Electromagnetic (EM) waves are whizzing around you at all times, and their study represents an entire crucial area of physics. Understanding, classifying and describing the various forms of electromagnetic radiation has helped NASA and other scientific entities push human technology into and beyond previously unexplored territory, often in dramatic ways. Yet only a tiny fraction of EM waves are visible to the human eye.

In physics, a certain amount of math is inevitable. But the nice thing in the physical sciences is that the math tends to be logically "neat" – that is, once you're familiar with the basic equations of classical mechanics (i.e., usually big, visible stuff moving around), the equations of electromagnetism look familiar, just with different variables.

To best understand electromagnetic fields and waves, you should have a basic knowledge of Maxwell's equations, derived by James Clerk Maxwell in the second half of the 1800s. These equations, from which the general solution for EM waves is derived, describe the relationship between electricity and magnetism. By the end, you should also understand what it means to "be" a wave – how ​these​ particular waves are a little different.

Maxwell's equations formalize the relationship between electricity and magnetism and describe all such phenomena. Building on the work of physicists such as Carl Gauss, Michael Faraday and Charles-Augustin de Coulomb, Maxwell discovered that the equations produced by these scientists relating electric and magnetic fields were fundamentally sound, but imperfect.

If you're unfamiliar with calculus, don't be discouraged. You can follow along quite nicely without solving a thing. Just remember that integration is nothing more than a clever form of finding the area under a curve in a graph by adding up incredibly tiny slices of that curve. Also, while the variables and terms may not mean much at first, you'll refer back to them repeatedly throughout the article as the "lights" continue to brighten for you on this vital topic.

​Maxwell's first equation​ is derived from ​Gauss' law​ for electric fields, which states that the net electric flux through a closed surface (such as the outside of a sphere) is proportional to the charge inside:

Here, the upside-down triangle ("nabla" or "del") represents a three-dimensional gradient operator, ​ρ​ is the charge density per unit volume and ​ε​₀ is the electrical ​permittivity of free space​.

​Maxwell's second equation​ is Gauss' law for magnetism, in which, unlike the case with electric fields, there is no such thing as a "point magnetic charge," or a ​magnetic monopole​. Instead, magnetic field lines appear as closed loops. The net magnetic flux through a closed surface will always be 0, which results directly from magnetic fields being dipolar.

The law states in effect that every line from a magnetic field ​B​ entering a chosen volume in space must exit that volume at some point, and that is the next magnetic flux through the surface is therefore zero.

​Maxwell's third equation​ (Faraday's law of magnetic induction) describes how an electric field is created by a changing magnetic field. The funny "∂" means "partial derivative" and implies fluctuation. Odd symbols aside, the relationship shows that a change in electric flux both results from and obligates a ​non-constant​ magnetic field.

​Maxwell's fourth equation​ (the Ampere-Maxwell law) is the wellspring for the others, for Maxwell's correction to Ampere's failure to account for non-steady currents rippled through the other three equations with correction factors of their own. The equation is derived from Ampere's law and describes how a magnetic field is generated by a current (moving charge), a changing magnetic field or both.

Here, ​μ​₀ is the permeability of free space. The equation shows how the magnetic field inside a given area around the current in a wire ​J​ changes with that current and with the electric field ​E​.

Once Maxwell had formalized his understanding of electricity and magnetism with his equations, he looked for various solutions to the equations that might describe new phenomena.

Since a changing electric field generates a magnetic field and a changing magnetic field generates an electric field, Maxwell determined that a self-propagating electromagnetic wave could be generated. Using his equations, he determined that the speed of such a wave would have a speed equal to the speed of light. This turned out to be no coincidence, and led to the discovery that light is a form of electromagnetic radiation!

In general, waves are oscillations in a medium that transfer energy from one place to another. Waves have a wavelength, period and frequency associated with them. The speed ​v​ of a wave is its wavelength ​λ​ times its frequency ​f​, or λf = v.

The SI unit of wavelength is the meter, though nanometers are more frequently encountered because these are more convenient for the visible spectrum. Frequency is measured in cycles per second (s^-1) or ​hertz​ (Hz), after Heinrich Hertz. The period ​T​ of a wave is how long it takes to complete one cycle, or 1/f.

For the case of an EM wave, unlike the situation with mechanical waves, ​v​ is constant in all situations, which means that ​λ​ varies ​inversely​ with ​f​. That is, higher frequencies imply shorter wavelengths for a given ​v​. "High frequency" also implies "high-energy"; that is, electromagnetic energy ​E​ in joules (J) is proportional to ​f​, via a factor called Planck's constant ​h​ (= 6.62607 × 10^-34 J).

• The equation for a wave is ​y = A sin(kx − ωt)​, where ​A​ is amplitude, ​x​ is the displacement along the x-axis, ​k​ is the wave number 2π/k, and

​ω​

is the angular frequency 2π/T. 

An electromagnetic wave consists of an electric field (​E​) wave oscillating in a plane perpendicular (at right angles) to a magnetic field (​B​) wave. If you imagine yourself as an EM wave waling ("propagating") across a level floor, the ​E​ wave component oscillates in a vertical plane through your body and the ​B​ wave oscillates within the horizontal floor.

Since electromagnetic radiation acts as a wave, then any particular electromagnetic wave will have a frequency and wavelength associated with it. Another constraint is that, since the speed of electromagnetic waves is fixed at c = 3 × 10⁸ m/s, the speed at which light travels in a vacuum (also used for the speed of light in air for close approximations). Lower frequency is therefore associated with longer wavelengths and vice versa.

EM waves do not require a medium such as water or gas through which to propagate; hence, they can traverse the vacuum of empty space itself at the fastest speed in the entire universe!

Electromagnetic waves are produced across an enormous range of frequencies and wavelengths. Starting with low frequency (lower energy) and thus longer wavelength, the various types of EM radiation are:

• ​Radio waves​ (about 1 m and longer): Radiofrequency EM radiation spans about 20,000 to 300 billion Hz. These "fly" not only around the world but deep into space, and their harnessing by Marconi at the turn of the 20th century revolutionized the world of human communication.
• ​Microwaves​ (about 1 mm to 1 m): These can also penetrate into space, but they are useful in weather applications because they can also penetrate clouds. 
• ​Infrared waves​ (700 nm to 1 mm): Infrared radiation, or "infrared light," is the stuff of "night-vision" goggles and other visual-enhancement equipment.
• ​Visible light​ (400 nm to 700 nm): Light waves in the visible spectrum span a tiny fraction of the electromagnetic wave frequency and wavelength range. Your eyes, after all, are the fairly conservative product of what nature needs them to collect for everyday survival.
• ​Ultraviolet light​ (10 nm to 400 nm): Ultraviolet radiation is what causes sunburn and probably skin malignancies as well. Nevertheless, tanning beds wouldn't exist without it.
• ​X-rays​ (about 0.01 nm to 10 nm): This higher-energy radiation is an incredible diagnostic aid in medicine, but this must be balanced against their potential to cause physical harm themselves in higher exposures.
• ​Gamma rays​ (< 0.01 nm): As you'd expect, this is very high-energy and hence potentially lethal radiation. Were it not for the Earth's atmosphere blocking most of it, life in its current form would not have been able to get going billions of years ago. They are used to treat especially aggressive tumors.

Because electromagnetic radiation has both the properties of a wave, and will act like a wave when measured as such but also acts like a particle (called a ​photon​) when measured as such, we say that it has particle-wave duality.

A steady current produces a steady magnetic field, while a changing current induces a changing magnetic field. If the change is steady and cyclical, the waves (and associated fields) are said to oscillate, or "wiggle" rapidly to and fro in a plane.

The same essential principle works in reverse: An oscillating magnetic field induces an oscillating electric field.

Electromagnetic waves result from this interplay between electric and magnetic fields. If a charge moves back and forth along a wire, it creates a changing electric field, which in turn creates a changing magnetic field, which then self-propagates as an EM wave, capable of emitting photons. This is an instance of two transverse waves (and fields) intersecting each other to form another transverse wave.

• Atoms and molecules can absorb and emit specific frequencies of electromagnetic radiation consistent with their associated quantized energy levels.

People often confuse these two types of waves simply because they're so familiar with listening to the radio. But radio waves are, as you now know, a form of electromagnetic radiation. They travel at the speed of light and transmit information from the radio station to your radio. However, that information is then converted into the motion of a speaker, which produces sound waves, which are ​longitudinal​ waves in the air (like those in a pond after it's been disturbed by a thrown rock).

• Sound waves travel at approximately 343 m/s in air, which is much slower than radio waves, and they require a medium through which to travel.

A phenomenon called the Doppler frequency shift in EM radiation allows astrophysicists to tell whether objects in space are moving toward us or away from us, because a stationary object emitting EM waves will show a different pattern than one that is moving, relative to a fixed observer.

A technique called spectroscopy allows chemists to determine the composition of gases. Earth's atmosphere shields the biosphere from the most harmful ultraviolet radiation and other higher-energy radiation such as gamma rays. Microwave ovens for cooking food have allowed college students to prepare meals in their dorms. Cell phone and GPS signals are a relatively recent yet already critical addition to the list of technologies reliant on EM energy.