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

Plane monochromatic waves      cont.

Waves propagating in arbitrary directions a → can be obtained by converting the wave number k into a vector with the direction of the propagating wave, k a → . This is the wave vector k → , with | k → |= k=2π /λ .

The electric and magnetic field of waves propagating in a direction given by the orientation of k →  is then:

E → ( r → ,t)= E → o sin( k → ⋅ r → −ωt )          B → ( r → ,t)= B → o sin( k → ⋅ r → −ωt )

Which relation exists between E → and B → ? They are connected together in the third and fourth Maxwell equation. E.g., the third equation reads:

∇× E → =− ∂ B → ∂t

We choose an electromagnetic wave propagating in direction x. Since the field vectors are orthogonal to x, they reduce in Cartesian coordinates to:

E → =( 0, E y , E z )            B → =( 0, B y , B z )

With these vectors, the third Maxwell equation becomes:

y-component:            ∂ E z ∂x = ∂ B y ∂t
z-component:            ∂ E y ∂x =− ∂ B z ∂t

(the x-components vanishes since Bx=0).

Question 2: Third Maxwell equation in Cartesian components ↓

To solve the y-component, we choose a sinusoidal electric field Ez:

E z = E z,o sin( kx−ωt )

With the partial derivative with respect to x, ∂ E z ∂x , one obtains for the y-component of the magnetic field:

B y =∫ ∂ E z ∂x dt =− k ω E z

In the same way, solving the z-component of the Maxwell equation yields:

B z = k ω E y

Both component equations can be combined into a vector equation:

B → = k ω a → × E →

where a → is again a unit vector pointing in direction of the wave propagation. The relations prove that

  • E → and B → and the direction of propagation of the wave are all orthogonal (what we found already above), and
  • E → and B → have in every point identical phase (e.g., zero-crossings, maxima...), as shown in the graph in chapter 1, section electromagnetic waves.