Supplement 1.3: The Speed of Electromagnetic Waves

The phase velocity of monochromatic waves

The speed of a monochromatic wave can be easily calculated. We consider the electric field E of a wave moving in the direction of the wave vector k :

E ( r ,t)= E o sin( k r ωt )

The speed of the wave is obtained by setting

E ( r ,t)=const. ,

i.e., looking at positions r at times t of a fixed value of the field. Since E o =const. in the case of plane waves, this is equivalent with a constant argument of the sine function:

k r ωt=const.

The velocity is calculated by differentiating | r |=r versus t:

dr dt = ω k

This is the phase velocity c of the wave. In case of electromagnetic waves this is the speed of light. With ω=2πf and k= 2π /λ it follows:

c= ω k =fλ

We can learn more about the dependence of the speed of light on electric and magnetic parameters by solving the wave equation

Δ E = ε o μ o 2 E t 2

with                                 E ( r ,t)= E o sin( k r ωt )

(using the equations for B leads to an identical result).

Equations

Second differentiation of E versus space and time leads to

k 2 E = ε o μ o ω 2 E

and hence:                             c o = ω k = 1 ε o μ o

This is the vacuum speed of light co, which is a fundamental constant in physics. With the vacuum permittivity εo=8.854·10-12 A·s/(V·m) and the vacuum permeability μo=1.256·10-6 V·s/(A·m) one obtains:

co=2.998·108 m/s

or 300 000 km/s, approximately.

The speed of light c in matter is smaller than the speed of light in vacuum:

c= ω k = 1 εμ = 1 ε r ε o μ r μ o ,

with the relative permittivity εr and permeability μr of the material. This equation is called Maxwell's relation. For transparent materials, it is μr≈1. For water and glass at frequencies of visible light, it is εr≈1.8 and 2.25, which explains their lower speed of light as indicated in chapter 1, section electromagnetic waves on page 2.

The refractive index n of matter is given by n= ε r μ r , and hence:

c= c o n

With this result the electromagnetic wave equations are:

Δ E = 1 c 2 2 E t 2            Δ B = 1 c 2 2 B t 2

The squared speed of light connects the second order spatial and temporal derivatives of the electric and the magnetic field quantities.