Plane waves
مُقدَّم| English | المعنى |
|---|---|
| plane wave | الموجة المستوية |
| intrinsic impedance | المعاوقة الذاتية |
| Poynting vector/ˈpɔɪntɪŋ ˈvektə/ | Poynting vector |
| skin depth | عمق الجلد |
| good conductor | موصل جيد |
Wi-Fi, radar and sunlight
- Wi-Fi, radar and sunlight differ only in frequency. Far from their sources, all of them are the same simple solution of Maxwell's equations: the plane wave.
- This lesson reads a plane wave's numbers off one formula, then sees what happens when it enters a metal.
The plane wave
- In a plane wave 平面电磁波, $\mathbf{E}$, $\mathbf{B}$ and the direction of travel are all at right angles, and $\mathbf{E}$ and $\mathbf{B}$ rise and fall together.
- Faraday's law fixes their sizes: for $E = E_{0}\cos(\omega t - kz)$, $B = \dfrac{k}{\omega}E = \dfrac{E}{c}$.
In a plane wave in free space, E and B reach their peaks at the same moment.
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They are in phase in a lossless medium: $B = \dfrac{E}{c}$ at every point and time.
What is the wavelength of a 2.4 GHz Wi-Fi signal in free space, in cm?
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$\lambda = \dfrac{3 \times 10^{8}}{2.4 \times 10^{9}} = 0.125$ m = 12.5 cm.
Impedance of free space
- The ratio of $E$ to $H = \dfrac{B}{\mu_{0}}$ is the intrinsic impedance 波阻抗 $\eta_{0} = \sqrt{\dfrac{\mu_{0}}{\varepsilon_{0}}} = 377\,\Omega$.
- It is a property of the medium, not a resistor: nothing is dissipated.
A plane wave has E = 377 V/m. What is H?
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$H = \dfrac{E}{\eta_{0}} = \dfrac{377}{377} = 1$ A/m.
The Poynting vector
- The Poynting vector 坡印亭矢量 $\mathbf{S} = \mathbf{E} \times \mathbf{H}$ points along the direction of travel. Its size is the power per square metre the wave carries.
- For a plane wave $S = EH = \dfrac{E^{2}}{\eta_{0}}$.
Worked example: a 1 MHz wave
- $E = 100\,\dfrac{\text{V}}{\text{m}}$ at 1 MHz: $B = \dfrac{100}{3 \times 10^{8}} = 0.333\,\mu\text{T}$ and $\lambda = \dfrac{c}{f} = 300\,\text{m}$.
- $S = \dfrac{100^{2}}{377} = 26.5\,\dfrac{\text{W}}{\text{m}^{2}}$. Check: $H = \dfrac{100}{377} = 0.265\,\dfrac{\text{A}}{\text{m}}$, and $E \times H = 26.5$ again.
A plane wave has E = 10 V/m. What is its power density, in W/m²? Give 3 decimal places.
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$S = \dfrac{E^{2}}{\eta_{0}} = \dfrac{100}{377} = 0.265$ W/m².
Skin depth in a conductor
- In a good conductor 良导体 the wave dies away as $e^{-z/\delta}$, with skin depth 趋肤深度 $\delta = \sqrt{\dfrac{2}{\omega\mu\sigma}}$.
- Copper ($\sigma = 5.8 \times 10^{7}\,\dfrac{\text{S}}{\text{m}}$): $\delta = 9.3\,\text{mm}$ at 50 Hz, but only $2.1\,\mu\text{m}$ at 1 GHz. High-frequency current flows in a thin surface layer.
$\delta \propto \dfrac{1}{\sqrt{f}}$: from 50 Hz to 1 GHz the frequency rises twenty million times, and $\sqrt{2 \times 10^{7}} \approx 4500$, which is exactly how much thinner the skin becomes. This is why thin metal shields work at radio frequencies.
Copper's skin depth is 9.3 mm at 50 Hz. At 5 kHz (100 times higher) it is
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$\delta \propto \dfrac{1}{\sqrt{f}}$: 100 times the frequency gives $\dfrac{1}{10}$ the depth.
Three common slips
- Using $c$ in a dielectric. There the speed is $\dfrac{1}{\sqrt{\mu\varepsilon}}$, slower than $c$.
- Mixing up the ratios. $\dfrac{E}{H} = 377\,\Omega$ is an impedance; $\dfrac{E}{B} = c$ is a speed.
- Expecting E and B in phase everywhere. That holds in lossless media only; in a conductor they shift.
Match each ratio to its value in free space.
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One is an impedance, one a speed, and the last turns field into power density.
In glass with εr = 2.25, light travels at c.
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$v = \dfrac{c}{\sqrt{2.25}} = \dfrac{c}{1.5} = 2 \times 10^{8}$ m/s: the refractive index is 1.5.
Recap
- Plane wave: $\mathbf{E} \perp \mathbf{B} \perp$ travel, in phase, $B = \dfrac{E}{c}$.
- $\eta_{0} = 377\,\Omega$; power density $S = \dfrac{E^{2}}{\eta_{0}}$.
- Skin depth $\delta = \sqrt{\dfrac{2}{\omega\mu\sigma}}$ shrinks as $\dfrac{1}{\sqrt{f}}$.