Electromagnetic Waves

Maxwell’s four equations of electromagnetism (Gauss’s law for E, Gauss’s law for B, Faraday’s law, and Ampère’s law corrected with Maxwell’s displacement current) together predict something no single equation does alone: a changing electric field creates a magnetic field, and a changing magnetic field creates an electric field, so the two can sustain each other and propagate through empty space with no charges or currents needed at all.

This self-sustaining disturbance is an electromagnetic wave. Maxwell’s equations fix its structure completely: the electric field $E$, the magnetic field $B$, and the direction of travel are all mutually perpendicular, and the fields oscillate in phase with each other:

The ratio of their peak magnitudes is fixed by the wave speed itself:

The speed itself falls directly out of the equations, in terms of only two constants that had already been measured independently from static charges and currents, the permittivity $\varepsilon_0$ and permeability $\mu_0$ of free space:

Like any periodic wave, this speed is also the product of frequency and wavelength, and because $c$ is a fixed constant, this relationship holds for every electromagnetic wave that exists, regardless of its frequency:

It is tempting to think of “radio waves,” “light,” and “gamma rays” as different things, because we meet them in completely different contexts: you tune a radio, you see by light, and gamma rays show up in nuclear physics and cancer treatment. The exam-relevant truth is sharper than that: they are the same physical wave, governed by the same Maxwell’s equations, travelling at the same speed $c$, differing from each other in exactly one number, frequency (equivalently, wavelength).

Push that number across its full range and the scale involved is hard to overstate. A long-wave radio station can broadcast at a wavelength of several kilometres. A gamma ray’s wavelength can be smaller than an atomic nucleus, roughly $10^{-15}$ m. That is a span of more than twenty orders of magnitude in wavelength alone, and yet $f\lambda$ is exactly $c$ at every single point along that range, not approximately, not “for most of the spectrum,” but identically, because $c$ is fixed by $\varepsilon_0$ and $\mu_0$ alone and has nothing to do with how the wave was generated.

The exam trap this produces: students memorise typical wavelengths or frequencies for each band and then try to use them as if they were independent facts to recall, rather than simply computing whichever one is missing from $\lambda = c/f$. If a question gives you a frequency, you don’t need to remember which band it’s in, compute $\lambda$ and then check it against the band boundaries. The boundaries themselves are approximate and a little fuzzy in nature; the relationship $c=f\lambda$ is not.