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__steady-state temperature distribution__in an absorbing medium using the steady-state heat conduction equation:

(1) [itex]-\nabla\cdot\left(K\nabla T\right) = q_{gen}[/itex]

where

*T*is the material's temperature,

*K*is the material's thermal conductivity, and

*q*is the power generated per unit volume (which has units W/(s[itex]\cdot[/itex]m

_{gen}^{3}). In this case,

*q*is dependent on a

_{gen}*monochromatic sinusoidal electromagnetic wave*that passes through the medium and is partially absorbed. How do we find out

*q*, given an electromagnetic field distribution and linear macroscopic material properties?

_{gen}What textbooks I can find that discuss heat generation starting with the macroscopic form of Maxwell's Equations:

(2) [itex]\nabla\cdot\textbf{B}=0[/itex], [itex]\nabla\cdot\textbf{D}=0[/itex], [itex]\nabla\times\textbf{H}=\sigma\textbf{E}+\frac{ \partial \textbf{D}}{\partial t}[/itex] ,[itex]\nabla\times\textbf{E}=-\frac{\partial \textbf{B}}{\partial t}[/itex], with constitutive equations [itex]\textbf{D}=\epsilon\textbf{E}[/itex] and [itex]\textbf{B}=\mu\textbf{H}[/itex],

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 [itex]\hat{\epsilon} \left( \omega \right) = \epsilon'\left( \omega \right)+i\epsilon''\left( \omega \right)[/itex] and [itex]\hat{\mu}\left( \omega \right) = \mu'\left( \omega \right)+i\mu''\left( \omega \right)[/itex], which depend on frequency [itex]\omega[/itex]. The imaginary terms cause dissipation of the electromagnetic field into the medium as it travels. This changes eq.2 to:

(3) [itex]\nabla\cdot\textbf{B}=0[/itex], [itex]\nabla\cdot\textbf{D}=0[/itex], [itex]\nabla\times\textbf{H}=\frac{ \partial \textbf{D}}{\partial t}[/itex] ,[itex]\nabla\times\textbf{E}=-\frac{\partial \textbf{B}}{\partial t}[/itex], with constitutive equations [itex]\textbf{D}=\hat{\epsilon}\textbf{E}[/itex] and [itex]\textbf{B}=\hat{\mu}\textbf{H}[/itex].

The big change is the removal of the

*σ*

**E**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) [itex]-\nabla\cdot\textbf{S} = \nabla\cdot\left(\textbf{E}\times\textbf{H}\right)[/itex]

(4b) [itex]-\nabla\cdot\textbf{S} = \left(\nabla\times\textbf{E}\right)\cdot\textbf{H} - \textbf{E}\cdot\left(\nabla\times\textbf{E}\right)[/itex]

(4c) [itex]-\nabla\cdot\textbf{S} = \left(\textbf{H}\cdot\frac{\partial \textbf{B}}{\partial t} + \textbf{E}\cdot\frac{\partial \textbf{D}}{\partial t}\right)[/itex].

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:

[itex]Re\left\{\textbf{E}\right\} = \frac{1}{2}\left(\textbf{E}+\textbf{E}^*\right)[/itex], [itex]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)[/itex],

[itex]Re\left\{\textbf{H}\right\} = \frac{1}{2}\left(\textbf{H}+\textbf{H}^*\right)[/itex], [itex]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)[/itex]

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) [itex]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)[/itex]

(5b) [itex]q = \frac{\omega}{2}\left(\epsilon''\left| \textbf{E} \right|^2 + \mu''\left|\textbf{H}\right|^2\right)[/itex].

Since

*q*represents the dissipated power/volume, eq.5b should be able to be used as

*q*=

*q*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.

_{gen}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) [itex]q = \left\langle\sigma \textbf{E} \cdot \textbf{E} \right\rangle=\frac{1}{2}\sigma \left| \textbf{E} \right|^2[/itex],

and dielectrics heat differently:

(7) [itex]q = \left\langle \textbf{E} \cdot \frac{ \partial \textbf{D}}{ \partial t} \right\rangle = \frac{1}{2}\omega\epsilon''\left| \textbf{E} \right|^2[/itex] (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) [itex]\epsilon’\left(\omega\right)=\epsilon_{o}\left(n^2-\kappa^2\right)[/itex], [itex]\epsilon’’\left(\omega\right)=2\epsilon_{o}n\kappa[/itex]

At 1.55µm, we have

*σ*= 1/(2.255x10

^{-8}Ωm) =

**4.435x10**at 25ºC, with n = 0.55 and κ = 11.5.

^{7}S/m*ωε’’*is then:

[itex]\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}[/itex]

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

*q*from absorbed electromagnetic wave energy, and the differences between dielectrics and metals in trying to find this value?

_{gen}