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Light as an electromagnetic wave, the spectrum from radio to gamma rays, antennas, polarizers and Malus's law, and Brewster's angle.
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By the end of this lesson you will be able to relate frequency and wavelength across the spectrum and follow polarized light through filters.
From the last lessons you know that light interferes and diffracts, so it is a wave, and that transverse waves oscillate across their direction of travel. From the magnetism unit you know that changing magnetic fields make electric fields. This lesson puts those together: light is a wave of electric and magnetic fields.
| Term | What it means |
|---|---|
| Electromagnetic wave | Oscillating electric and magnetic fields, perpendicular to each other and to the travel. |
| Electromagnetic spectrum | The full range, from radio to gamma rays, all traveling at $c$. |
| Polarization | The direction in which a wave's electric field oscillates. |
| Malus's law | $I = I_0\cos^2\theta$ for polarized light through a polarizer at angle $\theta$. |
| Brewster's angle | $\tan\theta_B = n$: reflected light is completely polarized. |
| Antenna | A conductor that sends or receives electromagnetic waves, often a quarter wavelength long. |
In the 1860s James Clerk Maxwell showed that changing electric fields make magnetic fields and changing magnetic fields make electric fields, so the two can sustain each other as a wave traveling through empty space at
$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00 \times 10^8\ \text{m/s}, \qquad c = f\lambda.$$
That was exactly the speed of light: light is an electromagnetic wave. Radio, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays are the same wave at different frequencies. Because the fields are transverse, light can be polarized; a polarizer passes $\cos^2\theta$ of polarized light, Malus's law.
Another way: picture
Picture shaking a rope tied to a post, up and down, and passing it through a picket fence. With the slats vertical, the up-and-down wave slips through; turn the fence sideways and the wave is blocked. A polarizing filter is a fence for light's electric field, and only the part of the field along its slats gets through.
Another way: steps
Maxwell combined the laws of electricity and magnetism and found that they allowed waves of fields moving at a speed he could calculate from lab measurements of electric and magnetic forces. It came out equal to the measured speed of light. "We can scarcely avoid the inference," he wrote, "that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."
In 1887 Heinrich Hertz made such waves with sparks and detected them across his lab, showing they reflect, refract and interfere like light. Within a decade Guglielmo Marconi was sending radio signals across the Atlantic.
Electromagnetic waves span wavelengths from kilometers to less than the size of a nucleus. Radio waves, meters to kilometers long, carry broadcasts; microwaves, centimeters long, cook food and run radar and Wi-Fi; infrared, micrometers long, is emitted by warm objects; visible light runs from $700$ nm red to $400$ nm violet.
Shorter still, ultraviolet causes sunburn, X-rays pass through soft tissue, and gamma rays come from nuclei. All travel at the same speed in vacuum; only the frequency, and so the wavelength and the energy each photon carries, differ.
Light from the Sun or a bulb is unpolarized: its electric field points every which way across the beam, changing randomly. A polarizing filter, made of long molecules lined up in one direction, absorbs the field component along the molecules and passes the perpendicular one. On average it passes half the unpolarized light.
Light already polarized passes a second filter according to Malus's law, $I = I_0\cos^2\theta$. At $0°$ all passes; at $90°$, none. Strangely, inserting a third filter at $45°$ between two crossed ones lets a quarter of the polarized light through, because each filter reorients the field.
Light reflecting from water, glass or a road is partly polarized, with its field mostly horizontal. At one angle, Brewster's angle, $\tan\theta_B = n$, the reflected light is completely polarized; for water that is about $53°$ from the vertical.
Polarized sunglasses have their axes vertical, blocking the horizontally polarized glare while passing most other light. Anglers wear them to see beneath the water's surface, and photographers use polarizing filters to deepen blue skies and cut reflections from windows.
Checking an answer. Intensities must never grow through a polarizer. Crossed polarizers alone must pass nothing. Longer wavelengths must mean lower frequencies.
The speed $c = 1/\sqrt{\mu_0\varepsilon_0}$ follows from Maxwell's equations and is the same for all frequencies in vacuum. In materials, the speed drops to $c/n$ and can depend slightly on frequency, giving dispersion.
Malus's law follows from projecting the electric field onto the polarizer's axis: the field passing is $E\cos\theta$, and since intensity goes as the field squared, the intensity passing is $I\cos^2\theta$. Averaging $\cos^2\theta$ over all random directions gives one half, which is why unpolarized light is halved.
An antenna is a conductor in which electrons are driven back and forth, radiating an electromagnetic wave, or pushed back and forth by one arriving. Antennas work best when their length matches the wavelength, often a quarter or a half.
An FM radio station at $100$ MHz has a $3$ m wavelength, so a car's whip antenna is about $75$ cm long. AM stations near $1$ MHz have $300$ m wavelengths, so their towers are tall and use the ground as part of the antenna. Phones at a few gigahertz fit antennas of a few centimeters inside their cases.
Liquid-crystal displays, in laptops, watches and many televisions, use polarizers. Light from a backlight passes a first polarizer, then a layer of liquid crystal, then a second polarizer at $90°$. The liquid crystal twists the light's polarization by $90°$ so that it passes the second filter.
Applying a voltage to a pixel straightens the crystals, stopping the twist, and the second polarizer blocks the light: the pixel goes dark. Tilt your head while wearing polarized sunglasses and look at an LCD screen: at some angles it goes black, because its light is polarized.
Just beyond violet lies ultraviolet light, with enough energy per photon to damage skin cells and DNA. The ozone layer, $15$ to $35$ km up, absorbs most of the Sun's shorter ultraviolet. In the 1970s and 1980s, chemicals called CFCs thinned it, creating an ozone hole over Antarctica.
The 1987 Montreal Protocol, which the United States helped lead, phased out CFCs, and NASA satellite measurements show the ozone layer slowly recovering. The National Weather Service's UV index forecasts how strong ultraviolet will be each day, advising sunscreen when it is high.
Astronomers observe the sky across the whole spectrum, since each band reveals different things. Radio telescopes like the Very Large Array in New Mexico map cold gas and the jets of black holes. Infrared telescopes like the James Webb Space Telescope see through dust to newborn stars and the most distant galaxies.
X-ray telescopes such as NASA's Chandra, in orbit because Earth's atmosphere blocks X-rays, see gas at millions of degrees swirling into black holes. Combining images from many bands, all the same kind of wave, gives the fullest picture of the universe.
A microwave oven's magnetron produces waves at $2.45$ GHz, a wavelength of about $12$ cm. The waves reflect from the metal walls and form a standing wave inside, with hot spots at the antinodes, half a wavelength apart, and cool spots between them.
Water molecules in food, being electric dipoles, twist back and forth in the field and heat up. The turntable carries food through hot and cold spots for more even cooking. The door's metal mesh has holes much smaller than $12$ cm, so the microwaves cannot pass, while visible light, with its much shorter wavelength, passes freely.
Sunlight scattering off air molecules is not evenly spread across colors: short wavelengths scatter far more strongly than long ones, about sixteen times more for violet than for red. Blue light scattered out of sunbeams toward us from every direction is the blue of the sky; at sunset, when sunlight crosses much more air, the blue is scattered away and the reds remain.
Scattered skylight is also polarized, most strongly in the band of sky ninety degrees from the Sun. Turn polarized sunglasses while looking at that part of the sky and it darkens and brightens. Bees and some birds see this polarization pattern and use it as a compass even when the Sun is hidden behind clouds, and Viking navigators may have used certain crystals to do the same.
Physics teachers across the United States run a kitchen experiment: take the turntable out of a microwave oven, put in a plate covered with chocolate or cheese, and heat it for about twenty seconds. The microwaves form a standing wave, and the food melts in spots at the antinodes, half a wavelength apart.
Measuring spots about $6.1$ cm apart gives a wavelength of $12.2$ cm. The oven's label gives the frequency, $2.45$ GHz, so $c = f\lambda = 2.45 \times 10^9 \times 0.122 \approx 3.0 \times 10^8$ m/s. With a ruler and a snack, students reproduce one of the fundamental constants of nature within a few percent, confirming that microwaves are light of a longer wavelength.
Sunlight reflecting from roads, car hoods and water is strongly polarized horizontally, especially near Brewster's angle, $53°$ from the vertical for water. That glare is what makes driving into a low sun or looking across a lake so uncomfortable.
Polarized sunglasses, invented by Edwin Land, who founded Polaroid in Massachusetts in 1937, have vertical transmission axes, crossed with the glare. Horizontally polarized light is blocked almost completely by Malus's law, while unpolarized light from the scene is merely halved. Anglers see fish beneath the surface, drivers see the road, and skiers see texture in bright snow. Pilots, though, often avoid them, because they can black out LCD instrument displays.
It is natural to think radio waves, light and X-rays are different kinds of things. They are all electromagnetic waves traveling at the same speed in vacuum, differing only in frequency and wavelength, which is why a single set of equations describes them all.
A related error is to think a polarizer only blocks light, so adding one can never brighten anything. Because each polarizer reorients the light's field as well as dimming it, a filter placed between two crossed polarizers lets light through that was completely blocked before.
A station broadcasts at $98.5$ MHz. Find the wavelength.
$\lambda = \dfrac{3.00 \times 10^8}{98.5 \times 10^6} = 3.05\ \text{m}$
$c = f\lambda$.
Find a quarter-wave antenna's length.
$\dfrac{3.05}{4} = 0.76\ \text{m}$
A car's whip antenna.
Find the period.
$T = \dfrac{1}{98.5 \times 10^6} = 1.02 \times 10^{-8}\ \text{s}$
About ten nanoseconds.
Find how far the wave travels in one period.
$c \times T = 3.05\ \text{m}$
One wavelength, as it must.
Find the time to reach a listener $50$ km away.
$t = \dfrac{5.0 \times 10^4}{3.00 \times 10^8} = 1.7 \times 10^{-4}\ \text{s}$
A sixth of a millisecond.
Unpolarized light of $400$ W/m² passes a polarizer at $0°$. Find the intensity.
$I_1 = 200\ \text{W/m}^2$
Halved.
A second polarizer at $90°$ follows. Find the intensity.
$I = 200\cos^2 90° = 0$
Crossed: nothing passes.
Insert a third at $45°$ between them. Find the intensity after it.
$I_2 = 200\cos^2 45° = 100\ \text{W/m}^2$
Malus's law.
Find the intensity after the last.
$I_3 = 100\cos^2 45° = 50\ \text{W/m}^2$
$45°$ from the middle axis.
Compare with the crossed pair alone.
$50 > 0$
Adding a filter let light through.
Explain the paradox.
$\text{each filter reorients the field}$
Polarizers project, not just block.
Sunlight reflects from a lake, index $1.33$. Find Brewster's angle.
$\theta_B = \tan^{-1} 1.33 = 53.1°$
From the vertical.
Find the refracted angle.
$90° - 53.1° = 36.9°$
Perpendicular to the reflected ray.
Check with Snell's law.
$\sin 53.1° = 1.33 \sin 36.9° \Rightarrow 0.800 = 0.798$
Consistent.
Find the Sun's height above the horizon for full polarization.
$90° - 53.1° = 36.9°$
Morning or afternoon sun.
State the glare's polarization.
$\text{horizontal}$
Parallel to the lake's surface.
Choose the sunglasses' axis.
$\text{vertical}$
Crossed with the glare.
Write Malus's law.
$I = I_0\cos^2\theta$
Already polarized.
Substitute the values.
$I = 300 \times \cos^2 60° = 300 \times 0.25$
$\cos 60° = 0.5$.
Evaluate the intensity.
Unpolarized light of intensity $600$ W/m² passes through two polarizers whose axes are $45°$ apart. What intensity emerges, in W/m²?
Complete the worked solution: sunlight in air reflects from acrylic, index $1.49$. With $c = 3.00 \times 10^8$ m/s, find the Brewster angle in degrees, at which the reflected light is completely polarized; the angle of the refracted ray then, in degrees; and the speed of light in the material, in units of $10^8$ m/s.
Find the Brewster angle.
$\theta_B = \tan^{-1} n =$ b
From the normal.
Find the refracted angle.
$\theta_r = 90° - \theta_B =$ r
Reflected and refracted rays are perpendicular.
Find the speed in the material.
$v = \dfrac{c}{n} =$ v
In units of $10^8$ m/s.
Relate this to sunglasses.
$\text{glare is horizontally polarized}$
Vertical-axis lenses block it.
Match each part of the electromagnetic spectrum to a use.
| broadcasting music and news | heating food and weather radar | thermal cameras and TV remote controls | imaging bones inside the body | |
|---|---|---|---|---|
| radio waves | ||||
| microwaves | ||||
| infrared | ||||
| X-rays |
a GPS satellite transmits at $1575.42$ MHz. With $c = 3.00 \times 10^8$ m/s, fill in the wavelength in m, the length of a quarter-wave antenna in m, and the wave's period in ns.
| value | |
|---|---|
| wavelength (m) | |
| quarter-wave antenna (m) | |
| period (ns) |
A radio signal travels along a foam coaxial cable with velocity factor 0.80, where waves move at that fraction of $c = 3.00 \times 10^8$ m/s. Write the wavelength along it, in meters, as a function of the frequency $f$ in MHz.
Answer:
Unpolarized light of intensity $500$ W/m² meets two crossed polarizers, at $0°$ and $90°$, with a third inserted between them at $45°$. What intensity emerges, in W/m²?
Answer: W/m²
A student removes a microwave oven's turntable, heats a chocolate bar briefly, and finds melted spots $6.0$ cm apart, where the standing wave's antinodes are. The oven's label reads $2.45$ GHz. What speed of light does this give, in units of $10^8$ m/s?
Answer: 10⁸ m/s
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A radio signal travels along a twin-lead cable with velocity factor 0.85, where waves move at that fraction of $c = 3.00 \times 10^8$ m/s. Write the wavelength along it, in meters, as a function of the frequency $f$ in MHz.
Answer:
You can work with electromagnetic waves. Explain to someone how adding a third polarizer can let more light through.
23. Your turn: polarized light of $300$ W/m² passes a polarizer at $60°$ to its polarization. What intensity passes?, step 3
$I = 75\ \text{W/m}^2$
A quarter passes.