Electromagnetic Waves

Electromagnetic Waves

Maxwell’s equations: wave equation

Light is, according to classical theory, the flow of electromagnetic (EM) radiation
through free space or through a medium in the form of electric and magnetic fields.
Although electromagnetic radiation covers an extremely wide range, from gamma rays
to long radio waves, the term “light” is restricted to the part of the electromagnetic
spectrum that goes from the vacuum ultraviolet to the far infrared. This part of the
spectrum is also called optical range. EM radiation propagates in the form of two
mutually perpendicular and coupled vectorial waves: the electric field E(r, t) and the
magnetic field H(r, t). These two vectorial magnitudes depend on the position (r)
and time (t ). Therefore, in order to properly describe light propagation in a medium,
whether vacuum or a material, it is necessary in general to know six scalar functions,
with their dependence of the position and the time. Fortunately, these functions are not
completely independent, because they must satisfy a set of coupled equations, known
as Maxwell’s equations.
Maxwell’s equations form a set of four coupled equations involving the electric
field vector and the magnetic field vector of the light, and are based on experimental
evidence. Two of them are scalar equations, and the other two are vectorial. In their
differential form, Maxwell’s equations for light propagating in free space are:
∇E = 0 (2.1)
∇H = 0 (2.2)
∇ ×E = −μ0
∂H
∂t
(2.3)
∇ ×H = ε0
∂E
∂t
(2.4)
where the constants ε0 = 8.85 × 10−12 m−3 kg−1 s4 A2 and μ0 = 4π × 10−7 mkgs−2
A−2 represent the dielectric permittivity and the magnetic permeability of free space
respectively, and the ∇ and ∇x denote the divergence and curl operators, respectively.
For the description of the electromagnetic field in a material medium it is necessary
to define two additional vectorial magnitudes: the electric displacement vector D(r, t)
and the magnetic flux density vector B(r, t). Maxwell’s equations in a material medium,
involving these two magnitudes and the electric and magnetic fields, are expressed as:


∇D = ρ (2.5)
∇B = 0



∇ ×E = −∂B
∂t
(2.7)
∇ ×H = J + ∂D
∂t
(2.8)
where ρ(r, t) and J(r, t) denote the charge density and the current density vector
respectively. If in the medium there are no free electric charges, which is the most
common situation in optics, Maxwell’s equations simplify in the form:
∇D = 0 (2.9)
∇B = 0 (2.10)
∇ ×E = −∂B
∂t
(2.11)
∇ ×H = J + ∂D
∂t