In a homogeneous, isotropic medium with permeability \(\mu\) and permittivity \(\varepsilon\),
Maxwell’s equations imply a wave equation whose solutions propagate with speed
\[
c=\frac{1}{\sqrt{\mu\varepsilon}}.
\]
In vacuum, \(\mu=\mu_0\) and \(\varepsilon=\varepsilon_0\), giving the familiar \(c_0\approx 3\times 10^8\ \mathrm{m/s}\).
Frequency and wavelength
For a monochromatic wave,
\[
\lambda=\frac{c}{f},\qquad \omega=2\pi f,\qquad k=\frac{2\pi}{\lambda}.
\]
Plane-wave field ratio
For a plane wave, the electric and magnetic field amplitudes satisfy
\[
\frac{E_0}{B_0}=c,
\qquad\text{equivalently}\qquad
B_0=\frac{E_0}{c}.
\]
Refractive index (university extension)
Writing \(\varepsilon=\varepsilon_0\varepsilon_r\) and \(\mu=\mu_0\mu_r\), the refractive index becomes
\[
n=\frac{c_0}{c}=\sqrt{\varepsilon_r\mu_r}.
\]
For many dielectrics \(\mu_r\approx 1\), so \(n\approx \sqrt{\varepsilon_r}\).
Energy density
The instantaneous electromagnetic energy density is
\[
u=\frac{\varepsilon E^2}{2}+\frac{B^2}{2\mu}.
\]
For a sinusoidal plane wave, the time-average total energy density is
\[
\langle u\rangle=\frac{\varepsilon E_0^2}{2}
\]
(with equal average electric and magnetic contributions).
Polarization
Polarization describes how the transverse electric field vector evolves in time at a fixed point:
linear (fixed direction), circular (constant magnitude rotating), and elliptical (general rotating ellipse).
Website tip: When \(\lambda\) is between about \(380\text{ nm}\) and \(750\text{ nm}\), it falls in the visible spectrum (approximate color labeling).
