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 $c = 3\cdot10^{8}\ \text{m/s}$.

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

A home router broadcasts Wi-Fi in two bands: $2,4\ \text{GHz}$ and $5\ \text{GHz}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$). 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 centimetres.
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.

A violet laser pointer emits light with a wavelength of $\lambda = 400\ \text{nm}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Express the wavelength in metres, in standard form.
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.

An X-ray photograph uses X-rays with a wavelength of $\lambda = 0,05\ \text{nm}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Express the wavelength in metres, in standard form.
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.

In 1849 Fizeau measured the speed of light on Earth's surface for the first time using a toothed wheel. A beam of light passed through the gap between two teeth, travelled $d = 8,63\ \text{km}$ to a mirror and returned; if on its return it encountered a tooth instead of a gap, the beam disappeared from the observer's view. The wheel had $720$ teeth and $720$ equal gaps (given: $c = 3\cdot10^{8}\ \text{m/s}$). Observe the setup:

1. a) Calculate the time taken for light to travel to the mirror and back.
2. b) The first eclipse occurs when, during that time, the wheel rotates half a division, that is, $1/1440$ of a turn. Calculate how many revolutions per second the wheel must turn to produce it, and express it also in revolutions per minute.
3. c) Comment on what makes this method ingenious: what quantity is measured directly using the wheel, and how is timing a millionth-of-a-second interval avoided?

Quantum physics closed the historical debate: light exchanges energy in packets called photons, each with energy $E = h \cdot f$ (given: $h = 6,63\cdot10^{-34}\ \text{J}\cdot\text{s}$).

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 $f = 3\cdot10^{18}\ \text{Hz}$.
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 carried out with controlled doses and lead protection.

The colour of a star reveals its surface temperature: the spectrum of a hot body reaches its emission maximum at a wavelength given by Wien's law, $\lambda_{max} \cdot T = 2,9\cdot10^{-3}\ \text{m}\cdot\text{K}$, with the temperature $T$ in kelvin. The graph shows the emission curve of the Sun.

1. a) The Sun has a surface temperature of $5800\ \text{K}$. Calculate the wavelength at which it emits with maximum intensity, express it in nanometres and state which band of the spectrum it belongs to.
2. b) Betelgeuse, the red supergiant in Orion, has a surface temperature of about $3300\ \text{K}$. Repeat the calculation and explain why it appears reddish to us.
3. c) Applying the law, predict the dominant colour of a star of $10000\ \text{K}$ and justify your answer based on the position of its emission maximum in the spectrum.