Supplement 1.2: Solving Maxwell's Equations for Electromagnetic Waves    (1/3)

The wave equations

In supplement 1.1 Maxwell's equations were written as follows:

∇⋅ E → = ρ ε            ∇⋅ B → =0
∇× E → =− ∂ B → ∂t            ∇× B → =μ j → +εμ ∂ E → ∂t

In vacuum there are no electric charges and currents, and hence no charge densities ρ and current densities j → , and the relative electric and magnetic matter properties εr and μr are unity:

ρ=0            j → =0            εr=μr=1

Then Maxwell's equations reduce to the very symmetric form:

∇⋅ E → =0            ∇⋅ B → =0
∇× E → =− ∂ B → ∂t            ∇× B → = ε o μ o ∂ E → ∂t

Taking the curl of the third equation and exchanging the time derivate and ∇ operator (which is allowed with continuous fields),

∇×( ∇× E → )=−∇× ∂ B → ∂t =− ∂ ∂t ( ∇× B → )

and substituting in the fourth equation yields:

∇×( ∇× E → )=− ε o μ o ∂ 2 E → ∂ t 2
Equations ↓

As shown in vector analysis, the following relation holds:

∇×( ∇× E → )=∇( ∇⋅ E → )−( ∇⋅∇ ) E →

With Maxwell's first equation, ∇⋅ E → =0 , and introducing the Laplace operator (or: delta operator) Δ=∇⋅∇ which is the second order spatial derivative, it follows for the electric field:

Δ E → = ε o μ o ∂ 2 E → ∂ t 2

In the same way, taking the curl of the fourth equation, and then substituting it into the third and considering the second one, it follows an analoguous equation for the magnetic field:

Δ B → = ε o μ o ∂ 2 B → ∂ t 2

Both equations combine the second spatial and temporal derivatives of the electric and magnetic field, respectively. They describe the properties of these field quantities in vacuum.

In other words, electric and magnetic fields subsist in the absence of electric charges and currents if their spatial and temporal behaviour corresponds to these equations. We will see below that this holds with the fields that characterise electromagnetic waves. Therefore, both equations are called wave equations.

Question 1: The Laplace operator ↓