Theory of microwave-based heating

It is well-known that microwaves can pose significant, and in some cases dangerous, thermal hazards. Here, we provide a theoretical analysis of microwave heating to describe the thermal effects of microwaves on matter.

Dielectric heating

High-energy microwave beams can lead to serious burns due to the dielectric heating they cause. This comes from the interaction of the rapidly-oscillating electric fields of these microwaves with polar molecules, causing the molecules to rapidly rotate back-and-forth (dipole rotation). Among the most well-known polar molecules is, of course, water; hence this mechanism is well-known for heating foods in a microwave oven.

Consider a molecule with electric dipole moment $\mathbf{p}$ placed in an electric field $\mathbf{E}$. The potential energy $U$ of this configuration is given by:

$$U=-\mathbf{p}\cdot \mathbf{E},\mathbf{p}\approx q\mathbf{d}$$

Where $q$ is the total number of valence electrons of the molecule, and $\mathbf{d}$ is the distance between the positive and negative charge centers. For water molecules, $|\mathbf{p}|\approx 6.2\times 10^{-30}\mathrm{C}\cdot \mathrm{m}$¹. Microwaves are electromagnetic waves, whose electric field can be described in terms of plane waves $\mathbf{E}={\mathbf{E}}_0\cos \omega t$, where $\omega$ is the frequency of the oscillations. Assuming that the electric field and the dipole moment are aligned such that $\mathbf{p}\cdot \mathbf{E}=pE$ where $p=|\mathbf{p}|$ and $E=|\mathbf{E}|$ we obtain:

$$U=-pE=-p\cos \omega t$$

Hence we see that the energy stored is directly related to the dipole moment, and vanishes for $p\to 0$. Microscopically, we may describe dielectric heating using the model of dipole rotations, whereas macroscopically it manifests as a complex-valued dielectric constant. The dielectric constant ${\epsilon }_r$ (more formally called the relative permittivity) is in general complex-valued and may be expressed as:

$${\epsilon }_r={\epsilon }_r^{\prime }+i{\epsilon }_r^{\prime \prime }$$

Where ${\epsilon }_r^{\prime }$ is the real part of the dielectric constant, while ${\epsilon }_r^{\prime \prime }$ is the imaginary part and is responsible for dissipation of energy (as heat) due to dielectric heating. The real and imaginary parts of the dielectric constant are related by:

$$\tan \delta =\frac{{\epsilon }_r^{\prime \prime }}{{\epsilon }_r^{\prime }}$$

Where $\tan \delta$ is called the loss tangent; for instance, water has a loss tangent of $\tan \delta =0.157$ and ${\epsilon }_r’=76.7$ at 3 GHz²; these values rise to $\tan \delta =1.12$ and ${\epsilon }_r^{\prime }=80$ at 3.58 GHz³ (for reference, household microwaves usually operate at 2.45 GHz). The power density (power per unit volume) caused by dielectric heating is given by:

$$\frac{dP}{dV}=\omega {\epsilon }_r^{\prime \prime }{\epsilon }_0|\mathbf{E}{|}^2=\omega {\epsilon }_r^{\prime }\tan \delta {\epsilon }_0|\mathbf{E}{|}^2$$

Note: The dielectric “constant” is not really a constant, since it varies throughout the microwave range and is in general a function of frequency (thus the more proper name for it is the dielectric function). In the general case, we can use the Kramers-Kronig relation ${\epsilon }_r^{\prime \prime }(\omega )=-\frac{1}{\pi }\mathcal{P}{\int }_{-\infty }^{\infty }\frac{{\epsilon }_r^{\prime }(\omega )}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$ to find the complex part of the dielectric function if the real part (as a function of frequency) is known; here $\mathcal{P}$ denotes the Cauchy principal value.

Since dielectric heating is linearly-proportional to the frequency, it is the dominant heating mechanism at high frequencies for polar molecules (and other polarizable media). Another effect, known as

Joule heating

Joule heating is another mechanism for electromagnetic heating, which arises due to the heat dissipated through conductors. The power density due to Joule heating can be classically described using the differential form of Ohm’s law:

$$\frac{dP}{dV}=\mathbf{J}\cdot \mathbf{E}$$

Assuming a medium with conductivity $\sigma$, then $\mathbf{J}=\sigma \mathbf{E}$, hence the power density (power per unit volume) is given by:

$$\frac{dP}{dV}=\sigma |\mathbf{E}{|}^2$$

Joule heating is the mechanism that enables electric stoves, electric ovens, and other types of electrically-powered heating elements. However, since the power density varies linearly with the conductivity, it most strongly affects conductive materials and has less effect on water, which has a conductivity on the order of $10^{-4}\text{ S/m}$⁴.

Radiative heating

Lastly, all electromagnetic radiation also carries (and transmits) intrinsic energy. Classically, this is the energy stored by the electromagnetic field; quantum-mechanically, it is the energy carried by photons, whose energies are quantized and given by $E=\hslash \omega$. The effect of radiative heating can be modelled using the Stefan-Boltzmann law:

$$u_{\text{em}}\equiv \frac{dP}{dA}=e_m{\sigma }_s{T}^4$$

Where ${\sigma }_s=5.67\times 10^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$ is the Stefan-Boltzmann constant, $T$ is the temperature, $e_m$ is the emissivity of the material upon which the laser beam is incident (it ranges between 0 and 1), and finally $u_{\text{em}}$ is the electromagnetic power density (power per unit area), given by:

$$u_{\text{em}}=|\mathbf{E}\times \mathbf{H}|\propto |\mathbf{E}{|}^2$$

For laser beams, the electromagnetic power density can be particularly high, not simply because of a laser’s power output (which for industrial lasers can be in the kW range, and for pulsed lasers can be in the MW or even GW/TW range), but also due to the extremely-focused nature of the beam, a feature that is almost universal among laser beams. Since the power density satisfies $u_{\text{em}}\propto A^{-2}$ (that is, it is inversely-proportional to the cross-sectional area of the beam), the power densities of a laser can be extremely high (at least for the region directly illuminated by the laser beam) if the beam cross-section is made small. However, the dependence of radiative heating on the emissivity $e_m$ means that heat transfer is minimal for materials with low emissivity (like metals or vacuum itself); by contrast, there is significant heat transfer for water, which has an emissivity of 0.96 (which is nearly 1, the highest possible emissivity).

Footnotes

1. See http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html ↩

2. See Table I of https://www.microwavejournal.com/articles/3038-liquid-sensing-at-radio-frequencies ↩

3. See Determination of the Relative Permittivity and Loss Tangent of Liquid Samples Using a Miniature Low-Cost Microwave Sensor (Olan-Nuñez et. al. 2023) ↩

4. See https://en.wikipedia.org/wiki/Conductivity_(electrolytic) and https://sciencedemonstrations.fas.harvard.edu/presentations/conductivity-water ↩