The Physics
Hypertextbook
Opus in profectus

# The Nature of Light

## Problems

### practice

1. Here is an excerpt from Galileo's report of his attempt to determine the speed of light in a vacuum.

Let each of two persons take a light contained in a lantern, or other receptacle, such that by the interposition of the hand, the one can shut off or admit the light to the vision of the other. Next let them stand opposite each other at a distance of a few cubits and practice until they acquire such skill in uncovering and occulting their lights that the instant one sees the light of his companion he will uncover his own…. Having acquired skill at this short distance let the two experimenters, equipped as before, take up positions separated by a distance of two or three miles and let them perform the same experiment at night, noting carefully whether the exposures and occultations occur in the same manner as at short distances; if they do, we may safely conclude that the propagation of light is instantaneous; but if time is required at a distance of three miles which, considering the going of one light and the coming of the other, really amounts to six, then the delay ought to be easily observable….

In fact I have tried the experiment only at a short distance, less than a mile, from which I have not been able to ascertain with certainty whether the appearance of the opposite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid….

Galileo Galilei, 1638

1. Estimate the time for a light wave to travel the distance in Galileo's speed of light experiment. (Nota bene: Un miglio italiano corrisponde a 1,873 chilometri.)
2. How does this compare to the reaction time of a typical human?
2. Use Rømer's method and Rømer's numbers to determine the speed of light in a vacuum. If you're comfortable reading 17th century French, here's the paragraph that reports Rømer's measurement of a 22 minute delay as the light from Jupiter's moon Io traverses the extra distance equal to the diameter of Earth's orbit (represented by HE on a diagram in the report).

Il ne s'ensuit pas pourtant que la lumière ne demande aucun temps : car après avoir examiné la chose de près, il a trouvé que ce qui n’était pas sensible en deux révolutions devenait très considérable à l'égard de plusieurs prises ensemble, et que par exemple quarante révolutions, observées du côté F, étaient sensiblement plus courtes que quarante autres, observées de l'autre côté en quelque endroit du zodiaque que Jupiter se soit rencontré ; et ce à raison de 22 pour tout l’intervalle HE, qui est le double de celui qu’il y a d'ici au soleil

Ole Rømer, 1676

I couldn't find any astronomical measurements from Rømer's day, so here are the currently accepted values.

quantity Earth Jupiter
distance to Sun (106 km) 149.6 778.6
orbital period (days) 365.25 4,331
length of day (hours) 24.0 9.9
1. Sketch the Sun, Earth, and Jupiter when…
1. the Earth is closest to Jupiter
2. the Earth is farthest from Jupiter
2. Determine the speed of light in a vacuum using Rømer's method.
3. A common measure of astronomical distances is the light year. This is the distance a beam of light would travel in a vacuum in one year. Determine the size of a light year in meters.
4. The Speed of Dark
What if one fine evening, as the Sun was setting and a full moon was rising, the Sun suddenly quit emitting light?
1. How soon after the Sun went black would we know about it on Earth?
2. Sketch the relative positions of the Sun, Earth, and Moon at the time described above.
3. How soon before or after we saw the Sun go dark would the Moon cease shining?
4. What is the speed of dark?

### numerical

1. A common measure of astronomical distances is the light year. This is the distance a beam of light would travel in a vacuum in one year (about 10 trillion kilometers). As an exercise calculate the size of a…
1. light day
2. light hour
3. light minute
4. light second
5. light millisecond
6. light microsecond
7. light nanosecond
8. light picosecond
9. light femtosecond
2. Cutting lasers now exist that emit pulses of laser light so brief that they must be measured in femtoseconds. One particular femtosecond laser emits radiation with a wavelength of 1053 nm for only 350 fs.
1. What type of electromagnetic radiation does this laser emit?
2. How many wavelengths of this radiation are present in each pulse?
3. Read the following passage from the Lunar and Planetary Institute.

The Laser Ranging Retroreflector experiment was deployed on Apollo 11, 14, and 15. It consists of a series of corner-cube reflectors, which are a special type of mirror with the property of always reflecting an incoming light beam back in the direction it came from. A similar device was also included on the Soviet Union's Lunakhod 2 spacecraft. These reflectors can be illuminated by laser beams aimed through large telescopes on Earth. The reflected laser beam is also observed with the telescope, providing a measurement of the round-trip distance between Earth and the Moon. This is the only Apollo experiment that is still returning data from the Moon….

Laser beams are used because they remain tightly focused for large distances. Nevertheless, there is enough dispersion of the beam that it is about 7 kilometers in diameter when it reaches the Moon and 20 kilometers in diameter when it returns to Earth. Because of this very weak signal, observations are made for several hours at a time. By averaging the signal for this period, the distance to the Moon can be measured to an accuracy of about 3 centimeters (the average distance from the Earth to the Moon is about 385,000 kilometers).

Lunar and Planetary Institute

Determine…

1. the angular spread of the laser beam used in this experiment
2. the round trip light time for the laser pulse sent to the moon
3. the error in the round trip light time measurements…
1. in absolute terms (in nanoseconds)
2. in relative terms (in parts per trillion)
4. Visible light ranges in wavelength from 400 nm (deep violet) to 700 nm (dull red). Compute the corresponding range of frequencies?
1. What is the frequency near the lower limit where λ ~ 400 nm?
2. What is the frequency near the upper limit where λ ~ 700 nm?
5. Some cars with autonomous driving capabilities use lidar to determine the distance to objects around them using pulses of laser light. The word lidar is an acronym of light detection and ranging and was intended to mimic the words radar (radio detection and ranging) and sonar (sound navigation and ranging). A detector near the laser measures the time between when a pulse was sent out and when a reflection was received back. Some systems also compare the wavelength of the reflected laser light to the wavelength transmitted. Optical data for one particular lidar system are given in the table below.
Automotive lidar system Source: Banner Engineering
characteristic value
wavelength 905 nm
pulse duration 3 ns
pulse power 8 W
laser class 1 EN 60825-1
beam divergence 0.12°
1. What type of electromagnetic radiation does this laser emit?
2. What is the energy in one pulse of this laser?
3. If the time of flight for one pulse to hit an object and return is 40 ns, how far away is the object from the car?
4. If the reflected laser light has a wavelength of 904.99994 nm, what additional information does this tell you about the distance between the object and the car?

### investigative

1. A common measure of astronomical distances is the light year. This is the distance a beam of light would travel in a vacuum in one year. For smaller distances I would like to propose an alternate unit — the light nanosecond. What is your height in light nanoseconds?
2. Determine the one-way transit time for a signal received on Earth today if it originated from…
1. a satellite in geosynchronous orbit
2. the Moon
3. the Sun
4. Voyager 1 or Voyager 2
5. Proxima Centauri
6. the Large Magellanic Cloud
7. the Andromeda Galaxy (a.k.a. M81)
8. the edge of the observable universe
3. Calculate the "round trip light time" for the following astronomical objects. That is, how long would it take a signal to travel from the Earth to the object and back?
1. the Moon
2. the Sun
3. Mars (when closest to the Earth)
4. Pluto (when farthest from the Earth)