information:extraction
Vacuum equations, electromagnetic waves and speed of light

In a region with no charges (ρ = 0) and no currents (J = 0), such as in vacuum, Maxwell's equations reduce to: ∇ ⋅ E = 0, ∇ × E + ∂ B ∂ t = 0, ∇ ⋅ B = 0, ∇ × B − μ 0 ε 0 ∂ E ∂ t = 0. {\displaystyle {\begin{aligned}\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{aligned}}}

Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain μ 0 ε 0 ∂ 2 E ∂ t 2 − ∇ 2 E = 0, μ 0 ε 0 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}\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{aligned}}}

The quantity μ 0 ε 0 {\displaystyle \mu _{0}\varepsilon _{0}} has the dimension (T/L). Defining c = (μ 0 ε 0) − 1 / 2 {\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}}, the equations above have the form of the standard wave equations 1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0, 1 c 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}{\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{aligned}}}

Already during Maxwell's lifetime, it was found that the known values for ε 0 {\displaystyle \varepsilon _{0}} and μ 0 {\displaystyle \mu _{0}} give c ≈ 2.998 × 10 8 m/s {\displaystyle c\approx 2.998\times 10^{8}~{\text{m/s}}}, 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 old SI system of units, the values of μ 0 = 4 π × 10 − 7 {\displaystyle \mu _{0}=4\pi \times 10^{-7}} and c = 299 792 458 m/s {\displaystyle c=299\,792\,458~{\text{m/s}}} are defined constants, (which means that by definition ε 0 = 8.854 187 8... × 10 − 12 F/m {\displaystyle \varepsilon _{0}=8.854\,187\,8...\times 10^{-12}~{\text{F/m}}}) 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, ε, and relative permeability, μ, the phase velocity of light becomes v p = 1 μ 0 μ r ε 0 ε r, {\displaystyle v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}},} which is usually less than c.

In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in 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. In turn, that electric field creates a changing magnetic field through 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 c.