Physics · Electricity and magnetism · Grade 11

Microwaves and Wi-Fi

How can chocolate in a microwave oven be used to measure the speed of light?

The situation

The microwave oven in the kitchen, the Wi-Fi router in the living room, the phone in your pocket — all work with electromagnetic waves. These waves are invisible, but they can be measured: take the rotating plate out of the oven and put chocolate inside, and it melts not evenly but in spots. By measuring the distance between the spots with a ruler, you can find the speed of light! How — and why does 5 GHz Wi-Fi not reach the bedroom?

Topic

Electromagnetic waves. Wi-Fi, mobile networks and the microwave oven

Key ideas

Electromagnetic wave
Standing wave
c = λf

Formulas

c = λ f

Wave speed = wavelength × frequency. In a vacuum (and almost so in air) $c = 299 792 458$ m/s.

d = \frac{λ}{2} \Rightarrow c = 2 d f

In a standing wave, the distance between neighbouring antinodes (the hottest spots) is half a wavelength.

E = E_{0} \sin \frac{m π x}{a} \sin \frac{n π z}{b} \cos ω t

A standing wave in the oven — mode $(m, n)$ . The heating power is proportional to $E^{2}$ .

f_{m n} = \frac{c}{2} \sqrt{{(\frac{m}{a})}^{2} + {(\frac{n}{b})}^{2}}

Mode frequency: an oven of size $a \times b$ “rings” in the mode that matches the magnetron frequency of 2.45 GHz.

L = 20 \lg \frac{4 π d f}{c} + \sum L_{w a l l}

Attenuation of the Wi-Fi signal (dB): spreading in free space and each wall along the way.

Theory

Electromagnetic wave

Electric and magnetic fields oscillate perpendicular to each other and propagate through space at speed $c$ — this is an electromagnetic wave. Radio waves, microwaves, light, X-rays and gamma rays are of the same nature and differ only in wavelength. In the “Spectrum” mode look at the scale from 1 pm to 1 km: each division differs by a factor of 10.

A standing wave in the oven

The metal walls of the oven reflect the wave; the forward and reflected waves add up to form a standing wave. At the antinodes $E$ is greatest and the chocolate melts there, while at the nodes ( $E = 0$ ) it stays solid. That is why an oven has a rotating plate. If the wave travelled only along $x$ , the spacing of the spots would be exactly $λ / 2$ . In reality the mode is two-dimensional: $1 / d_{x}^{2} + 1 / d_{z}^{2} = {(2 / λ)}^{2}$ , so $d_{x} = a / m$ is slightly larger. In a 20 L oven the error is ≈ 2 %, and in the (5, 2) mode of a 25 L oven ≈ 7 % — check it.

Wi-Fi and walls

Signal strength is measured in decibel-milliwatts (dBm): −3 dB means half the power, −10 dB means 10 times less. In free space the attenuation is $20 \lg (4 π d f / c)$ — if the frequency doubles, another 6 dB is lost. In addition, the wave is absorbed in a concrete wall, and more strongly at higher frequency: 16 cm of concrete “eats” ≈ 9 dB at 2.4 GHz and ≈ 15 dB at 5 GHz. 5 GHz is faster (a wider channel), but reaches less far and worse behind a wall. For a phone ≈ −67 dBm is a good signal, and below −80 dBm there is almost no connection.

Maxwell's theory and the plane wave

A changing magnetic field creates a vortex electric field — its lines do not start on charges but are closed: $\oint \vec{E} \cdot d \vec{l} = - d Φ_{B} / d t$ (this is the induced emf). A changing electric field (the displacement current $ε_{0} \partial \vec{E} / \partial t$ ), like a current, creates a vortex magnetic field; each gives rise to the other, and in the wave they are passed on as a chain; the full form of the equations is in the “Radiation from a dipole antenna” experiment. In free space they lead to the wave equation: $\partial^{2} E / \partial x^{2} = c^{- 2} \partial^{2} E / \partial t^{2}$ , $c = 1 / \sqrt{ε_{0} μ_{0}}$ . So an electromagnetic effect is not transmitted instantly but at speed $c$ : from the router to a phone 10 m away the signal arrives in ≈ 33 ns. The simplest solution is the plane monochromatic wave $E = E_{0} \cos (ω t - k x)$ : $\vec{E} ⟂ \vec{B} ⟂ \vec{v}$ , $B = E / c$ , and the direction of $\vec{E}$ is the polarisation. The energy density is $w = ε_{0} E^{2}$ (half in the electric and half in the magnetic field), the energy flow is the Poynting vector $\vec{S} = \vec{E} \times \vec{B} / μ_{0}$ , and the intensity is $I = \frac{1}{2} c ε_{0} E_{0}^{2}$ — in the “Wi-Fi” results window see $I$ and $E_{0}$ at the phone's position. The wave also carries momentum ( $p = W / c$ ) and exerts pressure on an obstacle (Lebedev's experiment — in the “Photon momentum and the Compton effect” experiment). For a plane wave $E^{2} - c^{2} B^{2} = 0$ and $\vec{E} \cdot \vec{B} = 0$ ; these two quantities are invariants — they are zero in every frame of reference, which means that light remains a wave for every observer. In 1888 Hertz reflected a wave from a zinc sheet on the wall of a lecture hall, produced a standing wave, and found $λ$ by measuring the distance between the nodes — you repeat just this in the oven.

In a medium and at a metal boundary

In matter, constitutive equations are added to Maxwell's equations: $\vec{D} = ε ε_{0} \vec{E}$ , $\vec{B} = μ μ_{0} \vec{H}$ , $\vec{j} = σ \vec{E}$ . In an alternating field conductivity becomes the imaginary part of the permittivity: $ε = ε^{'} - j ε^{''}$ , $ε^{''} = σ / (ω ε_{0})$ , and the refractive index is also complex: $n - j κ = \sqrt{ε}$ . $n$ slows the wave ( $v = c / n$ ), while $κ$ absorbs it: the amplitude falls as $e^{- k_{0} κ x}$ . The walls were calculated this way — in the results window see $n - j κ$ for concrete. The law of conservation of energy in a medium (Poynting's theorem): the flow into a volume = the increase in field energy + heat, the average per unit volume being $\frac{1}{2} ω ε_{0} ε^{''} E_{0}^{2}$ . It is this term that heats both the chocolate and the wall. Boundary conditions: at the boundary of two media the tangential component of $\vec{E}$ is continuous, and in an ideal metal it is zero — so there is always a node at the oven walls, the wave is reflected, and only modes with frequencies $f_{m n}$ “fit”. The magnetron sends the wave into the oven through a rectangular waveguide (a metal tube) — behind the mica cover in the wall. Inside the tube the wave is no longer purely transverse: in the fundamental $T E_{10}$ mode the magnetic field has a longitudinal component. In free space, though, an electromagnetic wave is always transverse. The link between dispersion and absorption (Kramers–Kronig) is in the “Laser, water and optical fibre” and “Scattering of light” experiments.

Model and simplifications

One fundamental mode is taken in the oven (a real oven is multimode), and its size is chosen so that the mode falls exactly at 2.45 GHz. The heating rate at an antinode is 1.2 K/s, cocoa butter melts at 34 °C, and the heat of fusion and the thermal conductivity of chocolate are taken into account. Wi-Fi: only the direct path is considered; reflection from walls and diffraction at doors are ignored, and the properties of the wall materials are approximate. The colour map is the signal at a height of 1 m. To make the wave motion visible it is slowed down by billions of times; the walls are cut at a height of 1.1 m.

Did you know?

According to the story, around 1945 the American engineer Percy Spencer, while working near a radar magnetron, noticed that the chocolate in his pocket had melted. After that he patented a way of heating food with microwaves.
A microwave oven and 2.4 GHz Wi-Fi work in the same frequency band (2.4–2.5 GHz), so 2.4 GHz Wi-Fi can slow down near a running oven.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Shake torch — Grade 10
Self-induction — Grade 10
Magnetic properties of matter — Grade 10
Plasma — the fourth state — Grade 10
Power transmission line — Grade 11
Radiation from a dipole antenna — Grade 11
Oscillating circuit (LC) — Grade 11
Alternating current and RLC — Grade 11
Radio communication and radar — Grade 11
Charged-particle accelerator — Grade 11