The oven that cooks in spots

The ripples that answer at half the shaking found a standing pattern on the surface of water, chosen from among those the container allows. Only some notes fit gave the simplest version of the idea: a wave confined between walls can have only certain wavelengths, and each has a pattern of nodes and antinodes. How many ways there are to vibrate counted the patterns in a box, and it did so to explain a cavity’s thermal radiation, by far the most consequential such count in physics.

The same count, done for a box 29 cm wide, turns out to explain why a chicken leg comes out with one scorched end and one frozen. A domestic microwave oven is a metal box with a source of electromagnetic waves of frequency 2.45 GHz, feeding it through a waveguide. Inside, the waves have a wavelength of 12.2 cm, about half the size of the box. It seems natural to think of the waves filling the box more or less evenly, like light in a sealed white room. That picture would be good for visible light, which has a wavelength a million times smaller than the box. At a wavelength comparable with the box it is entirely wrong.

How many ways a metal box can ring #

The electric field in a rectangular metal cavity has to have zero tangential component at each wall. That fixes the modes, each labelled with three integers $m$, $n$, $p$ counting half-wavelengths in each direction. The frequencies are

$$f_{mnp} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{p}{d}\right)^2},$$

with at most one index zero, and most combinations have two distinct polarisations.

Counting those one at a time, for a box of the usual kitchen size, gives 71 modes below 2.45 GHz. The smooth line in the figure is the leading term for large boxes, due to Hermann Weyl:

$$N(f) = \frac{8\pi V f^3}{3c^3}.$$

Everything about the box’s shape has gone from this; only its volume is left. That is the same count that gives the number of modes in the curve that would not come down, with the same factor, and it is a remarkably accurate approximation even for such a small box, considering it is intended for very large numbers. The stair steps of the true count weave around the curve. Near 2.45 GHz they are a little below it, with corrections from the surface area. The important thing is the size. At visible frequencies, the number of modes in the box would be about $10^{17}$, and the field is a sum of so many that it might as well be uniform. Here the number is less than a hundred.

Only those near the source’s frequency can be excited with any strength. The oven’s magnetron is a free-running oscillator, so its frequency is not stable: it varies by tens of megahertz with load and temperature, and over the band it occupies it can drive whatever modes lie nearby.

A handful of patterns #

Between 2.4 and 2.5 GHz, the box has seven modes when counted by distinct triples, some of them degenerate since a square base has two equal sides. So the field at a given moment is a sum of only a few standing waves. Each one is a pattern of nodes. It is impossible for any standing wave in a bounded box to avoid having them, since that is what a standing wave is, and the sum of a few of them has its own nodal structure. Some parts of the volume end up with more energy density and others with less. Those which sit near the nodes of everything, however, will receive almost no heat at all, and this is what is seen in a frozen pizza’s cold patch.

The figure shows the field squared, which is what produces heat, on a horizontal plane halfway up an empty oven for three of those modes averaged together. It has a clear grid of patches, about ten centimetres from one bright patch to the next. That pattern is an empty-oven calculation. Food changes it, because a dielectric with a large permittivity pulls the fields into itself and shifts the modes. All the same, experiments with fax paper, chocolate and thermal cameras routinely show such grids, of about the right size, in a stationary load.

This makes an appealing experiment for a school laboratory. Place a slab of chocolate or cheese on a stationary plate for a short time, and measure the distance between melted spots. Twice this should be the wavelength, 12.2 cm, and multiplying by 2.45 GHz gives the speed of light, within a few per cent. The physics is not as simple as this, for spots from several modes need not be exactly half a free-space wavelength apart: a mode’s wavelength along a wall is $2a/m$, which differs from a free-space wavelength by a factor. It also ignores the shift due to the load itself, and those effects together roughly account for the scatter that students find. The node that is not standing still discusses another case in which a node’s position does not correspond to one’s first expectation.

Designers try to reduce the problem in two ways. A rotating metal stirrer at the top of the oven changes the boundary conditions continually, so the mode frequencies shift and different modes are excited in turn. The other way is to move the food.

What a turntable can and cannot do #

A turntable moves each piece of food around a circle, so it averages the field over that circle. The averaging is perfect along the circle: all the points at one radius have the same time-averaged field, and the hot and cold spots along it disappear. The limit of what it can do is equally clear from the figure, since nothing moves any point in or out along a radius. So the pattern across a radius is retained, in smoothed form, as a set of rings. That is why food comes out with concentric rings of hot and cold, and why people have the experience of stirring it halfway through to help.

The centre of the plate deserves a special mention. A point on the axis stays put, so it has no averaging at all. For the modes drawn, the centre of the oven has a particular value, which is low in this example: the plate’s centre stays cold in many ovens. Many people have noticed that a dish of leftovers tends to be coldest at the middle. There are two other reasons for this. The penetration depth of the waves means that the outside of a large piece of food is heated first, and the centre relies on conduction. Curved surfaces can focus waves inside a dielectric, however, and a sphere of the right size can get a hot core, which is why an egg in its shell can explode, and why a drink can superheat. A dielectric sphere is an inefficient lens with a focal point inside itself, and an unstirred liquid has no way of passing heat outward before it boils.

The last step is conduction, which has to level out what remains over distances of centimetres. Thermal diffusivity in food is about that of water, or $1.4 \times 10^{-7}\ \text{m}^2/\text{s}$, and the equation that only runs forwards gives a levelling time for 2 cm of about 700 s. So the packet’s request for “standing time” of a minute or two only partially finishes the job, and a bigger stirring from the hand does more.

Why this frequency, and not a resonant one #

It is often said that microwaves work because 2.45 GHz is the resonant frequency of water. That is false, and the error has real consequences. The frequency that gets an answer and the constant that depends on how fast it is asked both note that water at this frequency is far from any sharp resonance. Water’s isolated-molecule rotational lines lie at much higher frequencies in the gas phase, with the lowest at 22 GHz, and liquid water’s rotational motion is a broad relaxation centred near 20 GHz at room temperature. Water at 2.45 GHz sits well down on the low-frequency shoulder of that broad curve.

That is good news for a cook, not bad. The absorption at resonance would be so strong that the waves would stop in the first millimetre, cooking the surface and leaving the inside cold, as a laser or a grill does. Going down from there buys depth. The band at 2.4–2.5 GHz was assigned in about 1947 as one of the international industrial, scientific and medical bands, and it was also a frequency at which tubes could be made cheaply. The other domestic-oven band at 915 MHz, used in large industrial ovens, reaches rather further into thick food.

Why a thawing dinner burns in places #

The physics of absorption in liquid water is rather simple. Peter Debye’s 1913 model describes a polar molecule in a viscous medium, with a single relaxation time $\tau$, about 8 ps at room temperature. The permittivity is

$$\varepsilon(\omega) = \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1 + i\omega\tau}.$$

Its imaginary part, which is the loss, is largest at $\omega\tau = 1$. At 2.45 GHz, $\omega\tau$ is well below 1, so the loss rises in proportion with $\tau$. Warming water lowers the viscosity and thus shortens $\tau$. So as water warms, it absorbs less, and the depth grows, from 1.9 cm at 25 °C to 6.7 cm at 90 °C. Hot water is more transparent to microwaves than cold water is. That is a kind of negative feedback, and a helpful one for liquids.

Ice has the opposite and very dangerous character. In ice the molecules are held in a lattice, and they can reorient only through defects, with a relaxation time that is many thousands of times longer, about 20 microseconds near the melting point. The relaxation has moved far below 2.45 GHz, and so ice’s loss at this frequency is thousands of times less than that of liquid water, with a penetration depth of several metres. Frozen food is nearly transparent to the oven, while any region that has melted absorbs strongly. So one spot begins to melt, and then it absorbs far faster than the ice around it, then heats further, and passes on its heat by slow conduction only. That is thermal runaway, in a frozen meal, and it produces pockets that are boiling next to ones that remain frozen.

Defrost settings work by turning the magnetron off for most of each cycle, in pulses of some seconds. A magnetron cannot usefully run at a lower continuous power, but a pulsed one allows conduction time between bursts, during which heat can move from the melted pockets into their frozen neighbours. Solvent-free fatty foods behave differently in general, and salt raises the loss greatly, by adding ionic conduction, which reduces penetration depth and heats the surface. Salty soup heats near its top surface for that reason.

What the frequency buys a large object #

The choice of frequency also sets the size of what can be cooked at all. Power falls by a factor of $e$ over one penetration depth, so a joint of meat 20 cm thick at 2.45 GHz receives very little in its centre: a depth of a couple of centimetres leaves five depths of decay to the middle if heated from outside, which is a factor of about 150, and conduction has to do the rest. Industrial tunnel ovens at 915 MHz reach about three times further for this reason. Their wavelength in air is 33 cm, so the cavities are bigger and the hardware costs more. A lower frequency would penetrate further still, and yet would need a larger box and heavier components, and it would be no use in a kitchen. So 2.45 GHz sits at a compromise among kitchen size, cost and depth.

The same comparison sets the difference between a small sample and a large one. A piece of food thinner than its penetration depth absorbs little of what passes through it, and most of the wave goes on through to reflect from the walls and return. A thick piece absorbs nearly everything that enters. In between, the absorption has resonances: at particular thicknesses, comparable with a wavelength inside the food, which is about 1.4 cm in water since its index is about 9, the waves reflected from the front and back surfaces add up, so that a slab of just the right thickness takes in much more than one twice as thick. Microwave heating can thus show very large differences between objects of slightly different sizes.

The oven as a resonant load #

An oven also has to keep its source working. A magnetron driving an empty oven would have nowhere to dump its power, and the field would build up to high values, since the walls absorb very little. This reflected power is returned to the magnetron and can damage it, which is why ovens warn against being run empty. The power absorbed by a load depends on how well it couples, and thus on how the load’s permittivity changes the modes near the source’s frequency. Since a small load shifts the modes less, it often heats more slowly per unit mass than would be expected, and can absorb only a small fraction of the available power. The field at which no electron reaches the anode deals with how a magnetron works, and the mismatch no network can remove with why any such load has a matching limit.

The door also relies on a simple piece of physics discussed in the pipe that will not carry a low note: a metal mesh with holes of about a millimetre is far below the cut-off for a wave of 12 cm, and so nothing at that frequency propagates through the hole. Visible light, with a much shorter wavelength, gets through easily, allowing the cook to see the food. A choke around the door’s edge provides a quarter-wave trap to keep leakage low. The whole appliance is therefore a case of designing with a wave’s wavelength in mind at each step, and everywhere the design choices come down to a comparison of a length with 12 cm.

The solid-state ovens now becoming available replace the magnetron with a transistor source whose frequency can be set. By choosing a frequency at which a particular mode is excited, a solid-state source can decide where the hot patches go, moving them across the food electronically. It can also measure reflected power and use that to infer how the load is changing. This is a tacit acknowledgement that a cavity’s modes are the central fact about the device.

What these calculations leave out #

The mode calculation uses an empty rectangular box with perfect walls. A real oven has a feed port, a stirrer and rounded corners, and above all a load, which changes everything by concentrating the field inside itself and shifting the frequencies. The hot-spot pattern drawn is for three of the modes in a chosen plane with equal weights, and a real oven would have some other mix depending on how the feed couples to each. So the figure shows the kind of pattern produced and not any specific oven’s map. The turntable calculation has the same limits and assumes a thin load following a circle. No conduction or stirring is included.

The Debye model uses a single relaxation time and only describes pure water. Real food contains dissolved salts, fats and bound water, and these have their own absorption. Ice at temperatures near melting also holds a liquid-like surface layer, whose absorption matters in a real thaw, and the precise depth quoted for ice depends strongly on temperature and on impurities. The penetration depth formula applies to a plane wave in a thick layer, whereas a real piece of food is comparable in size with the wavelength, and so internal focusing and standing waves within it make its heating far less uniform than a simple exponential.

Still open: whether the heating can ever be made uniform #

Heating evenly in a resonant cavity has remained a surprisingly hard problem for almost eighty years, even with extensive computation. A full electromagnetic and thermal simulation requires solving Maxwell’s equations with changing dielectric properties at every time step, and these properties depend on the temperature in a way that is only well measured for simple foods. Reports of optimum strategies differ widely. Industrial processes, in which a tunnel oven or a precisely controlled single mode can be used, can do much better than a domestic oven, and researchers are experimenting with sequences of solid-state frequencies to synthesise a desired heat pattern from the modes available. Whether this can provide a reliable, safe sterilisation of ready meals with an oven that a consumer can afford, where cold spots are a real health problem as they allow pathogens to survive, is still an open matter.

There is also a long-standing and more controversial claim in chemistry, that microwave heating speeds up some reactions beyond what temperature alone could explain, by a non-thermal effect. Careful measurements have found most of these to be due to hot spots or to bad temperature measurement, a finding very much in the spirit of what is drawn here. Some claims for special effects at interfaces or on solid catalysts remain under discussion.

A kitchen oven is a box that holds 71 modes below 2.45 GHz and only a few in its magnetron’s band, so the field is a sum of a few standing waves with real nodes several centimetres apart. A turntable smears the pattern into rings and leaves the centre alone. The frequency was chosen for depth and not for resonance, giving 1.9 cm in water, and ice, whose molecules cannot turn, absorbs thousands of times less. A melted patch then runs away from its frozen neighbours.