Amplitude of Electromagnetic Waves: UNIZOR.COM - Physics 4 Teens - Waves...

Notes to a video lecture on http://www.unizor.com

Electromagnetic Waves Amplitude

In the previous lecture Speed of Light of the current topic Electromagnetic Field Waves we discussed the relationship between electric permittivity of a medium ε (ε₀ for vacuum), its magnetic permeability μ (μ₀ for vacuum) and the speed of light in this medium (in vacuum speed of light is c and 1/c² equals to ε₀·μ₀).

In the same lecture we derived an expression for electric component of the simplest electromagnetic field oscillations in vacuum - monochromatic plane waves - as a function of time t and distance z from the source of oscillations along the Z-axis (direction of propagation of waves)
E(t,z) = E₀·sin(ω·(t−z/c))
where we assumed that the electric component E of an electromagnetic field regularly oscillates in the direction of X-axis, magnetic component B regularly oscillates in the Y-direction, electromagnetic waves propagate along Z-axis with speed c with angular frequency ω.

The expression for magnetic component B(t,z) of an electromagnetic field oscillations is similar and its derivation follows along exactly the same steps as the one above leading to the same formula:
B(t,z) = B₀·sin(ω·(t−z/c))

In this lecture we will derive a simple relationship between amplitudes of electric (E₀) and magnetic (B₀) amplitudes of the simplest electromagnetic field oscillations - monochromatic plane waves.

The third Maxwell equation that relates induced electric field to a changing magnetic field - the Faraday's Law - is

∇⨯E = −∂B/∂t

Recall the definition of a vector (cross) product of a pseudo-vector
∇={∂/∂x,∂/∂y,∂/∂z}
by any vector
V(x,y,z) = {V_x,V_y,V_z}
using unit vectors i, j and k along the coordinate axes:
∇⨯V =
= (∂V_z/∂y − ∂V_y/∂z)·i +
+ (∂V_x/∂z − ∂V_z/∂x)·j +
+ (∂V_y/∂x − ∂V_x/∂y)·k
(you can refresh this in the lecture "Curl in 3D" of this chapter of a course on UNIZOR.COM)

Considering a special characteristics of vector E during the simplest oscillations of electromagnetic field under consideration with only E_x(t,z)≠0,
∇⨯E =
= (∂E_z/∂y − ∂E_y/∂z)·i +
+ (∂E_x/∂z − ∂E_z/∂x)·j +
+ (∂E_y/∂x − ∂E_x/∂y)·k =
= (∂E_x/∂z)·j

Similarly, considering a special characteristics of vector B with only B_y(t,z)≠0,
−∂B/∂t = −(∂B_y/∂t)·j

Therefore, the third Maxwell equation in this case of a simple electromagnetic field takes the form
(∂E_x/∂z)·j = −(∂B_y/∂t)·j

Hence,
∂E_x/∂z = −∂B_y/∂t

Let's use the expressions for E(t,z) and B(t,z) as sinusoidal functions above (we dropped subscripts for brevity, as other components of the field electric and magnetic intensities are zero).
E(t,z) = E₀·sin(ω·(t−z/c))
B(t,z) = B₀·sin(ω·(t−z/c))

These functions should satisfy the above differential equation for the Faraday's Law.
∂E/∂z = −∂B/∂t

Therefore,
∂E/∂z=E₀·cos(ω·(t−z/c))·(−ω/c)
−∂B/∂t = −B₀·cos(ω·(t−z/c))·ω
from which follows the relationship between amplitudes of electric and magnetic components of these simple electromagnetic field oscillations

E₀ /c = B₀