Open access peer-reviewed chapter

Electromagnetism of Microwave Heating

Written By

Rafael Zamorano Ulloa

Submitted: December 9th, 2020 Reviewed: March 17th, 2021 Published: March 16th, 2022

DOI: 10.5772/intechopen.97288

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Detailed electrodynamic descriptions of the fundamental workings of microwave heating devices are presented. We stress that all results come from Maxwell equations and the boundary conditions (BC). We analyze one by one the principal components of a microwave heater; the cooking chamber, the waveguide, and the microwave sources, either klystron or magnetron. The boundary conditions at the walls of the resonant cavity and at the interface air/surface of the food are given and show how relevant the BC are to understand how the microwaves penetrate the nonconducting, electric polarizable specimen. We mention the application of microwaving waste plastics to obtain a good H2 quantity that could be used as a clean energy source for other machines. We obtained trapped stationary microwaves in the resonant cavity and traveling waves in the waveguides. We show 3D plots of the mathematical solutions and agree quite well with experimental measurements of hot/cold patterns. Simulations for cylindrical cavities are shown. The radiation processes in klystrons and magnetrons are described with some detail in terms of the accelerated electrons and their trajectories. These fields are sent to the waveguides and feed the cooking chamber. Whence, we understand how a meal or waste plastic, or an industrial sample is microwave heated.


  • microwave heating
  • resonant cavity
  • cooking chamber
  • waveguide
  • klystron
  • magnetron
  • boundary conditions
  • food-air interface
  • Lienard-Wiechert potentials
  • Jefimenko fields

1. Introduction

Microwaves are everywhere and permeate the universe. They reach earth constantly and we produce them in many medical, industrial, chemical, domestic, and research on magnetic and dielectric materials and in devices and equipment [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The modern communications technology uses them intensely, Wi-Fi, all around the world, every second, every day [14]. Many medical applications are concentrated in cancer treatments by giving hyperthermia to the cancerous cells while avoiding to damage healthy cells. Microwave ablation is widely used in many types of cancers, bone, cardiac arrhythmic tissue, thyroid glands, skin cancer, and many other damaged tissues [2, 3, 15, 16]. The apparatus is constituted basically by a microwave source, a waveguide that ends in an antenna that, as a needle, penetrates the tissue [2, 3, 15, 16]. Industrial applications go from thermally treating/curing polymers, rubber, and plastics to quickly heat cement and minerals, and to assist vulcanization [4, 5, 17, 18]. Chemical applications are mainly directed to organic and/or inorganic synthesis and accelerating reactions, and to search for novel synthesis routes and novel products. Chemical microwave heating has been used for decades [6, 7]. Research in magnetics and in dielectrics includes heat transfer and/or electromagnetic excitation of matter [10, 11, 12, 13]. Domestic technology uses them to heat up quickly and easily food, coffee, and water in microwave ovens (MWO) built for such function, see Figure 1(D) [8, 9]. By far, the two most commonly used microwave configurations include a source of microwaves, typically a klystron or a magnetron, a waveguide for these microwaves, and a resonant chamber where the microwaves are used to treat, to modify, to cure, to excite, or to heat up a sample put in the microwave chamber. There are two most common geometries of microwave chambers, cylindrical and rectangular [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Figure 1(A) shows an electric field pattern simulated inside a typical cylindrical cavity (3.4 cm of radius and 4.22 cm of height) used in a research equipment, in which magnetic samples are resonantly excited [19]. The magnetic field (not shown) is vertical (and orthogonal to E) and mostly concentrated along the z-axis at r = 0 and to its close vicinity. Figure 1(B) shows an electron paramagnetic resonance spectrometer that uses a cylindrical microwave chamber feed by a rectangular waveguide that collects the 9.4-GHz low-power microwaves produced by a klystron inside the box-labeled microwave bridge. Figure 1(C) is a calculated stationary electric field pattern from the solutions to Maxwell equations found in this work for a rectangular cavity. Figure 1(D) shows a typical domestic microwave oven of dimensions 26 cm × 30 cm × 34 cm.

Figure 1.

Microwaves in cylindrical and rectangular geometries. (A) Simulation of the electric field, circular, pattern formed inside a cylindrical microwave cavity. (B) The microwave bridge, the rectangular waveguide, and a cylindrical cavity of a commercial electron paramagnetic resonance spectrometer used to excite magnetic samples. Here, heating is not desired, and the microwave power used is 1 mW or less. (C) A stationary electric field pattern calculated in this work from the solutions to Maxwell equations. The pattern is calculated for the planesxandywith the coordinatezmaintained fixed at an arbitrary height. (D) A typical domestic microwave oven (MWO), open and showing its internal chamber heighth = 38 cm, and widtha = 32 cm andb = depth, 30 cm.

A universal advantage of microwave heating in industrial, medical, chemical, and domestic processes is that it does it quickly and efficiently. Yet, several investigations are pursued to find even faster and better microwave heating schemes and profiles [15, 16, 17].

The three main parts of these heating devices constitute a source, a waveguide, and a heating chamber. Figure 2(A) shows the essential parts of a microwave oven commonly used to heat food. In addition to the already large number of industrial microwave heating applications, very recently, microwaving plastic waste decomposition has been proposed as a central step in order to generate clean hydrogen, H2, out of heating a one-to-one mixture of triturated waste plastics with the catalyst FeAlOx [20]. Edwards et al. [21] have used microwaves to transform waste plastic bags, milk empty bottles, and other supermarket waste plastics, Figure 2(B), in a clean hydrogen energy source. A 1:1 mixture of the catalyst FeAlOx and waste plastics heated up with microwaves in a cylindrical cavity (as shown in Figure 2(C)) for 10, 30, and 60 s extracts 85–90% of molecular hydrogen that is sent through a column to be stored in a separate chamber for its eventual utilization as a clean energy source. The principle of operation is the same for the domestic microwave oven with a rectangular heating chamber and for this microwave heater that is used to transform waste plastics within a cylindrical heating chamber. This transformation process is clean and fast and could help to reduce drastically the world’s wide plastic contamination problem. Plastics invade mountains [22], forests, lakes, oceans, and cities [20, 21, 22]. As the hydrogen density content in plastic bags is around 14% per weight, its transformation into H2 and multifaceted fullerenes might offer an opportunity of clean energy production for the countries interested in producing clean H2 as a viable energy source for industrial and domestic usage and contribute to slowing down the global climate change [23]. Using tons and tons of garbage plastics, H2 can be produced in high quantities and then used as a clean industry and house energy supply, this way contributing to combat the global warming. We consider this kind of potential application relevant for the plastic industry and pollution problems and much more relevant for its contribution to a cleaner, greener planet. Microwave heating is used globally, also, to cook or just heat up meals, water, coffee, and pizza. These microwave ovens operate at a power of 1000 Watts; the meals are put into a rectangular resonant cavity that is the cooking chamber that gives them their familiar “box” appearance. In this box, microwaves are delivered unevenly (see uneven electric field pattern calculated here and shown in Figure 1(C)), to the meals, and they readily penetrate the matter, making electric dipoles (mainly from water and fatty molecules) to oscillate frenetically; then, this excitation energy is passed to the rest of the specimen as heat.

Figure 2.

Microwave heating. (A) A typical 1000 Watts, 2.45-GHz domestic microwave oven (MWO) cooking a chicken. (B) Plastic pollution derived from tons and tons of waste plastic bottles and supermarket bags. (C) Clean production of H2 by microwaving waste plastic bags and bottles mixed 1:1 with the catalyst FeAlOx within a cylindrical cavity, the input power is 1000 Watts, and H2 gas is liberated and carried toward a separate container. The cylindrical cavity operates in the TM010 mode.

In a few seconds or minutes, the specimen is hotter than at the start of the microwaving [5, 6, 7, 8, 9, 10, 11, 12, 13]. The utility of microwaves is multiple and of large range. In spite of the generalized use of microwave ovens, an informal survey has indicated to us that more than 90% of our STEM students do not really know how MWOs operate. The basic physics required to understand their workings is completed by the end of the undergraduate class work. This is telling us the degree of sophistication and depth that this technology carries, just as many other modern technologies do, as the Wi-Fi itself. A primal objective of this chapter is to describe its electromagnetic physics and to give the fundamentals. All comes from Maxwell equations. We emphasize the fact that most of the microwave heating devices (as the two devices shown in Figure 2) are composed of three essential electromagnetic parts: the production of the microwaves, their wave-guided structure that brings the microwaves from their origins toward the resonant cavity, being the third essential component, that is, the cooking chamber. Aforesaid, some devices use rectangular resonant cavities and others use cylindrical resonant cavities. We focus on rectangular cavities and just mention some results for cylindrical cavities.

We want to emphasize that the study of the electromagnetic functioning of a domestic, an industrial, or a laboratory MWO is an ideal technological case to see how the whole of the theory is applied. These microwave “boxes” contain all of the fundamental physics required to produce (in klystrons or magnetrons) microwaves, then input them into a loss-less waveguide, and finally get them to bath a rectangular heating chamber, or a cylindrical chamber, without appreciable microwave radiation being absorbed at the metallic walls. Their absorption is mainly carried out, precisely, by the specimen we want to heat up.

Our discussion has five parts: First, in Section 2, we deal qualitatively, in detail, with the fundamental constituents of a microwave heating system and the physical processes involved. Then, in Section 3, we analyze the boundary conditions, and in Section 4, we treat mathematically the resonant cavity. A description of the physics of this resonant cavity that keeps confined the microwaves all the time necessary for the food to warm up, or even be cooked, is given. We show that standing wave patterns are the solutions to Maxwell equations. Section 5 treats the rectangular waveguide and the general form of the traveling waves in them is obtained. The process of wave guiding the microwaves is the one that carries the microwaves from its source to the heating chamber. Then, in Section 7, we treat the production of microwaves in klystrons and/or magnetrons by radiating, accelerated, electrons moving in straight lines, or curved trajectories.


2. Fundamental constituents of a microwave heater

In formal terms, a domestic microwave oven [8, 9], or an industrial system [4, 5] or a laboratory prototype for clean extraction of H2 from microwaving waste plastics [20, 21], is constituted by three fundamental parts shown in Figure 2(A) and (C) and in Figure 3.

Figure 3.

Food being cooked in a microwave oven. (A) The meal is already hot and steam is actually getting out of the meat, and the green wavy arrow is pointing to the possibility that some microwaves get out of the oven. (B) The microwaves rebound from all six metallic surfaces of the “box” (yellow) and are reflected and transmitted from the surface of the specimen been heated (blue and red).

(A) The resonant cavity: Once the microwaves are inside the rectangular cooking chamber, Figure 2(A), or inside a cylindrical resonant cavity, Figure 2(C), these microwaves display themselves stationary wave patterns since they are confined within good conducting walls, see Figure 1(A) and (C). These waves rebound incessantly from these metallic walls without, practically, any energy loss. These wave patterns are specific for each geometry and each set of dimensions and the boundary conditions at the walls, as we show below.

In MWOs, the sixth wall is the see-through door that allows access to the interior of the chamber. The see-through window is covered by a metallic mesh with many small holes of r ≈ 1 mm radius that does not allow microwaves to escape; the condition λ> > rholds and, hence, it functions as a continuous metallic, highly reflecting, surface. The receptacle we see when we introduce the meal to be microwaved is the resonant cavity as shown in Figure 3. Electromagnetically, a rectangular resonant cavity, beneath the plastic covers, conforms to a metallic box of about 30 × 32 × 38 cm3 dimensions. A 3D standing wave pattern is self-established due to the boundary conditions that have to be fulfilled at the walls, see Figure 1(C).

Hence, maxima and minima and zeroes of the electric field Eand magnetic field Bappear at different locations, x, y, z, inside the cooking chamber. These microwaves bouncing continuously from the six walls bath constantly and heat our meal.

Can the microwaves, green wavy arrows in Figure 3, escape from the cooking chamber? Not in principle, unless some malalignment, or broken piece is there. Within the resonant cavity, the microwaves bounce back and forth from the metallic walls (yellow in Figure 3(B)) without any loss of electromagnetic energy.

Then, the microwaves hit the chicken at multiple points (blue-green wavy arrows), at the interface between food and air, reflection and refraction take place, and Snell’s law and Fresnel equations have to be fulfilled [24, 25, 26, 27, 28, 29]. Some microwaves are reflected (blue-green lines), and others are transmitted inside the chicken body (red wavy arrows). These red microwaves are responsible for heating, and they are the ones that transmit, quite efficiently, vertiginous motions vibrations, at 2.45 GHz, to the electric dipoles that are part of the meal (mostly water, but also some fatty molecules). These red microwaves penetrate several centimeters through the specimen. The electromagnetic energy carried out by the Poynting vector of these red microwaves, Srt=Ert×Hrt, is converted into frenetic Jiggling of these polar molecules and then converted into heat by their interactions with surrounding, neighboring molecules. Heat can be so high that some steam (water vapor) can be seen through the window in just a few seconds, see Figure 3(A). This process is the moment of energy conversion: from electromagnetic energy with 1000 Watts of power to motion, vibrations, mechanical energy. But that excited motion starts, rapidly, to pass to neighboring nonpolar atoms and molecules and locally, all the surrounding matter, starts jiggling more and more, which is heat. Microwave energy has been transformed into heat inside our meal being microwaved. In Figure 3, the wavy red lines represent the microwaves that get into the specimen and excite the electric dipoles within it. We show that at multiple points of incidence-reflection-transmission on the food-air interface, this bathing is by no means uniform since the microwaves distribute inhomogeneously inside the resonant cavity, see Figure 1(C). This is the reason why the specimen is placed on a rotating plate, so some homogeneous heating is achieved.

Normally, it is expected that the ceramic (or the plastic), from which the cup for coffee or water is made, does not get heated while the liquid inside it. In order to get such result, it is necessary to minimize the composition of electric dipoles in the structure of the ceramics, glass, or plastic that makes the cup or the dish, effectively rendering this object transparent to the microwaves.

(B) The metallic waveguide: The microwave radiation from the source is immediately channeled through a horn-like metallic collector toward a rectangular waveguide through multiple reflections on its conducting metallic walls, and the radiation is guided almost without attenuation to the resonant cavity of the MWO. A rectangular waveguide is shown in Figures 1(B) and 4(C). The good conductor quality of the waveguide is the responsible for no-attenuation microwaves at the waveguide walls even after multiple reflections. More detail on waveguides can be found in Refs. [24, 25, 26, 27, 28, 29]. The waveguide terminates in a “mouth” that connects with the cooking chamber. This way both parts are coupled.

Figure 4.

Constituent parts of a microwave oven, or a laboratory microwave heater. (A) A magnetron typically used to generate microwaves in a domestic MWO. (B) A microwave generator, klystron, that is frequently used in microwave laboratory equipment where low power is required. (C) A hollow rectangular waveguide of a, b cross section and coupled to another waveguide, of the same dimensions, from aforesaid (taken from Feynman Phys. Lectures, vol. II. [24]) both microwave trains join and interfere at the union of the metallic structures. (D) An example of microwaving solids and trapping gasses that are detached, from the specimen, in the process. Microwaving plastic waste mixed 1:1 with FeAlOx inside the transformation chamber, which is an aluminum TM010 resonant cavity (it is the analog of the microwave oven chamber). Principle of operation was modified from [21], not the actual experimental setup.

(C) The source of microwaves: For low-power applications, 1 mW or less, a klystron is used as the source of microwaves. For higher-power applications of microwave heating, around 1000 Watts, a magnetron is used. In both cases, it is a tube in which electrons are ejected from a hot cathode to a space where they get immediately accelerated and precisely, this acceleration produces electric and magnetic radiation fields, orthogonal to each other and to the propagation direction. Its power is proportional to the acceleration squared, Prad α a2, and is schematically depicted in Figure 4(A) and (B). Those accelerated electrons emit radiation at the same frequency of the acceleration that in this case is in the 2–3 GHz range.

Now let us be more quantitative. We start with the electromagnetic boundary conditions.


3. Microwave heating of food or of an industrial sample

When microwave radiation hits the surface of a specimen, what we have is incidence of electromagnetic fields on the interface between meal and air, two nonmagnetic, nonconducting media, and the laws of Snell and Fresnel of reflection and refraction have to be fulfilled. But, they will be obeyed once the boundary conditions for Eand Dand for Hand Bare fulfilled. Let us see what they are as follows:

3.1 Continuity of the normal component of the displacement field, D

Let us suppose we have the interface between any two media such as water-air, plastic-metal, raw meat-hot air, a catalyst-plastic, ceramic-coffee, and so on as shown in Figure 5, that is, the boundary. The boundary condition on Dis obtained from applying Gauss law to a very small cylinder (purple) of differential area and differential high that crosses the boundary as shown in Figure 5(A). Then, the Gauss integral Endaon the closed surface is decomposed into three integrals, one on S1 within medium 1, another one on surface S2 within medium 2, and a third integral on the lateral surface, which goes to zero because the high of the cylinder is as small as we wish; then after integrating the only two integrals we are left with produce (Dn1 − Dn2)S = σƒS[25, 26, 27, 28, 29]. When the interface carries no electric charge, as is usually the case with microwave heating, then σƒ = 0. Hence, (Dn1 = Dn2). And so, the normal component of the displacement vector is continuous through the interface of air-chicken!, or air-plastic, or air-ceramic, and so on.

Figure 5.

Boundary conditions onD. (A) Gaussian, very small, cylinder on the interface between two different media 1 and 2. The differenceDn1 − Dn2 between the normal components ofDis equal to the surface charge densityσƒ. When surface charge is zero, thenDn1 = Dn2, the normal components of the displacement field are continuous. (B) The same condition applies on the surface of meat when fieldDhits its surface inside a microwave oven, in this caseσƒ = 0 andDn1 = Dn2. In (B) the boundary interface is the skin of a chicken being microwaved.

Only in the case of the boundary between a conductor and a dielectric Dn = σƒ, being σƒthe free charge density on the interface as represented with the “+” signs in Figure 5(A). For the cases we are interested here, all the metallic walls in the cooking chamber (resonant cavity) and the waveguide are not charged, then σƒ = 0 and so Dn1 = Dn2.

On the other hand, when the D, field of the microwaves in the cooking chamber enters the surface, Dn2, of the piece of food (sample, specimen, system), as shown in Figure 5(B), it travels much more distance inside (several centimeters.) the sample and in its way excites electric dipoles (mostly water molecules and fat dipolar and other organic dipolar moieties) and gradually, but fast, transfers most of its energy to them. After a relaxation time period, ≈ 10−6 sec, the dipoles transfer all that juggling energy to vibrations of the bulk and appear as heat (measured as kBT).

3.2 Continuity of the tangential component of the electric field intensity

Consider the blue, rectangular, path shown in Figure 6(A) and (B) with two sides parallel to the boundary and arbitrarily close to it. The two vertical sides are infinitesimal. Stokes theorem states that ×Eda=Edl. If the vertical paths are as short as we wish, Etdoes not vary significantly over them and their integrals are zero. And the line integral of Edlis Et1LEt2L. By Stokes’s Theorem, this line integral is equal to the integral of ×Eover the surface enclosed by the path C [25, 26, 27, 28, 29].

Figure 6.

Boundary condition onE. (A) Closed path of integration crossing the interface between two different media 1 and 2. Whatever be the surface charge densityσf, the tangential components ofEon both sides of the interface are equal:Et1 = Et2. The tangential components ofEfield are continuous no matter what medium 1 is and what medium 2 is. (B) The same analysis for a piece of chicken, or a cup of coffee, or melting cheese in a microwave oven follows: The tangential components ofEare continuous.Et2contributes, at most, to some heating on the surface of the meal.

By definition, the enclosed area is zero. So, Et1LEt2L = 0, hence Et1 = Et2. The tangential component of Eis therefore continuous across the boundary. Applying this reasoning to the interface of a piece of food and the air in a microwave oven, as shown in Figure 6(B), everything follows and the tangential component of Ein the cooking chamber just above the surface of that matter is continuous with the tangential component of Ejust inside “the chicken.”

For the case of the metallic walls of the heating chamber and the walls of the waveguide, we have the case of a boundary between a dielectric (hot air) and a conductor, then E=0in the conductor and Et = 0 in both media. The magnetic part of the microwaves also obeys corresponding boundary conditions, namely: B1n = B2nand H1t = H2t, and B2nis quite capable of exciting magnetic dipoles inside the specimen, but food, beverages, water, and coffee do not possess magnetic moments, which are not magnetic. So, we do not treat here magnetic heating, even though it is a very active field of research [10, 11]. We concentrate on heating through electric dipoles inside the cooking chamber, as shown in Figures 3,5, and 6. Next, we describe more quantitatively the electromagnetics in the cooking chamber.


4. Microwave cooking chamber as a resonant cavity

Aforesaid, the empty cooking chamber in a microwave oven (MWO) is a closed rectangular space where, once microwaves are input, they bounce back and forth from metallic walls on the six sides and confine the electromagnetic waves in such space. This is an electromagnetic resonant cavity (ERC) in which electromagnetic waves (EMW) move in space and time periodically and, very importantly, forming standing wave patterns with nodes and anti-nodes. The cooking chamber is then electromagnetically a resonant cavity that imposes on the microwaves boundary conditions at the six walls. The Efield, just outside and parallel to each wall, Et, must be zero and the normal component of Bmust be continuous [25, 26, 27, 28, 29]. When food, water, coffee, cheese, or any other food are introduced in it, a dielectric medium with εε0and air with ≈ ε0 are now the composite dielectric that fills the resonant cavity, as shown in Figure 3. Dielectrics do not perturb considerably the standing wave patterns that form the microwaves inside the MWO.

Let us be more quantitative, and take a standard microwave oven of dimensions a = 30 cm, b = 32 cm, and c = 38 cm as shown in Figure 1(D). We will take the walls as perfect conductors as first approximation, boundary conditions on the six walls have to be fulfilled, and there will be multiple reflections at the metallic boundary surfaces. Figure 7 shows how a sinusoidal electromagnetic field wave bounces back from a perfect conducting surface [25, 26, 27, 28, 29].

Figure 7.

The standing wave pattern resulting from the reflection of a microwave at the surface of a good conductor wall of the cooking chamber. The curvy lines show the standing waves ofEandHat some particular time. The nodesEand ofHare not coincident but are spacedλ/4 apart as shown. Modified from [25].

An arbitrary standing wave pattern in the resonant cavity can then be obtained as an appropriate superposition of these standing waves. Let us consider the closed region (cooking chamber) with walls of sides a, b, and c, and with the origin at one corner as shown in Figure 3. The cavity is filled with a linear dielectric, food (material described by μ0 and ε). Both fields, Eand H, inside should obey Maxwell equations and each field component satisfies the wave equation; the solutions are stationary, confined, trapped microwaves. Applying separation of variables first to the time variable, it results in a solution of the form eiωt. Therefore, we have now, let us say, for


which, when substituted into the wave equation, leads to the Helmholtz equation 2ψ0+k02ψ0=0where k02=ω/υ2. This result is actually valid for any kind of coordinate system. Helmholtz equation is readily solved in rectangular coordinates by separation of variables. If we write, ψ0=XxYyZzand proceed with the standard separations, we obtain [28, 29].


where k1, k2, and k3 are the wave numbers in the x, y, zdimensions and are related to the frequency ωof the microwave field by the dispersion relation k12+k22+k32=k02=ω/υ2. Combining this ψ0rwith the temporal solution T(t), we get any one of the components of the Eand Hfield as


The boundary conditions that obey each Eand Hfield component are going to make the difference of the fields through the fact that the constants, C1, C2, …, C6, of each component will take different values, including zero. The boundary condition E1t = E2tat any metallic wall (Emetal = 0) makes for tangential components to be zero. Hence, Exwill be a tangential component and must therefore vanish at the faces y = 0 and band z = 0 and c. We see that this requires that C4 = C6 = 0 and that k2=/band k3=/cfor sink2b=0and sink3c=0, and where nand pare integers.

Then we have for Ex:


where C1=C1C3C5and C2=C2C3C5. Repeating this whole procedure for Eyand its boundary conditions and for Ezand its boundary conditions we get


and k1=/a. Substituting the expressions of k1, k2, k3 into the dispersion relation, we obtain all the possible frequencies of oscillation in this cooking chamber of a, b, cdimensions


Notice that k1 = km, k2 = kn, and k3 = kp. Each set of integers (n, m, p) define a mode of Efield. From this relation aforesaid, we see that frequency ω takes only particular values determined by m/a, n/b, and p/c. There are many combinations of (n, m, p) called modes and then corresponding kvalues and “mode” frequencies ωnmp. Each ωnmpis a mode of vibration of the electric field and of the Hfield (that we obtain below).

It is important to note that if any two of the integers m, n, pare zero, then the other corresponding two k1, k2, k3 are zero, and from the expressions for Ex, Ey, Ezwe see then that all three components of Eare zero. Hence, as a consequence all components of Hbecome zero since H=ξk×Eand no standing wave pattern is sustained in the cooking chamber. We define the vector wave number kwith components k1 = kn, k2 = km, and k3 = kp. The vector electric field must satisfy Maxwell’s equations and, in particular, the first Maxwell equation (Gauss Law in differential form) with ρƒ = 0. We must have εE=0. To apply divergence we construct ∂Ex/∂x, ∂Ey/∂y, ∂Ez/∂zwith the field components above, we obtain


The three terms on the left must sum up to zero. One way to have this zero is to ask for each term individually be zero, then we set k1C2+k2C4+k3C6=0and alsoC1=C3=C5=0. The only surviving constants are C2, C4, and C6 and the resulting field components are now


The amplitude of each of these waves is C2, C4, and C6. So, let us rename them as C2=E1, C4=E2, and C6=E3, we find that the last conditions can be written as [28, 29].


The expressions for the field components finally become


So that E1, E2, and E3 are the amplitudes of the respective components.

We now show, in Figure 8, a 3D plot of the Eycomponent for the mode n = 2, m = 4, p = 3, and a 3D plot of the Ezcomponent for the mode n = 3, m = 5, p = 2. It is immediately apparent that the number of maxima, minima, and nodes increases as the mode number (n, m, p) increases. Both plots show in the horizontal plane the projections of these maxima and minima. When thinking in the rectangular microwave cavity, this 2D plot represents the heating power at different spots at a z = constant plane (height in the microwave oven). This is just a very simplified picture of what the hot and cold spots are inside the 3D microwave chamber. The whole hot/cold distribution spots are the superpositions of many (n, m, p) electromagnetic standing wave patterns.

Figure 8.

Calculated and experimental electric field stationary wave patterns inside a rectangular cavity. (A) 3D plot of theEystationary pattern for the moden = 2,m = 1,p = 3, evaluated fromEq. (14)at an arbitraryzfixed value. (B) 3D plot of theEzstationary pattern for the moden = 2,m = 1,p = 3, evaluated fromEq. (15)at an arbitraryzfixed value. In both cases, the projection in thexyplane of the maxima, minima, and nodes is shown. (C) The experimental determination of the hot/cold spots in a rectangular chamber,a = 36 cm,b = 24 cm, andc = 26.5 cm. Note the alternating pattern of hot/cold spots in the stationary pattern (adapted from [30]).

The color code used in Figure 8(A) and (B) represents maxima (red) and minima (blue) of the electric field of the microwaves. Since the energy delivered to the sample goes as the square of the electric field, then red and blue extrema become hot spots and the nodes become cold spots and are located between the red and blue spots. In the 2D projection, the regions between the blue and red zones are the cold spots. Experimental measurements on the standing wave patterns in microwave ovens have been reported and nicely agree with the theoretical calculations [30, 31]. In Figure 8(C), we show one of those measurements carried out on three perpendicular planes, xz, yz, and xy. The agreement between our calculations, planar 2D plots, and the experimental hot/cold spots in Figure 8 is very satisfactory. The general pattern calculated and measured in the three planes consists of alternating maxima, minima, and nodes for each mode (n, m, p) as given by formulas (12)(15).

The results in Eqs. (12)(15) are very much like that one found for a plane wave [25, 26, 27, 28, 29]. Thus, for a given mode, a particular set (n, m, p), E0must be perpendicular to the vector k=/ax̂+/bŷ+/cẑ.

For cylindrical cavities, stationary electromagnetic wave patterns are also obtained. Cylindrical cavities are very frequently used in research and in industrial applications as we showed above in Figure 2(C) for decomposition of waste plastics into H2 and a set of fullerene solid compounds. Here, without any calculations, we show in Figure 9 the electric field stationary wave pattern that we simulated from the cylindrical solutions for the TM010 mode. It is shown as a manner of contrast with the stationary pattern that results in rectangular geometry.

Figure 9.

Simulation of the stationary pattern of the electric field inside a cylindrical resonant cavity, TM010 mode. (A) The electric field is concentric with minima close tor = 0, andE = 0 exactly atr = 0. The field is tangent to the metallic wall and very small atR = r. (B) A top view of the same electric field stationary pattern [19]. This stationary field configuration is established in cylindrical cavities, TM010 mode, used for research at low microwave powers to excite magnetic specimens.

Calculating the magnetic field, we start from our knowledge of Eand of the vector wave number k. From the third Maxwell equation, we have ×E=μHt=iωμH. For example, using the expressions for Ex, Ey, Ezjust found above,

we have


Since k2E3k3E2 is the xcomponent of k×E0, it is desirable to define a vector H0by H0=1ωμk×E0for, if we let its rectangular components be H1, H2, and H3, then we can write Hxas


Similarly, we find the other two components of Hto be




We see that Hx = 0 at x = 0 and x = a, that is, at the walls for which it is a normal component; similarly, Hyand Hzvanish at y = 0 and band z = 0 and c, respectively. Thus, the boundary conditions on Hhave been automatically satisfied once Ewas made to satisfy its own boundary conditions. Furthermore, it is easily verified that the two remaining Maxwell equations that we have not yet used are satisfied, that is, H=0and ×E=B/tOne needs to use kH0=0,as well as HxEzand the relation dispersion in its form kk=k02=ω2ν2=ω2με.Each component of Evaries as eiωt, while the components of Hare proportional to ieiωt=eiωt+12π.Thus, the electric and magnetic fields are not in phase in these standing waves but instead Hleads Eby 90° as shown in Figure 7. A given kcorresponds to a given mode, that is, a given set of integers m, n, pin k aforesaid. Now kEtell us that the vector E0 must be chosen to be perpendicular to k. However, there are two independent mutually perpendicular directions along which E0can be chosen and still be perpendicular to a k.

Thus, for each possible value of k, there are two possible independent directions of polarization of E0, so that there are two distinct modes for each allowed frequency given by ωnmp. This property is known as degeneracy and is a fundamental and important feature of electromagnetic standing waves. If a, b, care all different, then the various frequencies given by ωnmpwill generally be different. However, if there are simple relations among the dimensions, it is possible that different choices of the integers will give the same frequency so that we will also have degeneracy, but arising in a different manner. As an extreme example, consider a cube for which a = b = c, so that ωnmpreduces to ωυ2=πa2m2+n2+p2. Thus, all combinations of integers that have the same value of m2+n2+p2will have the same frequency and the modes will be degenerate.


5. The Poynting vector of the microwave fields inside the cooking chamber

Remembering that Srt=Ert×Hrt, taking the expressions of Eand HInside the cooking chamber, then we obtain Srt=EHk̂=ξυε/2E2k̂=ξυuk̂[25, 26, 27, 28, 29]. This result is the general one obtained in any electrodynamic circumstance, of course, and microwave ovens fulfill it. And the average Poynting vector is Sav=Power/Area=Energy/Areatime. What these expressions tell us is that microwave energy and microwave power inside the cooking chamber are traveling-moving, yes energy, and power, between the six walls in stationary wave patterns and in accord with the propagation vector kand carrying perpendicular to it the Eand Hfields. The microwave power deposited on a surface area of 1 mm2 is then Pav = Sav.A. Since Eand Hinside the microwave oven have nodes and anti-nodes, then Srtand consequently microwave power, Prthave nodes and anti-nodes at some positions along x, y, and z, and the heating is not uniform due to this standing wave feature of the microwave fields inside the cavity. Experimental results on the standing wave patterns have been reported and nicely agree with the theoretical calculations [30, 31]. Theoretically and experimentally standing micro-wave patterns are obtained. The reason of the rotating plate is to move in circular fashion the food to be heated and reach a more uniform microwave bathing on the food. Most of the time it is accomplished, but not always, as pizza fans report.

Now that we have a detailed treatment of the electromagnetic fields inside the cooking chamber, we want to develop some expressions for the Eand Hfields traveling on the waveguide from the magnetron toward the resonant cavity.


6. The waveguide in microwave heating systems, TE and TM modes

In research and technologically bound situations, the resonant cavities we saw above are feed with microwaves by means of waveguides connecting the source to the microwave cavity, see Figures 2 and 4. We can think of a waveguide as constructed from a cavity by taking the ± zwalls to infinity; then, the trapped stationary waves in the cavity can now travel indefinitely toward ± infinity as plane waves. As soon as we start taking the ±zwalls to infinity, we start liberating boundary conditions and in that dimension we are allowing free traveling waves. The remaining walls at x = 0, a, and y = 0, bcontinue limiting our bouncing waves along these dimensions. We will continue taking the bounding surfaces as perfect conductors. A question to ask at this point is; Is it possible to transfer electromagnetic energy along a waveguide, that is, a tube with open ends? From everyday experience, we already know that this is possible from the simple fact that we can see through long straight pipes. So, the answer is yes, but: How is this carried out? Solutions to the wave equations have the answer, but first we review quickly boundary conditions on perfect conductors.


7. Boundary conditions at the surface of a perfect conductor

We recall that a perfect conductoris one for which σ → ∞, more precisely, one for which the ratio Q = εω/σ → 0. Q1/501for common metals even at very high frequencies so that Q = 0 should be a good first approximation for metallic boundaries. Plane waves traveling freely along the zdirection take the form Exyexpiωtkz, where the Exypart has to be found but we already know that fulfills boundary conditions at the metallic walls. We remember that δ = (2/μσω)1/2 for a good conductor so that δ → 0 as σ → ∞. Therefore, the electric field is zero at any point in a perfect conductor since the skin depth is zero. Since the tangential components of Eare always continuous, we see that Etang=0just outside of the surface. In other words, Ehas no tangential component at the surface of a perfect conductor so that Emust be normal to the surface [25, 26, 27, 28, 29]. Binside the conductor is B=k/ωk×Eτso that Bwill also be transverse. Consequently, the transverse component of Binside will also vanish as σ → ∞. Since Bhas no normal component, the boundary condition B1n = B2nimplies that Bnorm=0just outside the conductor. Thus, at the surface of a perfect conductor and outside of it, Bhas no normal component; that is, it must be tangential to the surface. We see that allof the field vectors will be zero inside a perfect conductor. This simplifies greatly the general boundary conditions. To repeat: At the surface of a perfect conductor, Eis normal to the surface and Bis tangential to the surface. To put it another way, Ehas no tangential component while Bhas no normal component.


8. Propagation characteristics of waveguides

Figure 10 shows a waveguide that extends indefinitely in the zdirection and of arbitrary and constant cross section in the xyplane. We take the boundary walls as perfect conductors and the interior of the cavity is filled with a linear nonconducting medium described by μ0 and ɛ. If ψis any component of Eor B, we know that it satisfies the scalar wave equation 2ψ1υ22ψt2=0where υ²= l/μɛand υwould be the speed of a plane wave in the medium. Again, by separation of variables we easily find ψxyzt=ψ0xyeikgzωt.

Figure 10.

A waveguide made of a perfect conductor with arbitrary and constant cross section. A set of propagation vectorsk1,k2,k3, etc., are shown to impinge on different points on the metallic walls and reflect back following Snell law. Transmission is not depicted since perfect conducting walls are considered, and hence, the skin depth tends to zero, which implies zero transmission.

We note that this is nota plane wave since the amplitude ψ0is not a constant but depends on xand y, the cross section [28, 29]. The quantity kgis the guide propagation constant, or simply the kz constant of separation of the Z(z) componentof the whole solution and can be written as kg = 2π/λghere λgis the guide wavelength, that is, the spatial period along the guide, the zaxis.

Continuing with the separation of variables now for the xand yvariables, we obtain again a Helmholtz equation in 2D 2ψ0x2+2ψ0y2+kc2ψ0=0, where kc2=k02kg2and k0=ωυ=2πλ0. Writing kc=2πλcwe obtain a wavelength relation 1λc2=1λ021λg2. Therefore, we have found for a waveguide that we will get wave propagation onlyif k0 > kc, or λ0 < λc. For this reason, λcis called the cutoff wavelength. It is very common to state this result in terms of a cutoff frequency ωcdefined by kc=ωcυso that kg2 can also be written as kg2=1υ2ω2ωc2.Then, wave propagation is possible only if ω > ω2, that is, if the applied frequency is greater than the cutoff frequency.


9. Rectangular guide

This guide has a rectangular cross section of sides aand b, which we take to be located in the xyplane. It is relevant to mention that for either type of mode, TE or TM, we have to solve an Helmholtz equation and apply boundary conditions as aforesaid. We continue using separation of variables and write ψ0(x, у) = X(x)Y(y); then, the same arguments as used in resonant cavity section above lead us to the separated equations


so that, (d2X/d x2) + k12X = 0 and (d2Y/d y2) + k22Y = 0 the separation constants have been selected with minus sign since the solutions should be periodic. Hence, k12+k22=kc2is the dispersion relation in terms of the constants of separation for this 2D differential equation. Solving these in terms of sine and cosine functions, we find that ψ0xy=C1sink1x+C2cosk1xC3sink2y+C4cosk2y, where the C’s are constants of integration. This expression for ψ0contains a total of four constants.

Let us calculate for ТЕ modes. Here, we set Ez=0and write


With Ez=0, and with ×E=B/∂t, we find that when we substitute Hzinto Exand Eycoming from ×Eand after some algebra [28].


From the boundary conditions Exу=0=0and Exу=b=0and similarly, from the boundary conditions that Eyx=0must satisfy at x = 0 and x = a, then evaluating first for the zero values of xand у, we get


Notice that we have here established a 2D homogeneous Sturm-Liuville problem and we expect to obtain as solutions eigenvalues and eigenfunctions. From the aforesaid boundary conditions, we must have C1 = 0 and C3 = 0. Therefore, at this stage, Ex, Ey, Hzhave simplified to


We still have boundary conditions to satisfy at the two remaining faces. We see that the requirement Exxb=0leads to sin k2b = 0 so that k2b = where nis an integer. Similarly, Eyay=0gives the condition that k1a = with man integer. Thus, we have found the eigenvalues k1=aand k2=b, these are the eigenvalues of the solution. So that k12+k22=kc2shows that the allowed values of kc2 are kc2=kcmn2π2ma2+nb2. The cutoff wavelengths and frequencies can now be found by using kc2 above into our λc2 and ωc2 equations. The corresponding guide propagation constants are


The only quantity left undetermined is the arbitrary amplitude C2C4 of Hz. If we set C2C4 = H0, then we find that the amplitudes of a general ТЕ mode in a rectangular guide are as follows:


where kcand kgare as above. Multiplying each of these amplitude factors by the wave propagation term we get, for example, for the Exfield


Since i=ei1/2π, the exponential factor can be written expikgzωt+12π, which shows that Exleads Hzin time by 90°. Similarly, Hxand Hylead Hzby 90° while Eylag Hzby this same amount.

We now particularize to the simplest case which is also the most used. The TE10mode, we set m = 1 and n = 0 and we particularize the above equations for these particular values of mand n, we show now without calculations that:


The field amplitudes are


While Ex=Ez=0and Hy=0. Inserting these amplitudes into the complete Eand Hfield expressions above and taking the real parts of the resulting expressions, we find the only nonzero field components to be


We see that the values of the Eyare independent of y; hence, the electric field lines are straight lines with constant magnitude at a given value of xbut with a magnitude that does vary with xand is a maximum at the center where x = (a/2). The lines of Hxare straight with their maximum value at the center as well. The value of Hz, on the other hand, is zero in the center as has opposite signs on the two sides of the center.

TM modes. In this case, we set Hz=0and set Ezequal to the expression for ψ0 given above; hence, ×E=B/tis again applicable. This case is actually simpler because Ezcan be a tangential component and must vanish for x=0and aand y=and b. We have again an homogeneous Sturm-Liouville problem and expect eigenvalues and eigenfunctions. Proceeding in the same manner as before, we see that we must now have C2 = C4 = 0, while k1, k2, and kcare given, again as above. Thus, the TE and TM modes of a rectangular waveguide have the same set of cutoff wavelengths, the same eigenfrequencies, and cutoff frequencies; the field configurations can be expected to be different however. Setting C1C3 = E0, we find that (21) gives the starting point for the TM calculation to be Ez=E0sinmπxasinnπyb. We now use this Ezto calculate the rest of the field amplitudes following the above procedure. We note that m = n = 0 makes Ezand then all the other field components, zero; thus, there is no TM00 mode. Furthermore, if m = 0 or n = 0, Ez=0and all of the fields are zero. Thus, it is not possible to have a TMm0 or TM0nmode, in contrast to the TE case.

We have now calculated in detail the electric and magnetic fields that propagate in rectangular waveguides as the ones shown in Figures 1(B),2(A), and 3(C). The field patterns are stationary wave patterns in the xydirection and traveling waves along the zdirection as given by Eqs. (30)(36).

Let us proceed now to the last of the physical components of a microwave heater, the very source of 1000 Watts microwaves.


10. Radiation of accelerated point charges in klystrons and magnetrons

Klystrons and magnetrons produce microwaves that carry power; typically, klystrons are used when little power is needed, from 1 watt to milliwatts and even microwatts. Magnetrons are used in higher-power applications, 1000 Watts, or more. Clearly, magnetrons are better suited for microwave heating. Both rely on electrons being accelerated (these electrons are labeled em) and made move in periodic trajectories inside cylindrical chambers. Both devices are shown in Figure 11. Notice the motion of electrons in them. Straight trajectories in klystrons, w(t), Figure 11(A), and curved trajectories, Ç(t), in magnetrons, see Figure 11(B). To expose relevant physics of accelerated electrons, em, produced in klystrons and Magnetrons is to describe the Eradand Bradmicrowave radiation fields produced inside their structures and from these, the Poynting vector, Srt=Erad×Hradthat describes flux of such energy through space.

Figure 11.

Microwave sources, reflex klystron, and magnetron. (A) The basics of a klystron that produces accelerated electrons through an alternating electric potential difference ΔV12(ωt), theseemtravel the distanced; then, the acceleration is reversed,emtravel to the left now, and this repeats thousands of times at a GHz frequency. (B) In a magnetron hot electrons ejected from a central cathode, travel in circular-wavy trajectories inside a cylinder due to the Lorentz forceF=eE+υtxBêz, whereυt=çrt/t=ç˙rtis the velocity of theemelectrons,Eis the total electric field due to the perimetral charges (+, −), (+, −), (+, −), and the central −Qcharge.Bis a constant magnetic field (from a magnet) applied along theêzaxis. These two forces combined produce the curved-wave trajectoryçrt. (C) A complete diagram of the magnetron structure with the constant magnetic fieldBêz, the charge distribution (proper of magnetrons) that produces a totalEfield and the curved electron trajectoriesçr,t.

A microwave power, Prad=SArea, comes with this traveling energy. For klystrons, the accelerated electrons travel along straight lines, inside vacuum tubes, back and forth, due to voltage differences, ΔV12 ≥ 860 Volts, applied at the ends of the cylindrical tube (chamber), see Figure 11(A). So acceleration is linear and is azt=dvzdt=d2wztdt2=w¨zt, in which a=w¨is parallel to vand parallel to the tube axis, z,see Figure 11(A), and the acting force producing such acceleration is F=eE=eV12.For magnetrons, the trajectories of the accelerated electrons are wavy circular with average radius a ≤ r ≤ b, as shown in Figure 11(B) and (C). The wavy çrttrajectories of the accelerated electrons in magnetrons are the effect of a combined magnetic force Fmag=qç˙txBêzand the total electric force between these electrons and the charges located in pairs along the bradius, σ ± and − Qat the cathode. What we have now is, charged particles, em, moving along trajectories, wzt, in klystrons, and çzt, in magnetrons, both have velocities ẇzt,ç̇zt, and accelerations, w¨zt, ç¨zt. Charged particles in motion produce electric potentials and electromagnetic fields just as static charges do, except that here we have to calculate retarded potentials and retarded fields. For single charged particles, the resulting potentials are the well-known Lienard-Wiechert potentials [26, 27, 28, 29, 32]. We present them here now. We take an accelerated electron moving in a general trajectory given by çrt, or w(r,t) ≡ position of qat time t. Now, Vrt=14πϵ0ρrtr𝓇dτgives the electric potential at rat time t(Figure 12) [2, 26, 27, 28, 29].

Figure 12.

A moving charged particle following a trajectoryw(tr).rwtris the distance the “radiated field” from the moving electron must travel, and (t − tr) is the time it takes to make the trip, we shall callwtrtheretarded positionof the charge,ris the vector distance from the retarded position to the point the radiated electromagnetic wave arrived “to us” (now, at our timet2), which is atr, clearlyr=rwtr.

The retarded integration is not trivial, and the retardation in the q/𝓇term in the potential aforesaid throws in a factor q1rv/c, where vis the velocity of the charge at the retarded time, and ris the vector from the retarded position to the point rwhere we are standing and measuring [24, 26, 27, 28, 29, 32]. Then, ρrtrdτ=q1rv/c. It follows, then, that Vrt=14πϵ0qcrcrv. Meanwhile, since the current density of a rigid object is J=ρυ, we also have Art=μ04πρrtrvtrrdτ=μ04πv𝓇ρrtrdτ.

Or Art=μ04πqcv𝓇c𝓇v=vc2Vrt. These are the famous Lienard-Wiechert potentials for a moving point charge. By using E=VA/tand B=×A, the corresponding fields are evaluated. It seems to us that only two authors, Jefimenko and Griffiths, give detailed derivation of these fields. The differential operations should be carried out with great care as these authors do, and we refer to those calculations and just take their results here:


where u=cr̂v, and, very importantly, Evelα1/𝓇2and Eaccelα𝓇. And the magnetic field is


Bfollows the same time dependence as E.The first term in E(r, t) is called the velocity field, and the triple cross-product is called the acceleration field. With these potentials and Jefimenko fields, we are now in the position to describe more quantitatively the radiation produced in klystrons and magnetrons. Hence, the accelerated electrons, em, inside these devices produce Vand Apotentials and Eand Bfields. In klystrons, the radiation fields are, as they are, captured by the mouth of a waveguide and send to a resonant cavity in which they produce standing patterns of stationary microwaves for their use.

Finally, in magnetrons, the perimetral charges, σ ± experience Lorentz forces due to these Jefimenko fields, F=qEJ+ç·txBJ; hence, these charges move inside the conducting core behind radius b and around the cylindrical cavities, see Figure 9(C), and cut from the solid metal (usually copper). These moving charges, in turn, produce their own retarded potentials, Vσ(wt), Aσwtand fields, Eσwtand Bσwt. We end up with total fields, Etwtand Btwt, inside the magnetron space, including the cylindrical cavities (eight of them most of the time) behind the radius b. These cylindrical cavities are there, precisely, to trap microwaves in them and due to their perfect conducting walls, Etotwtand Btotwtreflected from them with almost no losses, and so these cavities sustain stationary microwave patterns of cylindrical geometry. The same process takes place in the eight cylindrical cavities distributed along the perimeter of radius b. With a simple wire antenna, microwaves are taken out of these cavities and sent to the entrance of the waveguide; then, these microwaves travel the short distance inside the waveguide and end up in the cooking chamber of the microwave oven; hence, our coffee absorbs so much the energy of these microwaves; the electric dipoles in the water vibrate and jiggle; frenetically, at 2.45 GHz, in a few seconds our coffee is hot and ready to drink.

11. Conclusions

In this chapter, detailed electrodynamic descriptions of the fundamental workings of microwave heating devices were given. We analyzed one by one the principal components of a microwave heater; the cooking chamber, the waveguide, and the microwave sources, either klystron or magnetron. The boundary conditions at the walls of the resonant cavity and at the interface between air and the surface of the food were stressed. It was shown how relevant the boundary conditions are to understand how the microwaves penetrate the nonconducting, electric polarizable specimen. In addition to microwave food, we mentioned the important application of microwaving waste plastics to obtain a good H2 quantity that could be used as a clean energy source for other machines and so contributing to a cleaner planet. We did use Maxwell equations to obtain trapped stationary microwaves in the resonant cavity and traveling waves in the waveguides. We showed 3D plots of a few lower Ex, Ey, Ezmodes calculated directly from the solutions obtained here and compared the general trend with experimentally obtained microwave heated patters inside rectangular cavities. The agreement is very good. We did simulate a single electro-magnetic field mode inside a cylindrical cavity in order to contrast with the stationary patterns obtained in rectangular cavities. The radiation processes in klystrons and magnetrons were stated in terms of the accelerated electrons produced. Then, using the Lienard-Wiechert potentials produced by these electrons, the Jefimenko fields were written. When all these are put together, we understand how a meal or a waste plastic, or an industrial sample, is microwave heated.


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Written By

Rafael Zamorano Ulloa

Submitted: December 9th, 2020 Reviewed: March 17th, 2021 Published: March 16th, 2022