Exercises: Nature of light: electromagnetic wave and 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 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.

Arrange the following bands of the electromagnetic spectrum in order from shortest to longest wavelength: microwaves, X-rays, visible light, radio waves, ultraviolet radiation, infrared radiation.

Then state which of them has the highest frequency and which the lowest, and what general relationship exists between wavelength and frequency.

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 $d = 3,84\cdot10^{8}\ \text{m}$ and light travels at $c = 3\cdot10^{8}\ \text{m/s}$. 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.

Maxwell predicted in 1865 that light is an electromagnetic wave and that its speed in a vacuum depends only on two constants of electromagnetism: $c = \frac{1}{\sqrt{\varepsilon_0 \cdot \mu_0}}$. Given: $\varepsilon_0 = 8,85\cdot10^{-12}\ \text{C}^2/\text{N}\cdot\text{m}^2$ and $\mu_0 = 4\pi\cdot10^{-7}\ \text{N/A}^2$.

1. a) Calculate the value of $c$ obtained from these data.
2. b) Compare it with the accepted value, $c = 3\cdot10^{8}\ \text{m/s}$, and explain why this agreement was decisive proof that light is an electromagnetic wave.

Visible light ranges in colour from red ($\lambda \approx 700\ \text{nm}$) to violet ($\lambda \approx 400\ \text{nm}$) (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Calculate the frequency of red light and that of violet light.
2. b) How many times higher is the frequency of violet than that of red? Express the result to two significant figures.

Radio telescopes study the universe through the $21\ \text{cm}$ line emitted by neutral hydrogen in the interstellar medium (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Express $21\ \text{cm}$ in metres and calculate the frequency of this radiation, in hertz and gigahertz.
2. b) State which band of the electromagnetic spectrum it belongs to and give a reasoned advantage of observing the cosmos in that band compared to visible light.

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.

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.

The LED of a remote control emits infrared radiation of $\lambda = 940\ \text{nm}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$; the human eye detects radiation between $380\ \text{nm}$ and $750\ \text{nm}$). The figure places the LED emission in the spectrum.

1. a) Calculate its frequency.
2. b) By comparing wavelengths with the visible range, explain why we do not see the flash from the LED even though a mobile phone camera records it on screen.

A geostationary telecommunications satellite orbits at $36000\ \text{km}$ above the equator. A transcontinental video call sends the signal up to the satellite, which forwards it to the receiving station on the surface (given: $c = 3\cdot10^{8}\ \text{m/s}$). Observe the diagram:

1. a) Express the altitude of the satellite in metres, in standard form.
2. b) Calculate the minimum time taken for the signal to travel up to the satellite and down to the receiving station.
3. c) Explain what effect this delay produces in a live conversation and why it is barely noticeable when the signal travels through a submarine cable.

The visible spectrum extends, approximately, from $\lambda = 380\ \text{nm}$ (violet) to $\lambda = 750\ \text{nm}$ (red) (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Calculate the frequency range of the visible spectrum, in standard form.
2. b) State which colour corresponds to the highest frequency end and which to the lowest, and write down the general relationship between wavelength and frequency for any electromagnetic wave in a vacuum.

A hospital germicidal lamp emits ultraviolet radiation of $\lambda = 254\ \text{nm}$ (UVC band), whereas a "black light" lamp emits $\lambda = 365\ \text{nm}$ (UVA band) (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Calculate the frequency of each type of radiation.
2. b) UVC destroys the genetic material of bacteria and viruses, whereas UVA is much less aggressive. Relate this behaviour to the calculated frequencies and to the position of each band within the electromagnetic spectrum.

A microwave oven heats food using electromagnetic waves of $f = 2,45\ \text{GHz}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$). Observe the diagram of the door:

1. a) Calculate its wavelength and express it in centimetres.
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 ($\lambda \approx 500\ \text{nm}$).

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) Summarise 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.

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 GPS system positions a receiver by measuring the time taken for the electromagnetic signal to arrive from each satellite, which orbits at an altitude of $20200\ \text{km}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$).

1. a) Calculate the time taken for the signal to travel from the satellite to the receiver.
2. b) If the receiver's clock accumulates an error of $30\ \text{ns}$, calculate the distance error that this time discrepancy introduces by applying $\Delta x = c \cdot \Delta t$.
3. c) In light of the result, justify why GPS satellites carry atomic clocks with nanosecond precision and describe what would happen to positioning accuracy if ordinary quartz clocks were used.

A weather radar emits microwave pulses of $f = 2,8\ \text{GHz}$ that are reflected by raindrops. The echo from a storm returns $0,50\ \text{ms}$ after the pulse is emitted (given: $c = 3\cdot10^{8}\ \text{m/s}$). Observe the diagram:

1. a) Calculate the distance to the storm, taking into account that the pulse travels twice that distance.
2. b) Calculate the wavelength of the radar.
3. c) Raindrops are a few millimetres in size. Explain why microwaves rather than visible light are used to detect precipitation tens of kilometres away, considering how each type of radiation interacts with clouds.

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.

In the vicinity of a transmitting antenna, the electromagnetic wave has a maximum electric field of $E_{max} = 300\ \text{V/m}$ and a frequency of $f = 90\ \text{MHz}$ (given: $c = 3\cdot10^{8}\ \text{m/s}$; in an electromagnetic wave $E = c \cdot B$).

1. a) Calculate the maximum value of the magnetic field, $B_{max}$.
2. b) Calculate the wavelength of the signal.
3. c) Sketch the wave at a given instant in your notebook: mark the direction of propagation and the orientations of $\vv{E}$ and $\vv{B}$, knowing that they are perpendicular to each other and to the direction of propagation, and that the latter is in the direction of the vector product $\vv{E} \wedge \vv{B}$.

Andromeda, the large galaxy closest to the Milky Way, is $2,5$ million light years away (given: $c = 3\cdot10^{8}\ \text{m/s}$; one year $\approx 3,16\cdot10^{7}\ \text{s}$).

1. a) Calculate how many metres there are in one light year.
2. b) Express the distance to Andromeda in metres and in kilometres, in standard form.
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.

Unit challenge. How does the invisible energy around you propagate? Your mission is to characterise the electromagnetic spectrum by connecting your kitchen with the solar radiation in your locality.

1. Experimental measurement: Remove the turntable from your microwave and heat a strip of cheese or chocolate for about $20\text{ s}$ until you see adjacent melted spots. Measure the internodal distance ($\lambda/2$) with a ruler ($\pm 0,1\text{ cm}$) in four repetitions. Record the data in a table with its uncertainties and calculate $c = \lambda \cdot f$ from the frequency indicated on the label ($f \approx 2450\text{ MHz}$).

2. Official consultation: Go to the AEMET website and find today's ultraviolet index (UVI) and maximum solar irradiance in your province, identifying the representative $\lambda$ of the UV-B range.

3. Data processing and final product: Plot a graph showing the dispersion of your $c$ values, determine the mean value with its uncertainty, and calculate the relative error with respect to $3,00 \times 10^8\text{ m/s}$. Write a short report contrasting the energy ($E = h \cdot f$) of a microwave photon with that of a solar UV photon.

Assessment criteria:

• Rigour in home data collection, uncertainty estimation, and quality of the table and graph.

• Conceptual correctness in calculating wave parameters ($c = \lambda \cdot f$) and photon energy ($E = h \cdot f$).

• Critical analysis of the experimental results against the theoretical value and AEMET data.