Editing Maxwell's equations (section)

== Vacuum equations, electromagnetic waves and speed of light ==
{{Further|Electromagnetic wave equation|Inhomogeneous electromagnetic wave equation|Sinusoidal plane-wave solutions of the electromagnetic wave equation|Helmholtz equation}}

[[File:Electromagneticwave3D.gif|thumb|This 3D diagram shows a plane linearly polarized wave propagating from left to right, defined by {{math|1='''E''' = '''E'''<sub>0</sub> sin(−''ωt'' + '''k''' ⋅ '''r''')}} and {{math|1='''B''' = '''B'''<sub>0</sub> sin(−''ωt'' + '''k''' ⋅ '''r''')}} The oscillating fields are detected at the flashing point. The horizontal wavelength is ''λ''. {{math|1='''E'''<sub>0</sub> ⋅ '''B'''<sub>0</sub> = 0 = '''E'''<sub>0</sub> ⋅ '''k''' = '''B'''<sub>0</sub> ⋅ '''k'''}}]]

In a region with no charges ({{math|1=''ρ'' = 0}}) and no currents ({{math|1='''J''' = '''0'''}}), such as in vacuum, Maxwell's equations reduce to:
<math display="block">\begin{align}
  \nabla \cdot \mathbf{E} &= 0, & \nabla \times \mathbf{E} + \frac{\partial\mathbf B}{\partial t} = 0, \\
  \nabla \cdot \mathbf{B} &= 0, & \nabla \times \mathbf{B} - \mu_0\varepsilon_0 \frac{\partial\mathbf E}{\partial t} = 0.
\end{align}</math>

Taking the curl {{math|(∇×)}} of the curl equations, and using the [[Vector calculus identities#Curl of curl|curl of the curl identity]] we obtain
<math display="block">\begin{align}
  \mu_0\varepsilon_0  \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} = 0, \\
  \mu_0\varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} = 0.
\end{align}</math>

The quantity <math>\mu_0\varepsilon_0</math> has the dimension (T/L)<sup>2</sup>. Defining <math>c = (\mu_0 \varepsilon_0)^{-1/2}</math>, the equations above have the form of the standard [[wave equation]]s
<math display="block">\begin{align}
  \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} = 0, \\
  \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} = 0.
\end{align}</math>

Already during Maxwell's lifetime, it was found that the known values for <math>\varepsilon_0</math> and <math>\mu_0</math> give <math>c \approx 2.998 \times 10^8~\text{m/s}</math>, then already known to be the [[speed of light]] in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the [[SI system|old SI system]] of units, the  values of <math>\mu_0 = 4\pi\times 10^{-7}</math> and <math>c = 299\,792\,458~\text{m/s}</math> are defined constants, (which means that by definition <math>\varepsilon_0 = 8.854\,187\,8... \times 10^{-12}~\text{F/m}</math>) that define the ampere and the metre. In the [[new SI]] system, only ''c'' keeps its defined value, and the electron charge gets a defined value.

In materials with [[relative permittivity]], {{math|''ε''<sub>r</sub>}}, and [[Permeability (electromagnetism)#Relative permeability and magnetic susceptibility|relative permeability]], {{math|''μ''<sub>r</sub>}}, the [[phase velocity]] of light becomes
<math display="block">v_\text{p} = \frac{1}\sqrt{\mu_0\mu_\text{r} \varepsilon_0\varepsilon_\text{r}},</math>
which is usually<ref group="note">There are cases ([[anomalous dispersion]]) where the phase velocity can exceed {{math|''c''}}, but the "signal velocity" will still be {{math|≤ ''c''}}</ref> less than {{math|''c''}}.

In addition, {{math|'''E'''}} and {{math|'''B'''}} are perpendicular to each other and to the direction of wave propagation, and are in [[phase (waves)|phase]] with each other. A [[sinusoidal]] plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through [[Faraday's law of induction|Faraday's law]]. In turn, that electric field creates a changing magnetic field through [[Ampère–Maxwell law|Maxwell's modification of Ampère's circuital law]]. This perpetual cycle allows these waves, now known as [[electromagnetic radiation]], to move through space at velocity {{math|''c''}}.