# Heat Generation by Electromagnetic Waves in Semi-Transparent Mediums

1. Sep 18, 2012

### rek

I am trying to figure out how I would go about finding the steady-state temperature distribution in an absorbing medium using the steady-state heat conduction equation:
(1) $-\nabla\cdot\left(K\nabla T\right) = q_{gen}$
where T is the material's temperature, K is the material's thermal conductivity, and qgen is the power generated per unit volume (which has units W/(s$\cdot$m3). In this case, qgen is dependent on a monochromatic sinusoidal electromagnetic wave that passes through the medium and is partially absorbed. How do we find out qgen, given an electromagnetic field distribution and linear macroscopic material properties?

What textbooks I can find that discuss heat generation starting with the macroscopic form of Maxwell's Equations:
(2) $\nabla\cdot\textbf{B}=0$, $\nabla\cdot\textbf{D}=0$, $\nabla\times\textbf{H}=\sigma\textbf{E}+\frac{ \partial \textbf{D}}{\partial t}$ ,$\nabla\times\textbf{E}=-\frac{\partial \textbf{B}}{\partial t}$, with constitutive equations $\textbf{D}=\epsilon\textbf{E}$ and $\textbf{B}=\mu\textbf{H}$,
with magnetic field B, electric field E, electric displacement field D, magnetizing field H, electrical conductivity σ, electric permittivity ε, and magnetic permeability μ.

They then take material dispersion into account, that is, treat the material's ε and μ as complex values $\hat{\epsilon} \left( \omega \right) = \epsilon'\left( \omega \right)+i\epsilon''\left( \omega \right)$ and $\hat{\mu}\left( \omega \right) = \mu'\left( \omega \right)+i\mu''\left( \omega \right)$, which depend on frequency $\omega$. The imaginary terms cause dissipation of the electromagnetic field into the medium as it travels. This changes eq.2 to:
(3) $\nabla\cdot\textbf{B}=0$, $\nabla\cdot\textbf{D}=0$, $\nabla\times\textbf{H}=\frac{ \partial \textbf{D}}{\partial t}$ ,$\nabla\times\textbf{E}=-\frac{\partial \textbf{B}}{\partial t}$, with constitutive equations $\textbf{D}=\hat{\epsilon}\textbf{E}$ and $\textbf{B}=\hat{\mu}\textbf{H}$.
The big change is the removal of the σE term from Ampere's Law. From what I can tell, σ (still constant) is somehow subsumed into ε''(ω), and as a value, is generally irrelevant unless focusing on frequencies near the static case (at which point it can be directly related to a metal's permittivity).

Derivation by Electrodynamics of Continuous Media: the rate of change of the energy in unit volume of the body is
(4a) $-\nabla\cdot\textbf{S} = \nabla\cdot\left(\textbf{E}\times\textbf{H}\right)$
(4b) $-\nabla\cdot\textbf{S} = \left(\nabla\times\textbf{E}\right)\cdot\textbf{H} - \textbf{E}\cdot\left(\nabla\times\textbf{E}\right)$
(4c) $-\nabla\cdot\textbf{S} = \left(\textbf{H}\cdot\frac{\partial \textbf{B}}{\partial t} + \textbf{E}\cdot\frac{\partial \textbf{D}}{\partial t}\right)$.
When integrated with respect to time, the resulting equation describes the difference between the internal energy per unit volume with and without the field. However, when in the presence of dispersion, the equation is instead related to the dispersion of energy into the medium. If eq.4d is averaged with respect to time, the result is the steady inflow of energy per unit time and volume from external sources to keep a field's amplitude constant (which is equivalent to the power/volume that is dissipated). Since eq.4d is quadratic in E and H, it must be written in real form. Conveniently, for a sinusoidal field, E and H are complex and have a e-iωt term, and as such they have the following forms:
$Re\left\{\textbf{E}\right\} = \frac{1}{2}\left(\textbf{E}+\textbf{E}^*\right)$, $Re\left\{\frac{\partial \textbf{D}}{\partial t}\right\} = \frac{1}{2}\left(-i\omega\hat{\epsilon}\textbf{E}+i\omega \hat{\epsilon}^*\textbf{E}^*\right)$,
$Re\left\{\textbf{H}\right\} = \frac{1}{2}\left(\textbf{H}+\textbf{H}^*\right)$, $Re\left\{\frac{\partial \textbf{B}}{\partial t}\right\} = \frac{1}{2}\left(-i\omega\hat{\mu}\textbf{H}+i\omega\hat{\mu}^* \textbf{H}^*\right)$
where * signifies the value's complex conjugate. Inputting the real forms of E, H, ∂D/∂t and ∂B/∂t into 4c gives terms with and without e±2iωt. When time averaged, the e±2iωt terms become zero, leaving:
(5a) $q = -\left\langle \nabla\cdot\textbf{S}\right\rangle = \frac{i \omega}{4} \left( \left( \epsilon^*-\epsilon \right) \left| \textbf{E} \right|^2 + \left( \mu^*-\mu \right)\left|\textbf{H}\right|^2\right)$
(5b) $q = \frac{\omega}{2}\left(\epsilon''\left| \textbf{E} \right|^2 + \mu''\left|\textbf{H}\right|^2\right)$.
Since q represents the dissipated power/volume, eq.5b should be able to be used as q = qgen in eq.1. In addition, the equation for would work for both metals and dielectrics, since eq.3 are meant to be universal over materials.

The web textbook Electromagnetic Fields and Energy has a different result. It works off Poynting's theorem and states that, for metals, the primary source of heat generation is Ohmic Conduction:
(6) $q = \left\langle\sigma \textbf{E} \cdot \textbf{E} \right\rangle=\frac{1}{2}\sigma \left| \textbf{E} \right|^2$,
and dielectrics heat differently:
(7) $q = \left\langle \textbf{E} \cdot \frac{ \partial \textbf{D}}{ \partial t} \right\rangle = \frac{1}{2}\omega\epsilon''\left| \textbf{E} \right|^2$ (which is equivalent to eq.5b when you consider that most dielectrics are essentially magnetically inert with μ''=0).
The book doesn't spend any time discussing how eq.7 is derived, and defines eq.6 from the Ohmic conduction law. It does state that eq.6 and 7 occur in different situations (eq.7 when conduction effects are negligible, and dipole polarization becomes dominant, eq.6 otherwise), but surely metals’ ε''(ω) becomes relevant at some point.

Let’s look at gold, for example. I’ve never seen any statement of σ varying with frequency, only temperature. We can determine ε(ω) in non-magnetic materials easily from experimental measurements of refractive index n and extinction coefficient κ by way of:
(8) $\epsilon’\left(\omega\right)=\epsilon_{o}\left(n^2-\kappa^2\right)$, $\epsilon’’\left(\omega\right)=2\epsilon_{o}n\kappa$
At 1.55µm, we have σ = 1/(2.255x10-8Ωm) = 4.435x107S/m at 25ºC, with n = 0.55 and κ = 11.5. ωε’’ is then:
$\omega\epsilon’’\left(\omega\right)=\frac{4\pi c}{\lambda_{o}}\epsilon_{o} n\kappa =\frac{ 4\pi\left(2.998\times10^{8} m/s\right)}{1.55\times10^{-6}m} \left(8.854\times10^{-12}F/m\cdot0.55\cdot11.5\right) = \textbf{1.361} \times \textbf{10}^{\textbf{5}}\textbf{S/m}$
There's a difference in two orders of magnitude in favor of σ, which is odd, considering that eq.5 seems to assume that σ gets incorporated into ε(ω).

So, what's going on? Can anyone shed some light into how to find qgen from absorbed electromagnetic wave energy, and the differences between dielectrics and metals in trying to find this value?

2. Sep 20, 2012

### rek

There's a typo in eq. 4b, it should be:
(4b) $-\nabla\cdot\textbf{S}=\left(\nabla\times\textbf{E}\right)\cdot\textbf{H}-\textbf{E}\cdot\left(\nabla\times\textbf{H}\right)$

So, what if I input eq.2 into 4b, instead of eq.3? Then I get:
(9) $\left\langle -\nabla\cdot\textbf{S} \right\rangle = \frac{1}{2} \left( \omega\mu'' \left|\textbf{H}\right|^2 + \left( \sigma+\omega\epsilon'' \left|\textbf{E}\right|^2 \right) \right)$
This essentially combines eq.6 and eq.7 together. For metals (or at least gold), this changes the coefficient in the example from 4.435x107S/m to 4.449x107S/m, changing the value only slightly compared to what Electromagnetic Fields and Energy suggests in eq.6, and since most dielectrics have σ=0 and μ''≈0, eq.9 reduces to eq.7 in that case.

Last edited: Sep 20, 2012
3. Sep 20, 2012

### Darwin123

There are few general reviews that combined classical electrodynamics with thermodynamics. Generally, it is implicitly assumed that conductivity relates to internal energy. However, seldom is it explicitly stated.
My conjecture is that the reason the two are not often discussed together is that classical electromagnetic theory logically contradicts some of the laws of thermodynamics. Here are two possible reasons.
There is no law in classical electrodynamics that insures that a system has to reach equilibrium eventually. In fact, a dynamics system can never reach equilibrium in the sense that everything stops moving. Classical thermodynamics postulates that the states never vary far from thermal equilibrium. There isn’t even a formal way to describe thermal equilibrium in classical electrodynamics.
The Third Law of Thermodynamics requires that the ground state of the system be unique. This is not consistent with classical electrodynamics, or with classical mechanics. The Third Law of Thermodynamics is consistent with quantum mechanics. However, classical electrodynamics does not include quantum mechanics. Note that if the Third Law of Thermodynamics is not satisfied, then the system doesn’t have to ever reach thermal equilibrium.
Mostly what happens is the scientist hypothesizes that the work done by the external electric field on the carriers of an electrically conducting medium equals one component of the change in internal energy.
The work done on the charge carriers becomes internal energy for an arbitrarily short time span. I have never seen a proof based on statistical mechanics that it is true. However, it agrees with my physical intuition. Therefore, I just hold my nose and make the hypothesis.

I found this link. This article at least places the laws of thermodynamics and classical electrodynamics together. I hope this helps.
http://web.mit.edu/kei1/www/CM231/transport.pdf
“In this section transport phenomena in crystals will be treated within the semiclassical Boltzmann transport formalism. This is a statistical treatment of the electrons, since the complex problem of many-body interacting electrons can not be solved, and even if it were, it would be too complex a solution for extracting transport properties which are average properties taken over large time and lengthscales. So a statistical treatment is needed to deal with such uctuating phenomena. Resistance of a metal, and in general resistance to ow is caused by scattering. In a metal or semiconductor, the main sources of scattering are mpurities and defects at low temperatures, and phonons at high temperatures, where their concentration increases linearly with T. Before studying the statistical formulation, we will discuss electron dynamics in a perfect crystal.”

This investigation states a hypothesis very close to what I suggested. It suggests that the external virtual work (conduction?) is equal to the internal virtual work (heating?).
http://146.6.104.11/~landis/Landis/Research_files/IJNLM2007.pdf
“A principle of virtual work for combined electrostatic and mechanical loading of materials
The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work.”

Yes, electrical conductivity varies with frequency. Variation with frequency is referred to as dispersion. Dispersion of electrical conductivity is studied in plasmas, metals and semiconductors. I recommend J. D. Jackson, Classical Electrodynamics, as the best reference on the dispersion of electrical conductivity in plasmas. However, he doesn’t discuss the thermodynamics of the system.

http://www.math.bgu.ac.il/~chaik/Pdf/jt83.pdf
“In homogeneous conductors, a dispersion of the conductivity occurs at frequencies of the order of the inverse momentum relaxation time. In an inhomogeneous
conductor, an external electric field gives rise to local charge bunches.”

http://144.206.159.178/ft/200/201985/14240996.pdf
“Dielectric dispersion and ac conductivity in—Iron particles
The dispersion of an electrically conductive phase within an insulating host medium, affects the overall performance of the heterogeneous system [1]. Furthermore, if the dispersed filler is in sufficient quantity, a conductive or semiconductive composite is formed [2,3]. In all the examined systems, ac conductivity exhibits a strong dispersion. At low frequencies conductivity tends to be constant, while at higher becomes strongly frequency dependant, varying approximately as a power of frequency.”

4. Sep 24, 2012

### rek

Shouldn't it be possible to just treat the electromagnetic wave as a heat source? The issue is only trying to decide what equation determines how that heat is produced, which is what equations 5, 6&7, or 9 try to describe: Ohmic Conduction and dielectric heating. Dielectric heating is most often discussed with regards to microwaves; maybe I could find some insight there.

Most texts that I have read, including Jackson, describe the dispersion of σ only in the limit of ω→0, where the dielectric constant can be directly related to the conductivity, in accordance with the Drude model. At what frequencies does the Drude model start becoming incorrect? This lecture by E.Y. Tsymbal has redefined ε=εL+iσ/ω (εL due to Polarization), then relating it directly to the index of refraction and extinction coefficient. Shouldn't Polarization PoχeE be allowed to be complex on its own (specifically complex χe), such that εLo(1+χe) is complex already without including σ in a redefined Maxwell-Ampere equation? Is it just that for metals, εLo(1+χe) is real, while for dielectrics, it isn't? Or is it that dielectrics only start absorbing electromagnetic field energy when they have a non-zero conductivity?

What does the Maxwell-Ampere equation look like when the wave experiences dispersion due to the material it's passing through?

Last edited: Sep 24, 2012
5. Sep 24, 2012

### Darwin123

The problem that I was referring to is in thermodynamics terminology. One can't "just" treat the electromagnetic wave as a "heat source" when the meaning of "heat" isn't specified. It gets worse. An electromagnetic wave can both do work and heat a material, whether or not it is a conductor or a dielectric. If the equation of state isn't specified, there is no way to tell for sure whether the energy absorbed by the material is in the form of work or heat.
I suggested that the current terminology has an implicit default. Unless other wise stated, the conductivity (σ) refers to the rate of change in internal energy. That is, the "E.j" where "j=σE" automatically refers to a rate of change in "internal energy".
So the word "heating" doesn't refer to either work or true heat. It means change in "internal energy." Whether the energy is transferred as "heat" or "work" depends on the statistical properties of the photons. Black body radiation is automatically "heat conduction" and coherent radiation is automatically considered "work".

I think that you are mistaken about texts only discussing dispersion in the limit ω→0. Jackson, certainly, discusses a more general theory of dispersion. In fact, the Drude model is not restricted to the limit ω→0.
The version of Jackson that I have refers to a dispersion in the conductivity in conductors and plasmas.
"Classical Electrodynamics" Third edition by John David Jackson (Wiley, 1999).
This book give an analysis of a material where a fraction f_0 of electrons in a molecule are "free" in the sense of having the resonanant frequency equal to zero (ω_0=0). On page 312, equation 7.58 says that:
σ=f_0 Ne^2/[m(γ_0 -iω)]
where f_0 is a coupling constant, e is the electric charge of a free carrier, ω is the frequency and γ_0 is a damping constant.
This is the obviously, this equation is far more general than a limit ω→0.

The discussion in the previous page (page 311) is a bit confusing with respect to the dispersion. It says that in the limit ω→0 there is a qualitative difference depending on whether the lowest resonant frequency is zero or nonzero. However, the rest of the discussion is not about the limit ω→0. The rest of the discussion is about the dispersion of the conductivity.
I think the Drude model is very general if one includes an arbitrary number of resonant frequencies. On page 310 of the same text, equation 7.51 and 7.52, it gives a general expression for the complex dielectric constant given by the Drude model with an arbitrary number of resonant frequencies. Afterwards, Jackson says:
"With suitable quantum-mechanical definitions of f_j, γ_j and ω_j, (7.51) is an accurate description of the atomic contribution to the dielectric constant."
Equation 7.51 is a general description of a medium where there is frequency dispersion, not wavelength dispersion. Equation 7.51 is also only valid for an isotropic medium.
The Drude models as described by equation 7.51 isn't exactly correct for anisotropic media (i.e., with birefringence) or media that has wave vector dispersion.
The Drude model is easily generalized in the case of a medium which is anisotropic. This means it is either birefringent or dichroic. This is discussed in certain quantum optics and solid state textbooks. However, I don't have the references so far.
I once attempted to do research on a system where the complex dielectric constant had a hypothetical variation with wavelength. Equation 7.51 would be slightly incorrect for such a system. However, I was unable to experimentally detect the wavelength dispersion. I don't know if anyone was able to successfully detect the wavelength dispersion after I left the project.

6. Sep 25, 2012

### DrDu

There are several points here.
The first is that with periodic fields it is usually easier to take mu=1 and epsilon to be dependent not only on omega but also on k . This is the usual approach in optics as discussed e.g. in Landau Lifgarbagez, Electrodynamics, or in this article :http://siba.unipv.it/fisica/articoli/P/PhysicsUspekhi2006_49_1029.pdf [Broken]
where also the connection to the case where both epsilon and mu are used is worked out.
The conductivity is then often defined to be $\sigma=\omega \epsilon''$.
Landau and Lifgarbagez also discuss explicitly the damping in media.
At least at non-zero frequency, there is no need to distinguish between metals and dielectrics in the formal treatment.
Ah, and finally, Jackson is not a good source for electromagnetics in matter, probably because he is a particle theorist.
One of the references for the experts is

Last edited by a moderator: May 6, 2017
7. Sep 26, 2012

### rek

The use of the term ω→0 is a bit of a misnomer in some of the books, as it isn't when ω=0; what I believe they mean is a general 'low frequency region' where Drude's model holds, which is what Jackson talks about in that section. The question is whether the frequency I'm working at (infrared) makes Drude's model inapplicable, and I am starting to believe isn't the case. What makes me edgy is the Maxwell-Ampere Law:
$\nabla\times\textbf{H}=\sigma\textbf{E}+\frac{ \partial \textbf{D}}{\partial t}$ where $\textbf{D}=\epsilon_o(1+\chi_e)\textbf{E}$, and $\chi_e$ is allowed to be complex.
This means that even before you reduce the Maxwell-Ampere equation to $\nabla\times\textbf{H}=\hat{\epsilon}(\omega) \textbf{E}$, there can be an imaginary term in the dielectric constant not related to the conductivity. Again, I may not need to worry about this, because Drude's model probably applies for the materials in the frequency I'm interested in.

Luckily, I don't need to worry about anisotropic media either.
Thank you for the clarification. I think that, for now, I am going to presume that the change in energy is transferred entirely as heat.

If I try and simulate the two-dimensional heat conduction over layers of material with different thermal conductivities (air, dielectrics, and metal), the thermal conduction equation becomes:
$-\nabla\cdot\left(K(x,y)\nabla T(x,y)\right)=q_{gen}(x,y)$,
correct? As such, I can't just simply change the equation to
$\nabla^2 T(x,y)=-q_{gen}(x,y)/K(x,y)$
to make my simulation simpler.

Last edited by a moderator: May 6, 2017
8. Sep 29, 2012

### DrDu

No, omega to 0 is not a misnomer and it is not about the validity of the Drude approximation which is another topic. The point is that in a scheme which only considers epsilon (possibly wave number dependent) epsilon will have a pole at omega =0.