer wishes to take a photo of the Eiffel Tower (300 m tall) from across the Seine River, a distance of 500 m from the tower. What focal length lens should she use to get an image that is 20 mm high on the film?

Solution:

  1. The object distance is so large compared to the size of the camera, that we can neglect $\frac{1}{d_o}$ in the lens equation:
    • $\frac{1}{f}\approx \frac{1}{d_i}$
    • $d_i\approx f$
  2. Use magnification:
    • $m=-\frac{d_i}{d_o}$ $=\frac{h_i}{h_o}=$ $\frac{-20\,{\text{mm}}}{300\,{\text m}}$ $=-6.67\times 10^{-5}$ (single-lens camera image is inverted, so $h_i$ is negative)
  3. From here,
    • $f\approx d_i=-md_o$ $=33.3\,$mm

Questions:

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    • [Workshop leaders, post your answers here]

Questions 6.3-6.9

  • In her bag, a photographer is carrying three exchangeable camera lenses with focal lengths of 400, 50, and 28 mm. Which lens should she use for:
    1. wide angle shots,
    2. everyday use, and
    3. telephoto work.
  • When snorkeling, you wear goggles in order to see clearly. Why is your vision blurry without the goggles?
  • A nearsighted person notices that he is able to see objects more clearly when he is underwater than in air. Why might this be true?
  • For human eyes, about 70% of the refraction occurs at the cornea; less than 25% occurs at the two surfaces of the lens. Why? Is the same thing true for fish eyes?
  • A slide projector forms a real image of the slide on a screen using a converging lens. If the bottom half of the lens is blocked by covering it with opaque tape, does the bottom half of the image disappear, or does the top half disappear, or is the entire image still visible on the screen? If the entire image is visible, is anything different about it?
  • Each retina has a blind spot with no rods or cones, located where the optic nerve leaves the retina. The blind spot is not usually noticed because the brain fills in the missing information. To observe the blind sport, draw a cross and a dot, about 10 cm apart, on a sheet of white paper. Cover your left eye and hold the paper far from your eyes with the dot on the right. Keep you ere focused on the cross as you slowly move the paper toward you face. The dot disappears when the image falls on the blind spot. Continue to move the paper even closer to your eye; you will see the spot again when its image moves off the blind spot.

Answers:

Questions:

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Workshop 5 (May 1,2)

Refraction and Lenses

Solutions WS.5

Equations and Relations WS.5

  1. Index of refraction
    • $v=\frac{c}{n}$
  2. Snell's Law
    • $n_1\sin\theta_1=n_2\sin\theta_2$
  3. Total internal reflection
    • $\sin\theta_c=\frac{n_2}{n_1}$
  4. Total polarization:
    • $\tan\theta_B=\frac{n_2}{n_1}$
      • $\theta_B$ is Brewster's angle
  5. Thin Lens Equation
    • $\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$
      • $d_o$ is the object distance
      • $d_i$ is the image distance
  6. Magnification
    • $m=-\frac{d_i}{d_o}=\frac{h_i}{h_o}$
Convex Lenses
Location Orientation Size Real/Virtual
Beyond 2F inverted reduced real
At 2F inverted same real
Between F and 2F inverted enlarged real
Just Beyond F inverted approaching infinity real
Just Inside F upright approaching infinity virtual
Between F & Lens upright enlarged virtual
Concave Lenses
Location Orientation Size Real/Virtual
Anywhere upright reduced virtual

Problem 5.1

A fish hovers beneath the still surface of a pond. If the sun is 33$^\circ$ above the horizon, at what angle above the horizontal does the fish see the sun? ($n_\text{water} =1.33$)

Solution:

  1. 51$^\circ$

Questions:

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Problem 5.2

The sunlight reflected from the surface of a lake is completely polarized. Calculate the angle of incidence of the sunlight

  1. in summer ($n_\text{water}=1.333$)
  2. in winter when the lake is frozen ($n_\text{ice}=1.309$)

Solution:

  1. 53.1$^\circ$
  2. 52.7$^\circ$

Questions:

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    • [Workshop leaders, post your answers here]

Problem 5.3

A beam of light strikes one face of a window pane with an angle of incidence of 30.0$^\circ$. The index of refraction of the glass is 1.52. The beam travels through the glass and emerges from a parallel face on the opposite side. Ignore reflections.

  1. Find the angle of refraction for the ray inside the glass.
  2. Show that the rays in air on either side of the glass (the incident and emerging rays) are parallel to each other.

Solution:

  1. 19.2$^\circ$

Questions:

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    • [Workshop leaders, post your answers here]

Problem 5.4

In an experiment a beam of red light of wavelength 645 nm in air passes from glass into air. The incident and refracted angles are $\theta_1$ and $\theta_2$, respectively. In the experiment, angle $\theta_2$ is measured for various angles of incidence, and the sine's of the angles are used to obtain the line shown in the following graph.

  1. Assuming an index of refraction of 1.00 for air, use the graph to determine a value for the index of refraction of the glass for the red light.
  2. Determine the frequency of the red light in air, its speed in the glass, and its wavelength in the glass.
  3. The index of refraction of this glass is 1.66 for violet light, which has a wavelength of 420 nm in air. Given the same incident angle $\theta_1$, show with a ray diagram the difference between the refracted red and violet rays.
  4. Determine the critical angle of incidence $\theta_C$ for the violet light in the glass in order for total internal reflection to occur.

Solution:

  1. $n_1\approx\frac{0.8}{0.5}$ $=1.60$
  2. For the red light,
    • $f=4.65\times 10^{14}\,$Hz
    • $v= 1.8\times 10^8\,\frac{\text m}{\text s}$
    • $\lambda_g= 4.03\times 10^{-7}\,$m
  3. The violet light will come out at a more shallow (larger from the normal) angle
  4. For the violet light,
    • $\theta_C=37^\circ$

Questions:

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Questions 5.1-5.2

  1. In the figure, light travels from material a, through three layers of other materials with surfaces parallel to one another, and then back into another layer of material a. The refractions (but not the associated reflections) at the surfaces are shown. Rank the materials according to their indexes of refraction, greatest first.
  2. A ray of light passes from air into water, striking the surface of the water with an angle of incidence of 45$^\circ$. Which of these quantities change as the light enters the water: wavelength, frequency, speed of propagation, direction of propagation?

Solution:

  1. The materials, in the order from highest to lowest index of refraction, are:
    • d,b,a,c
  2. all except the frequency

Questions:

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Problem 5.5

The plano-convex lens shown to the right has a focal length of 20 cm in air. An object is placed 60 cm ($3f$) from this lens.

  1. Is the image is real or virtual?
  2. What is the distance from the lens to the image?
  3. Determine the magnification of this image.
  4. The object, initially at a distance $3f$ from the lens, is moved toward the lens. On the axes to below, sketch the image distance as the object distance varies from $3f$ to zero.
  5. If the index of refraction of the lens were increased, would the focal length of the lens increase, decrease, or remain the same? Explain.

Solution:

  1. Real
  2. 30 cm
  3. $-\frac{1}{2}$
  4. Click here to view the plot
  5. decrease (the lens will have a tight focus)

Questions:

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Problem 5.6

A thin converging lens of focal length 8 cm is used as a simple magnifier to examine an object $K$ that is 6 cm from the lens.

  1. Draw a ray diagram showing the optical axis, the lens (at the origin), $K$ at $+6\,$cm, and the position of the image formed, $K'$. Be sure the height of $K'$ is correct relative to $K$.
  2. Is $K'$ real or virtual? Explain.
  3. Calculate the position (distance from the origin) of $K'$.
  4. How many times greater is the image height than the object height? (I.e. calculate the ratio of the image size to the object size, $\frac{h_{K'}}{h_K}\big).$
  5. Repeat (1)-(4) for a concave lens with $f = -8\,$cm.

Solution:

  1. convex lens:
    • image distance:
      • $-24\,$cm
    • the image is virtual
    • magnification
      • 4
  2. concave lens:
    • image distance:
      • $-3.43\,$cm
    • the image is virtual
    • magnification
      • 0.57

Questions:

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Questions 5.3-5.7

  1. What happens to the focal length of a symmetric, converging lens as you increase:
    1. the index of refraction $n$ of the lens
    2. the magnitude of the radius of curvature of the two sides
    3. the index of refraction nmed of the surrounding medium, keeping nmed less than $n$?
  2. A concave mirror and a converging lens (glass with $n$ = 1.5) both have a focal length of 3 cm when in air. When they are in water ($n$ = 1.33), are their focal lengths greater than, less than, or equal to 3 cm?
  3. Explain how the day is lengthened by atmospheric refraction.
  4. Using the principles of refraction, explain why a diamond is much more brilliant than a glass replica. Why are colors observed in the light from a diamond?
  5. The surface of water in a swimming pool is completely still. Describe what you would see looking straight up toward the surface from underwater.

Solution:

Questions:

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Workshop 4 (Apr 24,25)

Polarization and Mirrors

Solutions WS.4

Equations and Relations WS.4

  1. Transmission of polarized light
    • $I=I_0\cos^2\theta$
  2. Transmission of unpolarized light
    • $I=\frac{1}{2}I_0$
  3. Total polarization
    • $\tan\theta_B=\frac{n_2}{n_1}$
      • where $\theta_B$ is the Brewster's angle
  4. Focal length $f$ of a convex mirror (radius of curvature $R$)
    • $f=-\frac{1}{2}R$
  5. Concave mirror
    • $f=\frac{1}{2}R$
  6. Mirror equation
    • $\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$
      • $d_o$ is the object distance
      • $d_i$ is the image distance
  7. Magnification
    • $m=-\frac{d_i}{d_o}=\frac{h_i}{h_o}$
Concave Mirrors
Location Orientation Size Real/Virtual
Beyond C inverted reduced real
At C inverted same real
Between F & C inverted enlarged real
Just Beyond F inverted approaching infinity real
Just Inside F upright approaching infinity virtual
Between F & Mirror upright enlarged virtual
Convex Mirrors
Location Orientation Size Real/Virtual
Anywhere upright reduced virtual

Problem 4.1

A vertically polarized beam of light of intensity $100\frac{\text W}{\,{\text m}^2}$ passes through two polarizers. The transmission axis of the first polarizer is making an angle of 20$^\circ$ to the vertical and the second one is making an angle of 110$^\circ$ to the vertical.

  • (a) What is the transmitted intensity of this beam of light?

A third polarizer is added before the first polarizing sheet. The transmission axis of the polarizer is making and angle of 60° to the vertical.

  • (b) What is the transmitted intensity after the first two polarizers?
  • (c) What is the total transmitted intensity after all three polarizers?

Now the polarizer at 60$^\circ$ is moved in between the other two polarizers.

  • (d) What is the total transmitted intensity after all three polarizers in this case?
  • (e) What is the total transmitted intensity after all three polarizers if the initial light beam was unpolarized?

Solution:

  • (a) 0
    .
  • (b) $14.7\,\frac{\text W}{\,{\text m}^2}$
    .
  • (c) 0
    .
  • (d) $21.4\,\frac{\text W}{\,{\text m}^2} $
    .
  • (e) $12.1\,\frac{\text W}{\,{\text m}^2}$

Questions:

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Problem 4.2

Two rays, emitted from the same point, diverge with an angle of 15$^\circ$ between them. The rays reflect from a plane mirror. Draw a ray diagram and find the angle between the two rays after the reflection.

Solution:

  • 15$^\circ$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 4.3

Construct the images formed by a concave mirror when the object is

  • (a) beyond the center of curvature, C,
  • (b) at C,
  • (c) between C and the focal point F,
  • (d) at F, and
  • (e) between F and the mirror's surface.

Discuss the nature and relative size of each image.

Solution:

Questions:

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Problem 4.4

An object 2 cm in height is placed at the center of curvature C in front of a concave mirror. What is the height of its image?

Solution:

  • The same, inverted

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 4.5

An object 1.5 cm in height is placed 7 cm in front of a concave mirror with a radius of curvature of 5 cm.

  1. Draw a ray diagram to scale and measure
    1. the location
    2. the height
    3. the magnification of the image.
  2. Analytically calculate location, height, magnification and compare the values to the graphical result.
  3. Calculate location, height, and magnification if the object is placed 7 cm in front of a convex mirror with a radius of curvature of $-$5 cm.

Solution:

  1. Concave mirror:
    1. Location: 3.89 cm
    2. Height: $-$0.83 cm
    3. Magnification: $-$0.56
  2. Convex:
    1. Location: $-$1.84 cm
    2. Height: 0.39
    3. Magnification: 0.26

Questions:

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Problem 4.6

A dentist places a mirror 1.5 cm from your tooth. He sees an enlarged image 4 cm behind the mirror.

  • (a) What is the focal length of the mirror?
  • (b) What is the magnification ?
  • (c) Is the dentist’s mirror concave or convex?
  • (d) Is the image upright?

Solution:

  • (a) 2.4 cm
  • (b) $\frac{8}{3} \approx$ 2.67
  • (c) concave
  • (d) upright

Questions:

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    • [Workshop leaders, post your answers here]

Questions 4.1-4.9

  1. When you look at the back side of a shiny teaspoon at arm’s length you see yourself upright. When rotate the spoon and you look at front side you are upside down. Explain the nature of both images.
  2. Use the mirror equation to show that the image of an infinitely distant object is formed at the focal point of a spherical mirror.
  3. Use the mirror equation to show that the image of an object placed at the focal point of a concave mirror is located at infinity.
  4. Use the mirror equation to show that, for a plane mirror, the image distance is equal in magnitude to the object distance. What is the magnification of a plane mirror?
  5. Discuss the statement: One cannot “see” the surface of a perfect mirror.
  6. Why is the passenger’s side mirror in many cars convex rather than plane or concave?
  7. When a virtual image is formed by a mirror, is it in front of the mirror or behind it? What about a real image?
  8. If you look at mirror and see the image of clock, do the hands rotate clockwise or counterclockwise.
  9. Why is the receiving antenna of a satellite dish placed at a set distance from the dish?

Solutions:

Questions:

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Workshop 3 (Apr 17,18)

Doppler Effect and Electromagnetic Waves

Equations and Relations WS.3

workshop_3_solutions.pdf

  1. Speed, wavelength, and frequency of a wave
    • $v=\lambda\,f$
  2. Beats
    • $f_\text{beat}=\big|\,f_1-f_2\big|$
  3. Doppler Effect, moving observer:
    • $f'=\left(1\pm\frac{u_o}{v}\right)f$
  4. Doppler Effect, moving source:
    • $f'=\big(\!\frac{1}{1\mp\frac{u_s}{v}}\!\big)\,f$
      • $v$: speed of the wave
      • $u_o$: relative speed of the observer
      • $u_s$: relative speed of the source
  5. Doppler Effect for electromagnetic waves:
    • $f'=f\left(1\pm\frac{u}{c}\right)$
      • $u$: relative speed between source and observer
  6. Intensity as power per area:
    • $I=\frac{P}{A}$
    • $I=\frac{P}{4\pi r^2}$ (from a point source at a distance r)
  7. Energy density (not to confuse with relative speed in Doppler effect!):
    • $u=\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2$
  8. Ratio of electric and magnetic field:
    • $E=cB$
  9. Intensity in terms of the energy density $u$ and speed of light $c$:
    • $I=uc=\frac{1}{2}c\epsilon_0E^2+\frac{1}{2\mu_0}cB^2$
  10. Momentum in terms of total energy $U$ and speed of light $c$:
    • $p=\frac{U}{c}$
  11. Radiation pressure:
    • $P_\text{av}=\frac{I_\text{av}}{c}$

Problem 3.1

An auditorium has organ pipes at the front and at the rear of the hall. Two identical pipes, one at the front and one at the back, have fundamental frequencies of 264 Hz at 20$\,^\circ$C. During a performance, the organ pipes at the back of the hall are at 25$\,^\circ$C, while those at the front are still at 20$\,^\circ$C. What is the beat frequency when the two pipes sound simultaneously? (use: $v_{20\,^\circ{\text C}}=343\frac{\text m}{\text s}$, $\;\;v_{25\,^\circ{\text C}}=346\frac{\text m}{\text s}$, calculated with equation from thermodynamics: $v=\sqrt{\frac{\gamma RT}{M}}$, where M is the (average) molar mass, and $\gamma=\frac{c_p}{c_v}$ is the adiabatic constant).

Solution:

  • $f_\text{beat}=2.3\,$Hz

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.2

A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.50 times the speed of sound.

  1. What is the observed frequency if the observer moves toward the source?
  2. What is the observed frequency if the observer moves away from the source instead?

Solution:

  1. 1.5 kHz
  2. 0.5 kHz = 500 Hz

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.3

A source of sound waves of frequency 1.0 kHz is traveling through the air at 0.50 times the speed of sound.

  1. Find the frequency of the sound received by a stationary observer if the source moves towards her.
  2. Repeat if the source moves away from her instead.

Solution:

  1. 2 kHz
  2. 0.67 kHz = 670 Hz (approximately)

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.4

You drive in a your car at a speed of 50 km/h and ambulance approaches from behind at a speed of 80 km/h. When the ambulance is at rest its siren produces sound at a frequency of 1050 Hz.

  1. What is the frequency of the siren observed by you?
  2. What is the wavelength of sound reaching you?

Solution:

  1. $f_o = 1076\,$Hz
  2. $\lambda_s = 30.6\,$cm (in front of the ambulance)

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 3.1

The source and observer of a sound wave are both at rest with respect to the ground. The wind blows in the direction of source to observer. Is the observed frequency Doppler-shifted? Explain.

Answer:

  • No, since the number of waves between the source and the observer doesn't change over time in each case. All the emitted waves must be observed

Questions:

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Problem 3.5

A galaxy emits light at a wavelength of 656 nm. On earth the wavelength is measured to be 659.1 nm.

  1. What is the speed of the galaxy relative to the earth?
  2. Is the galaxy approaching or receding?

Solution:

  1. $u=1.41\times 10^6\,\frac{\text m}{\text s}$
  2. the galaxy is observed at a longer (“redder”) wavelength, so it is receding

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 3.2-3.3

  • What do cosmologist mean when the say the electromagnetic radiation from other galaxies is “red shifted”?
  • Can you distinguish between the case of a moving observer or moving source in the case of sound and electromagnetic waves?

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.6

When light having vibrations with angular frequency ranging from $2.7\times 10^{15}\frac{\text{rad}}{\text s}$ to $4.7\times 10^{15}\frac{\text{rad}}{\text s}$ strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light?

Solution:

  1. $T=2.3\times 10^{-15}\,$s to $1.3\times 10^{-15}\,$s
  2. $f=4.3\times 10^{14}\,$Hz to $7.5\times 10^{14}\,$Hz

Questions:

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    • [Workshop leaders, post your answers here]

Question 3.4

Microwave ovens, radio, radar, and x-rays utilize electromagnetic waves. Compare the energy, frequency and wavelengths of these waves to those of visible radiation.


Problem 3.7

A lightning flash is seen in the sky and 8.2 s later the boom of the thunder is heard. The temperature of the air is 12$\,^\circ$C. (use $v_{12\,^\circ{\text C}}=338\frac{\text m}{\text s}$) How far away is the lightning strike?

Solution:

  • a difference in signal arrival times is
    • $\frac{r}{v}-\frac{r}{c}=8.2\,$s
  • from this we find the distance
    • $r=2772\,$m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 3.5-3.6

  • During a thunderstorm, you can easily estimate your distance from a lightning strike. Count the number of seconds that elapse from when you see the flash of lightning to when you hear the thunder. The rule of thumb is that 5 seconds elapse for each mile distance. Verify that this rule of thumb is (approximately) correct. (One mile is 1.6 km and light travels at a speed of $c=3\times 10^8\frac{\text m}{\text s}$.)
  • For an xyz coordinate system, if the E-vector is in the z direction, and the B-vector is in the x direction, what is the direction of propagation of the electromagnetic waves?

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.8

The microwave background radiation leftover from the big bang has an average energy density of $4\times 10^{-14}\frac{\text J}{ {\text m}^3}$.

  1. What is the rms and maximum value of the electric and magnetic component of this radiation?
  2. Calculate the intensity of this radiation.

Solution:

  1. the field values are:
    • rms values
      • $E_\text{rms}=0.067\,\frac{\text V}{\text m}$
      • $B_\text{rms}=2.24\times 10^{-10}\,{\text T}$
    • max values
      • $E_\text{max}=0.095\,\frac{\text V}{\text m}$
      • $B_\text{max}=3.17\times 10^{-10}\,{\text T}$
  2. intensity
    • $I=1.2\times 10^{-5}\,\frac{\text W}{\,{\text m}^2}$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 3.9

The average intensity of the sunlight reaching the earth is $1390\frac{\text W}{ {\text m}^2}$.

  1. What is the average radiation pressure due to the sunlight?
  2. What is the maximum energy that a $5 \times 8\,$m solar panel could collect in 12 hours, if all sunlight is absorbed (the real efficiency of solar panels is much lower)?
  3. Calculate the average force exerted by the light on the solar panel assuming it absorbs all incoming light.
  4. Calculate the energy density of sunlight.

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 3.7-3.10

  • Suppose you triple the magnitude of the magnetic field of an electromagnetic wave. By what factor does the electric component of the wave change? By what factor changes the intensity of the wave?
  • In tennis a radar gun is often used to measure the speed of the ball. Describe how such a radar gun could work.
  • If a spaceship uses the radiation pressure of the sun, should it use sails that are reflection or absorbing?
  • If you move on a spaceship away from the sun, by how much does the intensity of the radiation decrease as you double the distance between you and the sun?

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Workshop 2 (Apr 10,11)

Waves on a String and Sound

Equations and Relations WS.2

  1. Speed, wavelength, and frequency of a wave
    • $v=\lambda\,f$
  2. Speed of a wave of a string:
    • $v=\sqrt{\frac{F_T}{\mu}}$, where $F_T$ is the tension force, and $\mu=\frac{m}{L}$ is the mass per length (string)
  3. Harmonic wave function
    • $y(x,t)=A\cos\left(\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t\right)$
    • $k=\frac{2\pi}{\lambda}\;$ is the wavenumber, not to be confused with the spring constant
    • and the angular frequency is $\omega=\frac{2\pi}{T}$
  4. Intensity as power per area:
    • $I=\frac{P}{A}$
    • $I=\frac{P}{4\pi r^2}\;$ (point source at a distance r)
  5. Intensity level $\beta$ of a sound wave:
    • $\beta=10\log\big(\frac{I}{I_0}\big)$
    • $I_0=10^{-12}\frac{\text W}{ {\text m}^2}$
  6. Standing waves on a string:
    • $f_n=nf_1=n\frac{v}{2L}$
    • $\lambda_n=\frac{\lambda_1}{n}=\frac{2L}{n}\;\;$ where $n=1,2,3...$ is the mode number
  7. Vibrating columns of air – closed at one end
    • $f_n=nf_1=n\frac{v}{4L}$
    • $\lambda_n=\frac{\lambda_1}{n}=\frac{4L}{n}\;\;$ where $n=1,3,5...$ is the mode number
  8. Vibrating columns of air – open at both ends
    • $f_n=nf_1=n\frac{v}{2L}$
    • $\lambda_n=\frac{\lambda_1}{n}=\frac{2L}{n}\;\;$ where $n=1,2,3...$ is the mode number
  9. $L$ is the length of the tube or string

Problem 2.1

When sound waves strike the eardrum, the membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of 50.0 $\mu$s. What are the frequency and angular frequency of the vibrating eardrum for this sound?

Solution:

  • $f=2\times10^4\,$Hz
  • $\omega\approx 1.3\times 10^5\,\frac{\text{rad}}{\text s}$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.2

High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x-rays do. To detect small objects, such as tumors, a frequency of around 5.0 MHz is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

Solution:

  • $T=2\times 10^{-7}\,$s
  • $\omega=3.14\times 10^7\,\frac{\text{rad}}{\text s}$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.3

A variable oscillator allows a laboratory student to adjust the frequency of a source to produce standing waves in a vibrating string. A 1.20-m length of string ($\mu$ = 0.400 g/m) is placed under a tension of 200 N. What frequency is necessary to produce three standing loops in the vibrating string? What is the fundamental frequency? What frequency will produce five loops?

Solution:

  • $f_3$ $=3\frac{v}{2L}=3\frac{\sqrt{\frac{F_T}{\mu}}}{2L}$ $=884\,$Hz
  • $f_1=295\,$Hz
  • $f_5=1473\,$Hz

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Question 2.1

The piano strings that vibrate with the lowest frequencies consist of a steel wire around which a thick coil of copper wire is wrapped. Only the inner steel wire is under tension. What is the purpose of the copper coil?

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.4

A piano string of length 1.50 m and mass density 25.0 mg/m vibrates at a (fundamental) frequency of 450.0 Hz.

  1. What is the speed of the transverse string waves?
  2. What is the tension?
  3. What are the wavelength and frequency of the sound wave in air produced by vibration of the string? The speed of sound in air at room temperature is 340 m/s.

Solution:

  1. $c=1350\,\frac{\text m}{\text s}$
  2. $F_T=46\,{\text N}$
  3. $\lambda=0.76\,$m

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.5

One of the harmonics of a column of air open at both ends has a frequency of 324 Hz and the next higher harmonic has a frequency of 378 Hz.

  1. What the frequency of the next higher harmonic?
  2. What is number, n, of this harmonic?
  3. What is the fundamental frequency of the air column?

Solution:

  1. $f=432\,$Hz
  2. $n=8$
  3. $f_1=54\,$Hz

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 2.2-2.3

  • If the length of a guitar string is decreased while the tension remains constant, what happens to each of these quantities?
    1. the wavelength of the fundamental
    2. the frequency of the fundamental
    3. the time for a pulse to travel the length of the string
    4. the maximum velocity for a point on the string (assuming the amplitude is the same both times)
    5. the maximum acceleration for a point on the string (assuming the amplitude is the same both times)
  • A cello player can change the frequency of the sound produced by her instrument by
    1. increasing the tension in the string,
    2. pressing her finger on the string at different places along the fingerboard, or
    3. bowing a different string.
    • Explain how each of these methods affects the frequency.

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.6

A wave on a string has equation $y(x,t) = (4.0\,{\text{mm}}) \sin (\omega t – kx)$, where $\omega = 6.0 \times 10^2\,$rad/s and $k = 6.0\,$ rad/m.

  1. What is the amplitude of the wave?
  2. What is the wavelength?
  3. What is the period?
  4. What is the wave speed?
  5. In which direction does the wave travel?

Solution:

  1. $A=4\times 10^{-3}\,$m
  2. $\lambda=1.05\,$m
  3. $T=1.05\times 10^{-2}\,$s
  4. $v=100\,$m/s
  5. in the $+x$ direction

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 2.7

  1. Plot a graph for $y(x,t) = 4.0\,{\text{cm}}\cdot\sin\!\big(378\,{\text s}^{-1}\!\cdot\!t$ $–\, 314\,{\text m}^{-1}\!\cdot\!x\big)$, where $t$ is in s and $x$ and $y$ are in cm, versus $x$ at $t\!=\!0$ and at $t\!=\!\frac{1}{480}\,$s and find the wavelength of the wave.
  2. For the same function, plot a graph of $y(x,t)$ versus $t$ at $x\!=\!0$ and find the period of the vibration.
  3. Calculate the speed of the wave.

Solution:

  • $\lambda=2\times 10^{-2}\,$m
  • $T=1.7\times 10^{-2}\,{\text s}$
  • $v=1.2\,\frac{\text m}{\text s}$
click: $0.04\,{\text m}\cdot\sin\!\big($ $–314\,{\text m}^{-1}\!\cdot x\big)$ click: $0.04\,{\text m}\cdot\sin\!\big(\frac{378}{480}$ $–\,314\,{\text m}^{-1}\!\cdot x\big)$ click: $0.04\,{\text m}\cdot\sin\!\big(378\,{\text s}^{-1}\!\cdot t\big)$

Questions:

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    • [Workshop leaders, post your answers here]

Problem 2.8

During a concert a single singer generates an intensity level of 55 dB at a certain location in the concert hall. With the whole choir singing the intensity level is 75 dB. Assuming that each singer generates the same intensity level, how many people are in the choir?

Solution:

  • $N=100$

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 2.4-2.8

  • Why must astronauts on the surface of the moon communicate with each other by radio? Can they hear another spacecraft as it lands nearby? Can they hear by touching helmets?
  • When an earthquake occurs, the S waves (transverse waves) are not detected on the opposite side of the Earth while the P waves (longitudinal waves) are. How does this provide evidence that the Earth’s solid core is surrounded by liquid?
  • Why is it that your own voice sounds strange to you when you hear it played back on a tape recorder, but your friends all agree that it is just what your voice sounds like? [Hint: Consider the media through which the sounds wave travels when you usually hear your own voice.]
  • Is the vibration of a string in a piano, guitar, or violin a sound wave? Explain.
  • Many real estate agents have an ultrasonic rangefinder that enables them to quickly and easily measure the dimensions of a room. The device is held to one wall and reads the distance to the opposite wall. How does it work?

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Workshop 1 (Apr 3,4)

Oscillations

Equations and Relations WS.1

  1. Frequency, period, angular frequency
    • $f=\frac{1}{T}$
    • $\omega=2\pi f$
  2. Position, velocity and acceleration of a harmonic oscillator
    • $x=A\cos\left(\frac{2\pi\cdot t}{T}\right)=A\cos(\omega t)$
    • $v=-A\omega\sin(\omega t)$
      • $v_\text{max}=A\omega$
    • $a=-A\omega^2\cos(\omega t)$
      • $a_\text{max}=A\omega^2$
  3. Spring period
    • $T=2\pi\sqrt{\frac{m}{k}}$
  4. Energy
    • $E=\frac{1}{2}kA^2$
    • $U=\frac{1}{2}kA^2\cos^2(\omega t)$
    • $K=\frac{1}{2}kA^2\sin^2(\omega t)$
  5. Simple pendulum period
    • $T=2\pi\sqrt{\frac{L}{g}}$
  6. Physical pendulum period
    • $T=2\pi\sqrt{\frac{I}{mgd}}$
  7. Underdamping
    • $A=A_0e^{-\frac{bt}{2m}}$
  8. Hooke's law:
    • $F=-kx$

Problem 1.1

A mass of 1 kg is attached to a spring and undergoes simple harmonic oscillations with a period of 1 s. What is the force constant of the spring?

Solution:

Questions:

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Problem 1.2

  1. If a simple pendulum has period T = 1.0 s and you double its length, what is its new period in terms of T ?
  2. If a simple pendulum has a length L = 1.0 m and you want to triple its frequency, what should be its length?
  3. Suppose a simple pendulum has a length L and period T on earth. If you take it to a planet where the acceleration of freely falling objects is ten times what it is on earth, what should you do to the length to keep the period the same as on earth?
  4. If you do not change the simple pendulum’s length in the previous part, what is its period on that planet in terms of T ?
  5. If a simple pendulum has a period T and you triple the mass of its bob, what happens to the period (in terms of T)?

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 1.3

What is the period of a pendulum formed by placing a horizontal axis (i.e. pivot point)

  1. through the end of a meterstick (100-cm mark)?
  2. through the 75-cm mark?
  3. through the 60-cm mark?

Assume g = 9.80 m/s2.
Hint: $I_\text{cm}=\frac{1}{12}ml^2$ , Parallel axis theorem: $I=I_\text{cm}+md^2$

Solution:

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Questions 1.1-1.3

  • A pendulum clock runs too slow and loses time. What adjustment should be made?
  • How can the principle of the pendulum be used to compute (a) length, (b) mass, and (c) time?
  • A pendulum is mounted in an elevator that moves upward with constant acceleration. Is the period greater than, less than, or the same as when the elevator is at rest? Why?

Questions:

  • [Students, post your questions here]
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Problem 1.4

A 1.5-kg mass oscillates at the end of a spring in SHM. The amplitude of the vibration is 0.15 m, and the spring constant is 80 N/m. If the mass is displaced 15 cm,

  • what are the magnitude and direction of the acceleration and force on the mass?

If the system is now operated on a frictionless horizontal surface,

  • what is total energy?
  • what is the maximum velocity?
  • what is the maximum acceleration?

Solution:

Questions:

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    • [Workshop leaders, post your answers here]

Questions 1.4-1.7

  • What effect will doubling the amplitude A of a body moving with SHM have on (a) the period, (b) the maximum velocity, and (c) the maximum acceleration?
  • Explain how the period of a mass-spring system can be independent of amplitude, even though the distance traveled during each cycle is proportional to the amplitude.
  • Explain why the velocity in SHM is greatest when the magnitude of the acceleration is the least.
  • A mass hanging vertically from a spring and a simple pendulum both have a period of oscillation of 1 s on Earth. An astronaut takes the two devices to another planet where the gravitational field is stronger than that of Earth. For each of the two systems, state whether the period is now longer than 1 s, shorter than 1 s, or equal to 1 s. Explain your reasoning.

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

Problem 1.5

5. A mass is vibrating at the end of a spring of force constant 225 N/m. The figure shows a graph of its position x as a function of time t.

  1. At what times is the mass not moving?
  2. How much energy did the system originally contain?
  3. How much energy did the system lose between t = 1.0 s and t = 4.0 s

Solution:

Questions:

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Questions 1.8-1.9

  • A bungee jumper leaps from a bridge and comes to a stop a few centimeters above the surface of the water below. At that lowest point, is the tension in the bungee cord equal to the jumper’s weight? Explain why or why not.
  • A ball is dropped from a height h onto the floor and keeps bouncing. No energy is dissipated, so the ball regains the original height h after each bounce. Sketch the graph for y(t) and list several features of the graph that indicate that this motion is not SHM.

Questions:

  • [Students, post your questions here]
    • [Workshop leaders, post your answers here]

physics_workshop.txt · Last modified: 2015/05/19 16:15 by wikimanager