Energy Flux Density
As we have established earlier (lectures Electric Field Energy and Magnetic Field Energy of topic Energy of Waves of the current part Waves of the course), the total electric + magnetic energy density of electromagnetic energy (that is, an amount of energy per unit of volume) is
PE+M = ½·[ε·E²+(1/μ)·B²]
where E(t,x,y,z) is an intensity of electric component and B(t,x,y,z) is a magnetic component's intensity of the field.
In the previous lecture Energy Continuity of the current topic Electromagnetic Field Waves we discussed a concept of electromagnetic field energy flux density vector, which, based on a concept of energy continuity, we have identified as Poynting vector
S = (1/μ)·(E⨯B)
Using the Poynting vector, we have represented the rate of change of electromagnetic energy flux as
−∂PE+M /∂t = ∇·S
In this form this expression consttutes the continuity equation of the flux density rate of change of electro-magnetic energy.
Let's examine the correspondents of both expressions, PE+M and S, in a simple case of flat sinusoidal monochromatic electromagnetic waves.
In the lecture E-M Waves Amplitude of the current topic we came up with a simple relationship between electric and magnetic components of the plane electromagnetic waves in vacuum.
If a sinusoidal in time (t) electromagnetic field propagates along Z-axis with speed c according to these equations
E(t,z) = E0·sin(ω·(t−z/c))
B(t,z) = B0·sin(ω·(t−z/c))
where electrical component E oscillates along X-axis and magnetic component B oscillates along Y-axis, then
E0 /c = B0
E(t,z)/c = B(t,z)
It's quite appropriate now to check if our formula for Poynting vector as an energy flux density checks in this simple case.
Replacing B² with E²/c² in the expression for PE+M and taking into consideration that the speed of light c in terms of electrical permittivity ε and magnetic permeability μ is expressed as
PE+M = ½·[ε·E²+(1/μ)·E²/c²] =
= ½·[ε·E²+(1/μ)·E²·ε·μ] =
Consider a unit area of 1m² and light flowing perpendicularly through it with speed c.
During the unit time interval of 1s the light going through this area will fill the volume
1(m²)·c(m/s)·1(s) = c(m³)
So, the amount of electromagnetic energy flowing through a unit area during a unit of time (that is, energy flux density j) is this volume times the calculated above density of the energy ε·E²
j = c·ε·E²
The above is the magnitude of an electromagnetic energy flux density vector directed along Z-axis.
In our special case vectors E (oscillating along X-axis) and B (oscillating along Y-axis) are perpendicular to each other.
Therefore, the magnitude of their vector product equals to a product of their magnitudes and the magnitude of Poynting vector is
|S| = (1/μ)·|E|·|B| =
= (1/μ)·E·B = (1/μ)·E·E/c =
Since c²=1/(ε·μ) and 1/μ=c²·ε,
|S| = c·ε·E²
which is exactly the same as j above calculated based on energy density.
The direction of Poynting vector is perpendicular to both electrical and magnetic components and, therefore, is along Z-axis.
As we see, Poynting vector in this special case fully corresponds in magnitude and direction to the energy flux density obtained by direct calculation based on energy density and speed of light.
This confirms (at least, in this simple case) that Poynting vector correctly represents the electromagnetic energy flux density.