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Light traveling in straight lines, shadows and pinhole images, the speed of light, and its slower speed and shorter wavelength in materials.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to use rays to explain shadows and images and compute light's speed, wavelength and travel time.
From earlier courses you know that light lets us see objects, that it travels very fast, and that shadows form behind opaque objects. From Physics 1 you know speed as distance over time and similar triangles from geometry. This lesson builds the ray model that the rest of the optics unit uses.
| Term | What it means |
|---|---|
| Ray | A straight line showing the direction light travels in a uniform medium. |
| Speed of light | $c = 3.00 \times 10^8$ m/s in vacuum. |
| Index of refraction | $n = c/v$, how many times slower light travels in a material. |
| Umbra and penumbra | The full and partial shadow of an object lit by an extended source. |
| Pinhole camera | A box with a tiny hole that forms an inverted image by straight rays. |
| Light-year | The distance light travels in a year, $9.46 \times 10^{15}$ m. |
In a uniform material, light travels in straight lines. Drawing those paths as rays is enough to explain shadows, pinhole images, mirrors and lenses, as long as the openings involved are much wider than light's wavelength.
Light's speed in vacuum is $c = 3.00 \times 10^8$ m/s. In a material it is slower:
$$v = \frac{c}{n},$$
where $n$, the index of refraction, is about $1.0003$ for air, $1.33$ for water and $1.5$ for glass. The light's frequency, set by its source, never changes; its wavelength shrinks to $\lambda_0/n$.
Another way: picture
Picture sunlight streaming through a gap in the clouds or a dusty attic window: the beams are straight, sharp-edged and diverge only slightly. Each beam is a bundle of rays. Where a ray meets an opaque object, it stops, and the space behind is in shadow; where it passes a small hole, it continues in the same straight line.
Another way: steps
A small source casts a sharp shadow: every ray from it either passes the edge of an object or is blocked. An extended source, like the Sun or a frosted bulb, casts a shadow with two parts. In the umbra no part of the source can be seen, and it is fully dark; around it, in the penumbra, part of the source is hidden, and it is partly lit.
Eclipses are shadows on a grand scale. During the total solar eclipse of April 8, 2024, the Moon's umbra swept from Texas to Maine; people in the path saw day turn to night, while those in the wide penumbra saw only a partial eclipse.
A tiny hole in one side of a dark box lets through only one ray from each point of an outside object. Rays from the top of a tree travel down through the hole to the bottom of the back wall; rays from its base travel up. The result is an inverted image.
Similar triangles give its size: the image height over the box depth equals the object height over its distance. A $10$ m tree $20$ m away forms a $7.5$ cm image in a box $15$ cm deep. Artists once used room-sized pinhole cameras, the camera obscura, to trace scenes accurately.
Galileo tried to time light between lanterns on distant hills and found it too fast to measure. In 1676 Ole Rømer noticed that Jupiter's moons ran late when Jupiter was far from Earth and early when near, and he estimated the time light takes to cross Earth's orbit.
Albert Michelson, the first American Nobel laureate in science, measured it with rotating mirrors, including a run between Mount Wilson and Mount San Antonio in California in 1926. Since 1983 the speed of light has been fixed at exactly $299{,}792{,}458$ m/s, and the meter is defined from it.
In glass or water, light is absorbed and re-emitted by the atoms' electrons, and the combined wave travels more slowly than in vacuum. The index $n = c/v$ measures how much: in water, light moves at $c/1.33$, three-quarters of its vacuum speed; in diamond, at less than half.
The frequency cannot change at the boundary, since each wave crest arriving must leave, so the slower speed shows up as a shorter wavelength: $\lambda = \lambda_0/n$. Red light of $633$ nm in air has a wavelength of $476$ nm in water. The next lessons show how this slowing bends light at boundaries.
Checking an answer. Speeds in materials must be less than $c$. Indices must be at least one. Pinhole images must be inverted and grow with box depth.
The ray model is an approximation of the wave nature of light, valid when objects and openings are much larger than the wavelength, about half a micrometer for visible light. For tiny openings light spreads out, which is diffraction, the subject of the physical optics lessons.
Rays are reversible: if light can travel from A to B along a path, it can travel from B to A along the same path. That is why if you can see someone in a mirror, they can see you.
Because light takes time to travel, looking far away means looking into the past. Sunlight is about eight minutes old when it reaches us. Light from the nearest star beyond the Sun, Proxima Centauri, left it over four years ago; it is $4.2$ light-years away.
The James Webb Space Telescope, operated by NASA with European and Canadian partners, sees galaxies whose light left them more than thirteen billion years ago, showing the universe as it was a few hundred million years after the Big Bang.
Most long-distance communication now travels as light pulses in glass fibers. In a fiber with index $1.47$, light moves at about two-thirds of $c$, so a signal takes about five microseconds per kilometer. A New York to Los Angeles route adds about twenty milliseconds each way.
For financial trading, where milliseconds matter, firms have built straighter fiber routes between Chicago and New York and even microwave towers, since radio in air travels at nearly the full speed of light, faster than light in glass.
Radio waves are light of long wavelength, traveling at $c$. Commands to a Mars rover take between three and twenty-two minutes to arrive, depending on where the planets are in their orbits. Engineers at NASA's Jet Propulsion Laboratory in Pasadena plan each day's drive in advance and send it in one batch.
The Voyager 1 probe, launched in 1977 and now in interstellar space, is so far away that its signals take more than twenty-two hours to reach the Deep Space Network's dishes in California, Spain and Australia.
A laser beam is the ray model made visible: an intense, narrow, nearly parallel bundle of rays. Surveyors use lasers to lay out straight lines for roads and buildings; construction crews use rotating laser levels to set floors flat.
Astronauts on Apollo 11, 14 and 15 left retroreflectors on the Moon. Observatories such as the one at Apache Point in New Mexico fire laser pulses at them and time the echoes, measuring the Earth-Moon distance to within millimeters and showing the Moon drifts away about $3.8$ cm a year.
We see an object because rays from each point on it enter our eyes. For a luminous object like a lamp, the rays start there; for most objects, they are rays from a light source that scatter off the surface in every direction. Our brains assume the rays traveled in straight lines and place the object back along them.
That assumption is what mirrors and lenses exploit. When rays are bent or reflected on the way, the brain still traces them straight back and sees an image where there is no object: the subject of the next four lessons.
Rays spreading from a small source cover a larger and larger sphere as they travel. The same light energy spread over a sphere four times the area at twice the distance means the brightness, the power per square meter, falls as one over the distance squared. A reading lamp one meter away gives four times the light of the same lamp two meters away.
Photographers use this when placing flashes, and lighting designers use it when spacing streetlights. Astronomers turn it around: knowing how much light a certain kind of star really gives off, they measure how faint it looks and work out how far away it is. Henrietta Leavitt at the Harvard College Observatory found such standard stars in 1912, and Edwin Hubble used them to show that the universe contains galaxies far beyond our own.
When NASA's Perseverance rover drives across Jezero Crater, no one steers it live. Mars is between $55$ and $400$ million km from Earth, so a radio command traveling at $3.00 \times 10^8$ m/s takes between three and twenty-two minutes to arrive, and any picture it sends back takes as long again.
Engineers at the Jet Propulsion Laboratory therefore plan each day's route from the previous day's images and send the commands in one batch. The rover uses its own cameras and software to spot rocks and slopes and pick a safe path, a necessity that the finite speed of light imposes. Future crewed missions will face the same delay in every conversation home.
Nearly all long-distance internet traffic in the United States travels as pulses of infrared light in glass fibers thinner than a hair. In the fiber's core, index about $1.47$, light moves at about $2.04 \times 10^8$ m/s. A $1145$ km route from New York to Chicago therefore takes about $5.6$ ms each way.
That delay limits how quickly distant computers can respond to each other. Online gamers notice it as lag; cloud companies build data centers near large cities to shorten it. Some trading firms pay for microwave links between Chicago and New Jersey because radio in air travels nearly fifty percent faster than light in glass, saving a couple of milliseconds.
It is natural to think light arrives the moment it is emitted. It travels at a finite speed, and over astronomical distances the delay is minutes, hours or years. In glass and water it is slower still, which is what makes lenses and prisms work.
A related error is to think light's frequency or color changes when it enters water. The frequency is set by the source and stays the same; the speed and wavelength change together, and the color we see depends on the frequency.
The Sun is $1.50 \times 10^{11}$ m away. Write the travel time.
$t = \dfrac{d}{c} = \dfrac{1.50 \times 10^{11}}{3.00 \times 10^8}$
Distance over speed.
Evaluate it in seconds.
$t = 500\ \text{s}$
Seconds.
Convert to minutes.
$t = 8.3\ \text{min}$
We see the Sun as it was.
Find the Moon's light time at $3.84 \times 10^8$ m.
$t = 1.28\ \text{s}$
Just over a second.
Find a laser echo's round-trip time to the Moon.
$2 \times 1.28 = 2.56\ \text{s}$
What Apache Point measures.
A small bulb is $0.50$ m from a $10$ cm tall figure; the wall is $2.0$ m from the bulb. Set up similar triangles.
$\dfrac{h_s}{2.0} = \dfrac{10}{0.50}$
Rays from the point source.
Solve for the shadow's height.
$h_s = 10 \times \dfrac{2.0}{0.50} = 40\ \text{cm}$
Four times larger.
Move the figure to $1.0$ m from the bulb. Find the new shadow.
$h_s = 10 \times \dfrac{2.0}{1.0} = 20\ \text{cm}$
Smaller as it nears the wall.
Replace the bulb with a frosted globe.
$\text{a blurred penumbra appears}$
Extended source.
Explain the sharp edge with a point source.
$\text{each edge ray is a single line}$
No partial shadow.
Relate this to shadow puppets.
$\text{near the light, large shadows}$
Puppeteers move closer to grow.
Green light, $532$ nm in air, enters glass of index $1.50$. Find its speed.
$v = \dfrac{3.00 \times 10^8}{1.50} = 2.00 \times 10^8\ \text{m/s}$
$c/n$.
Find its frequency.
$f = \dfrac{3.00 \times 10^8}{532 \times 10^{-9}} = 5.64 \times 10^{14}\ \text{Hz}$
Set by the source.
Find its wavelength in the glass.
$\lambda = \dfrac{532}{1.50} = 355\ \text{nm}$
Shorter.
Check $v = f\lambda$ in the glass.
$5.64 \times 10^{14} \times 355 \times 10^{-9} = 2.00 \times 10^8$
Consistent.
Find the time to cross $5.0$ cm of glass.
$t = \dfrac{0.050}{2.00 \times 10^8} = 2.5 \times 10^{-10}\ \text{s}$
A quarter of a nanosecond.
Compare with the same path in air.
$t_0 = 1.67 \times 10^{-10}\ \text{s}$
The glass delays it by half.
Write the speed formula.
$v = \dfrac{c}{n}$
Index as a ratio.
Substitute the values.
$v = \dfrac{3.00 \times 10^8}{1.25}$
Vacuum speed over index.
Evaluate the speed.
The index of refraction of acrylic is $1.49$. How fast does light travel in it, in units of $10^8$ m/s? Take $c = 3.00 \times 10^8$ m/s.
Complete the worked solution: yellow light of vacuum wavelength $589$ nm enters ice, index $1.31$. With $c = 3.00 \times 10^8$ m/s, find its speed in the material in units of $10^8$ m/s, its wavelength there in nm, and its frequency in units of $10^{14}$ Hz.
Find the speed in the material.
$v = \dfrac{c}{n} =$ v
In units of $10^8$ m/s.
Find the wavelength in the material.
$\lambda = \dfrac{\lambda_0}{n} =$ w
Shorter by the index.
Find the frequency.
$f = \dfrac{c}{\lambda_0} =$ f
Set by the source; unchanged.
Check with the material's values.
$v = f\lambda$
Speed equals frequency times wavelength.
Match each term to its meaning in the ray model.
| a straight path along which light's energy travels | a region an opaque object keeps rays from reaching | the part of a shadow where the whole source is hidden | the speed of light in vacuum divided by its speed in the material | |
|---|---|---|---|---|
| ray | ||||
| shadow | ||||
| umbra | ||||
| index of refraction |
A pinhole camera is a box $15$ cm deep. It is pointed at a tree $10$ m tall and $20$ m away. Fill in the height of the image in cm, the magnification, and the image height in cm if the box were twice as deep.
| value | |
|---|---|
| image height (cm) | |
| magnification | |
| image height with a doubled box (cm) |
Light crosses a thickness $L$, in meters, of crown glass, index $1.50$. Taking $c = 0.300$ m/ns, write the crossing time in nanoseconds as a function of $L$.
Answer:
A fiber-optic route from New York to Chicago is $1145$ km long, and the fiber's core has index $1.47$. How long does a light pulse take to travel it, in ms? Take $c = 3.00 \times 10^8$ m/s.
Answer: ms
Engineers at NASA's Jet Propulsion Laboratory send commands to a Mars rover by radio, which travels at $3.00 \times 10^8$ m/s. When Mars is $55$ million km away, how long does a command take to arrive, in minutes?
Answer: min
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Light crosses a thickness $L$, in meters, of crown glass, index $1.50$. Taking $c = 0.300$ m/ns, write the crossing time in nanoseconds as a function of $L$.
Answer:
You can use the ray model. Explain to someone why a pinhole camera's image is upside down.
23. Your turn: what is the speed of light in a liquid of index $1.25$?, step 3
$v = 2.40 \times 10^8\ \text{m/s}$
Eighty percent of $c$.