The nature of light: electromagnetic waves and the spectrum

Throughout history, two models concerning the nature of light have competed: the corpuscular model, defended by Newton at the end of the 17th century, and the wave model, proposed by his contemporary Huygens.

Match each statement with the model it belongs to:

1. a) Light is made up of small particles that travel in straight lines.
2. b) Light propagates as a wave disturbance, in a similar way to waves on the surface of water.
3. c) Reflection is explained as particles bouncing off a polished surface.
4. d) Light should exhibit characteristic wave phenomena, such as diffraction and interference.

A radio station broadcasts in FM at a frequency of $f = 98.5\ \text{MHz}$. Radio waves are electromagnetic waves that travel through a vacuum at $c = 3\times10^{8}\ \text{m/s}$.

1. a) Express the frequency in hertz, in scientific notation.
2. b) Calculate the wavelength of the signal by applying $c = \lambda \times f$.

A home router broadcasts Wi-Fi in two bands: $2.4\ \text{GHz}$ and $5\ \text{GHz}$ (given: $c = 3\times10^{8}\ \text{m/s}$). The figure shows what happens to a wave when it reaches an obstacle with an aperture.

1. a) Calculate the wavelength of each band and express it in centimeters.
2. b) In practice, the $2.4\ \text{GHz}$ band penetrates walls and furniture better. Using the calculated wavelengths and the phenomenon of diffraction, explain why the longer-wavelength signal gets around obstacles better.

The debate on the nature of light lasted for more than two centuries and marked the history of physics. The figure schematically shows Young's double-slit experiment described in section b).

1. a) Summarize what Newton's corpuscular model and Huygens' wave model consisted of, and state an optical phenomenon that both models were able to explain.
2. b) In 1801, Young presented his double-slit experiment: when light passed through two close slits, a pattern of alternating bright and dark fringes appeared on the screen. Explain which wave phenomenon this pattern reveals and why it proved decisive against the corpuscular model.
3. c) What did Maxwell prove in 1865 about the type of wave that light is, and what did he predict about its speed in a vacuum?
4. d) At the beginning of the 20th century, the photoelectric effect, interpreted by Einstein in 1905, could only be explained by assuming that light exchanges energy in discrete packets, photons. State in your own words the wave-particle duality with which modern physics describes light.

A violet laser pointer emits light with a wavelength of $\lambda = 400\ \text{nm}$ (given: $c = 3\times10^{8}\ \text{m/s}$).

1. a) Express the wavelength in meters, in scientific notation.
2. b) Calculate the frequency of this light.

According to Maxwell's theory, light is an electromagnetic wave. The figure represents an instant of that wave propagating through a vacuum.

1. a) State which two fields oscillate, perpendicular to each other and to the direction of propagation, in an electromagnetic wave.
2. b) Explain the fundamental difference between the propagation of light and that of sound: which of the two requires a material medium and which can travel through a vacuum? Give an astronomical example that demonstrates this.

The astronauts of the Apollo missions left retroreflectors on the Moon. A laser pulse is sent from an observatory on Earth, reflects off them and returns. The Earth-Moon distance is $d = 3.84\times10^{8}\ \text{m}$ and light travels at $c = 3\times10^{8}\ \text{m/s}$. Observe the diagram:

1. a) Calculate the time taken for the pulse to reach the Moon.
2. b) Calculate the total round-trip time measured by the observatory.

An X-ray photograph uses X-rays with a wavelength of $\lambda = 0.05\ \text{nm}$ (given: $c = 3\times10^{8}\ \text{m/s}$).

1. a) Express the wavelength in meters, in scientific notation.
2. b) Calculate the frequency of these X-rays.
3. c) Compare this wavelength with that of visible light ($\lambda \approx 500\ \text{nm}$) and explain why X-rays pass through soft tissue that light cannot pass through.

A microwave oven heats food using electromagnetic waves of $f = 2.45\ \text{GHz}$ (given: $c = 3\times10^{8}\ \text{m/s}$). Observe the diagram of the door:

1. a) Calculate its wavelength and express it in centimeters.
2. b) The oven door has a metal mesh with holes of about $2\ \text{mm}$: we can see inside, but the microwaves do not escape. Justify this by comparing the size of the holes with the wavelength of microwaves and with that of visible light ($\lambda \approx 500\ \text{nm}$).

Andromeda, the large galaxy closest to the Milky Way, is $2.5$ million light years away (given: $c = 3\times10^{8}\ \text{m/s}$; one year $\approx 3.16\times10^{7}\ \text{s}$).

1. a) Calculate how many meters there are in one light year.
2. b) Express the distance to Andromeda in meters and in kilometers, in scientific notation.
3. c) The light from Andromeda captured today by telescopes left the galaxy long before modern human beings existed. Explain why observing distant objects is equivalent to "looking back in time" and why astronomers prefer to express these distances in light years.

Quantum physics closed the historical debate: light exchanges energy in packets called photons, each with energy $E = h \times f$ (given: $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$).

1. a) Calculate the energy of a photon from a radio station broadcasting at $f = 100\ \text{MHz}$.
2. b) Calculate the energy of an X-ray photon of $f = 3\times10^{18}\ \text{Hz}$.
3. c) Calculate how many times more energetic the X-ray photon is than the radio photon and use that factor to justify two facts: we live surrounded by radio waves without suffering any ionisation damage, whereas X-rays are used only with controlled doses and lead protection.