Plane waves

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 $E = E_{0}\cos(\omega t - kz)$, $B = \dfrac{k}{\omega}E = \dfrac{E}{c}$.

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.

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.

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.

Three common slips

Recap