What Microwaves, X-Rays and Rainbows Have in Common

Sound needs air and water waves need water, yet sunlight crosses empty space to reach us. Light is an electromagnetic wave — changing electric and magnetic fields that keep each other going — and so are radio signals, microwaves and X-rays.

This part covers displacement current, the properties of electromagnetic waves, and the electromagnetic spectrum.

**Displacement current, $i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$, is the effective current produced by a changing electric field; adding it to the conduction current makes Ampere's law consistent everywhere, including in the gap of a charging capacitor.

The problem.** While a capacitor charges, current flows in the wires but no charge crosses the gap. Ampere's law for a surface cut by the wire gives $\oint \vec{B}\cdot d\vec{l}={\mu }_{0}i$, but for a surface passing through the gap it gives zero — a contradiction.

The resolution. Between the plates the electric flux ${\Phi }_E$ is increasing. Including ${\epsilon }_0\frac{d{\Phi }_E}{dt}$ as a current:

$$\oint \vec{B}\cdot d\vec{l}={\mu }_0\left(i_c+{\epsilon }_0\frac{d{\Phi }_E}{dt}\right)$$

Since ${\Phi }_E=\frac{q}{{\epsilon }_0}$ between the plates, the displacement current in the gap equals the conduction current $\frac{dq}{dt}$ in the wires.

Worked example. A capacitor is charged by a steady current of $0.15$ A. The displacement current between its plates is also $0.15$ A, so

$$\frac{d{\Phi }_E}{dt}=\frac{i_d}{{\epsilon }_0}=\frac{0.15}{8.85\times 10^{-12}}\approx 1.7\times 10^{10}{\text{ V m s}}^{-1}$$

An everyday example. A radio transmitter's antenna works because its rapidly changing electric fields create magnetic fields — the effect that displacement current describes.

The substance. A changing electric field produces a magnetic field, just as a changing magnetic field produces an electric field — together, these make electromagnetic waves possible.

**Electromagnetic waves consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel, so they are transverse; they need no medium, travel in vacuum at $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\approx 3\times 10^8$ m s${}^{-1}$, and carry energy and momentum.

Characteristics:

- Produced by accelerated charges**
- $\vec{E}$, $\vec{B}$ and the direction of travel are mutually perpendicular
- $\vec{E}$ and $\vec{B}$ oscillate in phase, with $\frac{E_0}{B_0}=c$
- No medium is needed; in a medium the speed is $\frac{c}{n}$
- They carry energy and momentum and exert radiation pressure

Worked example 1. Checking the speed:

$$c=\frac{1}{\sqrt{4\pi \times 10^{-7}\times 8.85\times 10^{-12}}}\approx 3.0\times 10^8{\text{ m s}}^{-1}$$

Worked example 2. A wave with peak electric field $60$ V m${}^{-1}$ has peak magnetic field

$$B_0=\frac{E_0}{c}=\frac{60}{3\times 10^8}=2\times 10^{-7}\text{ T}$$

An everyday example. Polarised sunglasses cut the glare from a wet road by blocking light whose electric field vibrates in one direction — something only transverse waves allow.

The substance. All electromagnetic waves travel at the same speed in vacuum; they differ only in frequency and wavelength.

In order of decreasing wavelength and increasing frequency, the spectrum runs from radio waves through microwaves, infrared, visible light and ultraviolet to X-rays and gamma rays, and each region has characteristic uses.

The regions:

- Radio waves — longer than about $0.1$ m; produced by oscillating currents in antennas; used for radio, TV and mobile communication
- Microwaves — about $0.1$ m to $1$ mm; used in radar, satellite links and microwave ovens
- Infrared — about $1$ mm to $700$ nm; given off by warm bodies; used in TV remote controls and night-vision devices
- Visible light — about $700$ nm to $400$ nm; the band our eyes detect, from red to violet
- Ultraviolet — about $400$ nm to $1$ nm; used in water purifiers and sterilisation; mostly absorbed by the ozone layer
- X-rays — about $1$ nm to $10^{-3}$ nm; used in medical imaging
- Gamma rays — shorter than about $10^{-3}$ nm; from nuclear processes; used in cancer treatment

Worked example. An FM radio station broadcasts at $100$ MHz:

$$\lambda =\frac{c}{f}=\frac{3\times 10^8}{100\times 10^6}=3\text{ m}$$

A microwave oven working at about $2.45$ GHz uses $\lambda =\frac{3\times 10^8}{2.45\times 10^9}\approx 12$ cm.

An everyday example. Your TV remote uses infrared, your phone uses radio waves, and a hospital machine uses X-rays — all the same kind of wave at different frequencies.

The substance. Higher frequency means more energy per photon, which is why ultraviolet, X-rays and gamma rays can damage living cells while radio waves cannot.