Electromagnetic waves

Radio, television, mobile networks, Wi-Fi, microwave ovens, X-ray machines and even sunlight are all electromagnetic waves. They differ from one another only in frequency (wavelength). In this topic we see how such a wave arises, its properties, the electromagnetic spectrum, and the principles of radio communication and radar.

§1. Maxwell’s theory and how a wave arises

Maxwell (1864) combined two ideas:

• a changing magnetic field creates a circulating electric field (electromagnetic induction);
• a changing electric field creates a magnetic field.

So once changing fields appear, they keep generating each other and travel through space — this is an electromagnetic wave. It is emitted by accelerating charges (for example, electrons oscillating in an antenna). A closed LC circuit hardly radiates, so an antenna is an open circuit. Hertz was the first to produce and detect such waves experimentally (1888).

From his equations Maxwell calculated the speed of the wave: c = 1/√(ε₀μ₀) ≈ 3·10⁸ m/s. This matched the measured speed of light, and Maxwell concluded that light is an electromagnetic wave too. Hertz, using a spark-gap dipole and a receiving loop (resonator), showed that the waves reflect, refract and form standing waves, and confirmed that they travel at the speed of light.

§2. Properties of electromagnetic waves

• The wave is transverse: E⃗, B⃗ and the propagation velocity v⃗ are mutually perpendicular.
• E and B oscillate in phase.
• No medium is needed — the wave travels through a vacuum; its speed in vacuum is

• In a medium the speed is lower: v = c/n, where n is the refractive index.
• The wave carries energy; it reflects, refracts, shows interference and diffraction, and can be polarised.

The basic relation:

When a wave enters another medium its frequency does not change; the speed and wavelength change: λ_medium = λ₀/n.

Worked example

An FM station broadcasts at 100 MHz. Wavelength?

1. ν = 100·10⁶ = 10⁸ Hz.
2. λ = c/ν = 3·10⁸/10⁸ = 3 m.

Worked example

Light with a vacuum wavelength of 600 nm enters glass (n = 1.5). Speed and wavelength in the glass?

1. v = 3·10⁸/1.5 = 2·10⁸ m/s.
2. λ = 600/1.5 = 400 nm; the frequency is unchanged: ν = 3·10⁸/(6·10⁻⁷) = 5·10¹⁴ Hz.

§3. Aerials and the energy of a wave

In a closed circuit the electric field is “locked” inside the capacitor and the magnetic field inside the coil, so it hardly radiates. If we pull the capacitor plates apart and straighten the coil into a wire, we get an open circuit, a dipole (aerial): the field fills the surrounding space and breaks away as a wave. The simplest aerial is a wire half a wavelength long (a half-wave dipole), in which the current oscillations build up by resonance. That is why aerials for high frequencies (mobile phones, Wi-Fi) are small, while long-wave radio masts are very tall.

An electromagnetic wave carries energy equally in its electric and magnetic fields. From the quantum point of view radiation is a stream of photons, each with energy

The shorter the wavelength, the greater the photon energy. Ultraviolet, X-rays and gamma rays can break bonds in molecules, so they are ionising radiation, whereas radio photons are far too weak for that.

Worked example

Find the length of a half-wave aerial for 100 MHz and the energy of one photon.

1. λ = c/ν = 3·10⁸/10⁸ = 3 m, aerial length λ/2 = 1.5 m.
2. Photon energy: E = hν = 6.63·10⁻³⁴·10⁸ = 6.63·10⁻²⁶ J.
3. For comparison, a 300 nm ultraviolet photon has E = hc/λ = 6.63·10⁻³⁴·3·10⁸/(3·10⁻⁷) = 6.63·10⁻¹⁹ J ≈ 4.1 eV, 10⁷ times more.

§4. The electromagnetic spectrum

In order of decreasing wavelength and increasing frequency:

1. Radio waves (longer than 1 mm) — radio, television, mobile networks.
2. Microwaves (roughly 1 mm to 1 m, part of the radio band) — radar, microwave ovens, Wi-Fi.
3. Infrared (760 nm – 1 mm) — thermal radiation, night vision, remote controls.
4. Visible light (380–760 nm): from red to violet.
5. Ultraviolet — sunburn, disinfection, vitamin D.
6. X-rays — medicine, studying crystals.
7. Gamma rays — nuclear processes, cancer therapy.

As frequency increases, the photon energy (E = hν) and penetrating power increase.

§5. Principles of radio communication

Sound frequencies (20 Hz – 20 kHz) are low, and such waves cannot be radiated efficiently. Therefore:

1. An oscillator produces a high-frequency "carrier" wave.
2. Modulation — the sound signal varies the carrier’s amplitude (AM) or frequency (FM).
3. The antenna radiates the modulated wave.
4. In the receiver, the circuit is tuned to the wanted station by resonance.
5. Demodulation (detection) — the sound signal is extracted from the high-frequency oscillation (using a diode) and sent to the loudspeaker.

Among the first to use radio in practice were A. S. Popov (1895) and G. Marconi.

§6. Radar

A radar sends a short pulse and receives the signal reflected from an object. The signal covers the distance twice:

The gap between pulses must be longer than the echo’s return time, so the maximum range is s_max = c/(2ν_pulse).

Worked example

A radar echo returns after 200 μs. Distance to the object?

1. t = 200·10⁻⁶ = 2·10⁻⁴ s.
2. s = 3·10⁸·2·10⁻⁴/2 = 3·10⁴ m = 30 km.

Worked example

The distance to the Moon is 3.84·10⁸ m. After what time does a radio echo return?

1. t = 2s/c = 2·3.84·10⁸/(3·10⁸).
2. t = 2.56 s.

§7. Intensity of radiation

Intensity is the energy a wave carries per second through 1 m² of a perpendicular surface (W/m²). For a point source radiating equally in all directions:

Doubling the distance cuts the intensity by a factor of 4. The intensity radiated by an oscillating charge is proportional to the fourth power of frequency (ν⁴) — this is why radio uses a high-frequency carrier.

Worked example

Intensity 10 m from a 100 W source?

1. I = 100/(4π·10²) = 100/1257.
2. I ≈ 0.08 W/m².

Check yourself

1. Frequency of a 300 m radio wave?

2. A mobile phone works at 900 MHz. Wavelength?

3. A radar echo returns after 0.4 ms. Distance?

4. 600 nm light enters water (n = 4/3). Speed and wavelength in water?

5. A radar’s pulse repetition frequency is 1000 Hz. Maximum range?

6. Which has the higher frequency, infrared or ultraviolet?

7. Energy of a photon of 600 nm light?

8. Length of a half-wave aerial for 900 MHz?

Common mistakes

• Thinking electromagnetic waves, like sound, need a medium.
• Forgetting to halve the time in radar problems.
• Thinking the frequency changes when a wave enters a medium: it is the speed and wavelength that change.
• Not converting MHz, nm, μs prefixes to SI.
• Calling an electromagnetic wave longitudinal: it is always transverse.
• Thinking radio waves and ultraviolet are equally dangerous: the ionising effect depends on photon energy, i.e. on frequency.