Electromagnetic Energy Continuity: UNIZOR.COM - Physics4Teens - Waves - ...

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

Electromagnetic Energy Continuity

Let us recall the definition of the divergence of a time-dependent vector field V(x,y,z,t) in three-dimensional Cartesian space (X,Y,Z), with three components V_x(x,y,z,t), V_y(x,y,z,t) and V_z(x,y,z,t) along each space coordinate presented in the lecture Divergence of the topic Electromagnetic Field Waves of the Waves part of this course:

divV(x,y,z,t) = ∇ · V(x,y,z,t) =
= ∂V_x(x,y,z,t)/∂x +
+ ∂V_y(x,y,z,t)/∂y +
+ ∂V_z(x,y,z,t)/∂z

In particular, in that lecture we used an example of magnitude and direction of air blown by the wind in a unit of time, a time-dependent vector field V(x,y,z,t), and its relationship with air density, a time-dependent scalar field ρ(x,y,z,t).

The final conclusion we came up with was an equation that connects the divergence of the above vector field at any point and time with the rate of change of the air density there as follows
∇ · V(x,y,z,t) = −∂ρ(x,y,z,t)/∂t


Constant Flux Density

Consider a trivial example.
If oil of density 800 kg/m³ uniformly moves along a pipeline with speed 6 m/s and the pipe cross section is 0.2 m², the rate of oil flow is
q = 800 kg/m³ · 6 m/s · 0.2 m²

In this case the flux density j, by definition, is the rate of oil flow per unit of time per unit of area of a pipe cross section, which equals to
j = 800 kg/m³ · 6 m/s =
= 4,800 (kg/s)/m²
For an area A = 0.2 m² of a pipe cross section the total rate of flow of oil through a pipe in a unit of time is q = j·A:
q = 4,800 (kg/s)/m² · 0.2 m² =
= 960 kg/s


Flux Density

For any such "flowing" substance at any point inside this substance we define flux density vector field (usually denoted j) as an amount of this substance "flowing" through a unit of area perpendicular to its direction during a unit of time or, in other words, the rate of change of the density of this substance at a given point.

Then, given some surface and knowing the value of flux density at each point of this surface, we can evaluate the quantity of a substance "flowing" through this surface per unit of time.

In the oil flow of the example above the flux density vector at any point inside a pipe at any moment in time is a vector directed along a pipe of a magnitude j=4,800(kg/s)/m² independent of space position inside a pipe and time.
Assuming the pipe is stretched along X-axis, this flux density vector is
j(x,y,z,t) = {j,0,0}
and its divergence is
∇·j(x,y,z,t) = 0
because all components of the flux density vector are constants.
Therefore, according to the formula about relationship between divergence and density
the derivative of oil density by time is zero, which means that the density of oil is constant.

The above examples of an air flow because of a wind or oil flow in a pipe can be generalized to other measurable substances that can "flow" - number of molecules, mass, electric charge, momentum, energy, etc.

Assume, ρ(x,y,z,t) represents a time-dependent density in space of some substance.
Further assume that a time-dependent vector field j(x,y,z,t) represent the flux density of this substance, that is an amount of substance going through a unit of area perpendicular to the flux vector in a unit of time.

Then an equation similar to the above air flow equation
∇ · j(x,y,z,t) = −∂ρ(x,y,z,t)/∂t
constitutes the continuity equation for this substance.


Electric Charge Flux Density

Let's examine the flow of electricity in terms of the flux density, which somewhat similar to an air wind.

Assume, we have certain time-dependent continuous distribution of electric charge in space with charge density ρ(x,y,z,t).

The amount of electricity going through some area in a unit of time is an electric current
I = dq/dt
Therefore, the electric current density (electric current per unit of area) is
J = dI/dA
where A is a perpendicular to electric current unit area, through which the current I is flowing.

The electric current density is an example of a flux density applied to electricity. Therefore, according to the same divergence theorem, the following continuity equation holds:
∇ · J(x,y,z,t) = −∂ρ(x,y,z,t)/∂t

The equation above represents a local law of conservation of electric charge.
The charge does not appear from nowhere, neither it disappear without a trace, but gradually increases or decreases with electric current density delivering additional charge into an area or taking charge from an area.

This law is stronger than a statement that the total amount of charge in a closed space remains constant because the latter does not exclude instantaneous transfer of charge from one point to another within a closed space without any trace in between.

As such, the equation above represents a continuity of electric charge.


Electromagnetic
Field Flux Density

The next step after analyzing an electric current flux density and its relation to electric charge continuity and local law of conservation is to discuss electromagnetic field energy flux density and corresponding continuity and local conservation law.

The first difficulty is to realize that electromagnetic field is not similar to air or electric charge - those are real material substances, while electromagnetic field is not "material" in the same sense as air molecules or electrons.

Nevertheless, electromagnetic field carries energy (see lectures Electric Field Energy and Magnetic Field Energy in the topic Energy of Waves of the Waves part of this course), electromagnetic field energy is transferred in space with some speed (with speed of light in vacuum), so it must have a flux.

So, our task is to express the electromagnetic field energy flux density vector in terms of characteristics that define this field - time-dependent vector fields E(x,y,z,t) of electric intensity and B(x,y,z,t) of magnetic intensity.

In the above mentioned lectures we have derived two formulas for electric P_E and magnetic P_B energy densities of electromagnetic field with electric intensity E and magnetic intensity B:
P_E = ½ε·E²
P_M = ½(1/μ)·B²
where ε is electric permittivity and μ is magnetic permeability of the medium where electromagnetic field exists (correspondingly, ε₀ and μ₀ for vacuum).

The total energy density of electromagnetic field is, therefore,
P_E+M = ½·[ε·E²+(1/μ)·B²]
Strictly speaking, we have to use vectors E and B instead of scalars E and B. Then the energy density would be
P_E+M =
= ½·[ε·(E·E)+(1/μ)·(B·B)]

Here are the steps that we will follow to accomplish our task of expressing the energy flux density vector field S(x,y,z,t) in terms of vector fields E(x,y,z,t) of electric intensity and B(x,y,z,t) of magnetic intensity.
.

The continuity equation relates the flux density with the rate of change of the density of whatever substance we talk about.
In our case we are analyzing the electromagnetic energy as such a substance and know how its density expressed in terms of intensities of electromagnetic field E and B.

Using the Maxwell equations, we can express the rate of E and B change (that is, their time derivative) in terms of divergence of B and E (see the Faraday Law as the equation #3 and Amper-Maxwell Law as the equation #4).

Finally, comparing the expression for the rate of change of energy density in term of divergence of E and B with an expression in terms of divergence of energy flux density vector, we will express the energy flux density vector in terms of vectors E and B.

To determine the rate of electromagnetic field energy density change, we have to differentiate the above expression for the energy density in terms of E and B by time:
∂P_E+M /∂t =
= ε·(E·∂E/∂t)+(1/μ)·(B·∂B/∂t)

Let's use the Faraday's Law (Maxwell equation #3 in this course) to express ∂B/∂t in terms of E
∇⨯E = −∂B/∂t

Let's use the Amper-Maxwell Law (Maxwell equation #4 in this course) to express ∂E/∂t in terms of B, assuming a simple case of the vacuum (no electric current within a field)
∇⨯B = μ·ε·∂E/∂t

Now the rate of change of the energy density looks like
∂P_E+M /∂t =
= ε·(E·(∇⨯B))/(μ·ε) −
− (1/μ)·(B·(∇⨯E)) =
= (1/μ)·[E·(∇⨯B)−B·(∇⨯E)]

At this point we can use a vector identity
Q·(∇⨯P)−P·(∇⨯Q) = ∇·(P⨯Q)
(see the proof of this identity in Problem 3 of Vector Field Identities notes from topic Electromagnetic Field Waves of the Waves part of this course)

Using this identity for
E=Q and
B=P,
we obtain an expression for the rate of change of the energy density in terms of a divergence of vector B⨯E:
∂P_E+M /∂t = (1/μ)·∇·(B⨯E)

Standard continuity equation has negative rate of change equated to divergence of the flux density.
Therefore, we can rewrite the above equation in a standard form, changing the sign on the left and order of vectors on the right, getting

−∂P_E+M /∂t = (1/μ)·∇·(E⨯B)

The equation above implies that vector
S = (1/μ)·(E⨯B)
represent the electromagnetic field energy density rate of change.
This vector S is called Poynting vector.

Poynting vector represents the direction and magnitude of the electromagnetic field energy flow.
It's perpendicular to both electric and magnetic intensity vectors.

The equation above represents the continuity of electromagnetic field energy and local Energy Conservation Law.
Not only the total amount of energy in a closed system remains constant, but also the energy movement is continuous, it gradually flowing from one location to its immediate neighborhood.