Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

All electromagnetic waves obey the same Maxwell equations, from the $50 Hz$ field of a power line to the gamma rays of a PET scanner. Only the frequency differs, or, equivalently, the free-space wavelength $λ_{0} = c / f$ . Figure 5.6.4 lays out twenty-two decades of frequency with some of their uses.

The electromagnetic spectrum on a logarithmic frequency scale, with the free-space wavelength \lambda_0=c/f below the axis. The radio part is divided into the ITU bands (VLF to EHF), above which come the infrared, the narrow visible window, the ultraviolet, X-rays and gamma rays. The labels mark typical uses. — **Figure 5.6.4.** The electromagnetic spectrum on a logarithmic frequency scale, with the free-space wavelength $λ_{0} = c / f$ below the axis. The radio part is divided into the ITU bands (VLF to EHF), above which come the infrared, the narrow visible window, the ultraviolet, X-rays and gamma rays. The labels mark typical uses.

The International Telecommunication Union names the radio bands by decades: band $N$ runs from $0.3 \times 1 0^{N}$ to $3 \times 1 0^{N} Hz$ , from VLF ( $N = 4$ , $3$ – $30 kHz$ ) to EHF ( $N = 11$ , $30$ – $300 GHz$ ). Above the radio bands the infrared reaches up to about $430 THz$ , visible light occupies the narrow window from $430$ to $750 THz$ (wavelengths from $700$ to $400 nm$ ), the ultraviolet extends to about $30 PHz$ and X-rays to about $30 EHz$ .

Same equations, different hardware

The equations are the same everywhere on the spectrum, but the hardware scales with the wavelength. A $50 Hz$ wave is $6000 km$ long, so a power line is electrically short and circuit theory suffices. An FM wave is $3 m$ long, which is the size of a car aerial, and the optical carrier of a fibre is $1.55 μ m$ long. From light upwards the photon energy $h f$ matters too, and the classical field picture of this book must be supplemented by quantum physics.

Application 5.6.1 · The radio spectrum and its allocation

The radio spectrum is a shared and finite resource. The ITU Radio Regulations, whose international Table of Frequency Allocations goes back to the Atlantic City conference of 1947, assign every band to services, and national regulators license the users. Table 5.6.1 lists the bands of Figure 5.6.4 with their free-space wavelengths and their typical users.

Table 5.6.1. The ITU radio bands, their free-space wavelengths $λ_{0} = c / f$ and typical services.

Band	$f$	$λ_{0}$	Typical services
VLF (4)	3–30 kHz	100–10 km	submarine links, navigation
LF (5)	30–300 kHz	10–1 km	time signals (DCF77 at 77.5 kHz), long-wave broadcasting
MF (6)	0.3–3 MHz	1 km–100 m	AM broadcasting (531–1602 kHz in Europe), maritime
HF (7)	3–30 MHz	100–10 m	short-wave broadcasting, amateur radio, RFID at 13.56 MHz
VHF (8)	30–300 MHz	10–1 m	FM broadcasting 87.5–108 MHz, air band, digital radio
UHF (9)	0.3–3 GHz	1 m–10 cm	TV, mobile phones, GPS (1575.42 MHz), Wi-Fi 2.4 GHz, ovens
SHF (10)	3–30 GHz	10–1 cm	5G at 3.5 GHz, Wi-Fi 5 GHz, satellite TV (10.7–12.75 GHz), radar
EHF (11)	30–300 GHz	10–1 mm	car radar 76–81 GHz, 60 GHz links, radio astronomy

Why does the band matter? Three reasons stand out.

Antenna size follows $λ$ . A resonant wire antenna is about half a wavelength long: the current on it is then one half of a standing wave, and the antenna looks like a resistor to the transmitter. An AM mast is therefore a tower, while a $77 GHz$ radar antenna fits on a chip (Figure 5.6.5).
Bandwidth grows with the carrier. An AM channel is $9 kHz$ wide, a Wi-Fi channel $20$ to $160 MHz$ .
Propagation changes with frequency: ground wave in the LF and MF bands, sky wave reflected by the ionosphere in the HF band (Section 6.2), line of sight from VHF upwards.

Wavelength and antenna size

λ_{0} = \frac{c}{f}, half-wave dipole \approx \frac{λ _{0}}{2} = \frac{c}{2 f} . (5.6.6)

Service	$f$	$λ_{0} /2$
AM radio	1 MHz	150 m
FM radio	100 MHz	1.5 m
GSM 900	900 MHz	16.7 cm
GPS L1	1575.42 MHz	9.5 cm
Wi-Fi, oven	2.45 GHz	6.1 cm
car radar	77 GHz	1.95 mm

The wavelength sets the antenna. The length \lambda_0/2 of a half-wave dipole falls from 150\,\mathrm{m} for an AM broadcast at 1\,\mathrm{MHz} to 1.95\,\mathrm{mm} for a car radar at 77\,\mathrm{GHz}, a factor of 77\,000. — **Figure 5.6.5.** The wavelength sets the antenna. The length $λ_{0} /2$ of a half-wave dipole falls from $150 m$ for an AM broadcast at $1 MHz$ to $1.95 mm$ for a car radar at $77 GHz$ , a factor of $77000$ .

Some bands are left free for unlicensed industrial, scientific and medical (ISM) use: $13.56 MHz$ , $27.12 MHz$ , $433.92 MHz$ , $915 MHz$ (in the Americas), $2.45 GHz$ , $5.8 GHz$ and $24.125 GHz$ . Microwave ovens, Wi-Fi and Bluetooth all share the band at $2.45 GHz$ (Application 5.6.2 and Worked example 5.6.2).

Quick check

How long is a half-wave dipole for a $433.92 MHz$ key fob or remote sensor?

$λ_{0} /2 = c /2 f = 3 \times 1 0^{8} / (2 \times 433.92 \times 1 0^{6}) = 34.6 cm$ . This is why such devices use shortened, coiled or printed antennas instead of a straight half-wave wire.

5.6.3 The electromagnetic spectrum