Electromagnetic Waves

Maxwell's equations in vacuum ($\rho=0$, $\vec{J}=0$) predict the existence of electromagnetic waves: disturbances of the $\vec{E}$ and $\vec{B}$ fields that propagate through space. Hertz generated them experimentally in 1887, confirming Maxwell's theory and paving the way for radio.

Derivation of the wave equation

Combining Maxwell's equations in vacuum yields: $$\frac{\partial^2\vec{E}}{\partial t^2} = c^2\nabla^2\vec{E}$$ This is the three-dimensional wave equation, where: $$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.998\times10^8\,\mathrm{m/s}$$ The same equation holds for $\vec{B}$. The propagation speed $c$ is a universal constant, independent of the reference frame.

Intuition: Maxwell's equations show that a time-varying electric field generates a magnetic field (displacement current) and a time-varying magnetic field generates an electric field (Faraday). This chain of mutual inductions is self-sustaining: an initial perturbation propagates as a wave, without needing a material medium (unlike sound). The EM wave is self-supporting: energy oscillates between the two fields.

Harmonic plane wave

The simplest solution is a monochromatic plane wave propagating along the $x$-axis: $$\vec{E} = E_0\cos(kx-\omega t)\hat{y},\qquad \vec{B} = \frac{E_0}{c}\cos(kx-\omega t)\hat{z}$$ where $k = 2\pi/\lambda$ is the wave number and $\omega = 2\pi f$ is the angular frequency.

Key properties:

• Transversality: $\vec{E}$ and $\vec{B}$ are perpendicular to each other and to the direction of propagation ($\vec{k}$). The wave is transverse.
• Constant ratio: $|\vec{E}|/|\vec{B}| = c$ at every point and instant.
• Same phase: $\vec{E}$ and $\vec{B}$ oscillate in phase: they reach maximum, zero, and minimum simultaneously.

Dispersion relation in vacuum: $\omega = ck$. In vacuum there is no dispersion: all frequencies travel at the same speed $c$.

Applications

• Wireless communications: Transmitting antennas generate modulated EM waves (AM, FM, Wi-Fi, 4G/5G) that travel at $c$ and are received by receiving antennas.
• Radar: Microwave pulses are reflected by distant objects; the round-trip time measures distance.
• Astronomy: Radio telescopes capture EM waves from stars, galaxies, and pulsars, revealing the invisible universe.

EM waves carry energy, linear momentum, and can take on different polarization states.

Polarization

Polarization describes the time evolution of the $\vec{E}$ vector at a fixed point in space:

• Linear: $\vec{E}$ oscillates along a fixed direction. Obtained by passing unpolarized light through a polaroid (polarizing filter).
• Circular: $\vec{E}$ rotates with constant amplitude, tracing a helix in space. Two perpendicular components $90°$ out of phase produce circular polarization (right-handed or left-handed).
• Elliptical: The general case, with varying amplitude during rotation. Any polarization state can be described as a combination of two orthogonal linear polarizations.

Intuition: Polarization is analogous to the oscillation direction of a taut string: if the string oscillates vertically, it is linear vertical polarization; if it traces a circle, it is circular polarization. For light, polarization is the basis of LCD screens, 3D glasses, and glare reduction. Malus's law states that the intensity transmitted through a polarizer rotated by $\theta$ is $I = I_0\cos^2\theta$.

Poynting vector and energy

The Poynting vector represents the energy flux carried by the wave (power per unit area): $$\vec{S} = \frac{1}{\mu_0}\vec{E}\times\vec{B}$$ The direction of $\vec{S}$ is the direction of energy propagation. For a plane wave: $|\vec{S}| = E^2/(\mu_0 c) = c\varepsilon_0 E^2$.

The average intensity (average power per unit area) is: $$I = \langle S\rangle = \frac{c\varepsilon_0 E_0^2}{2}$$

Example: Sunlight at Earth has $I \approx 1361\,W/m^2$ (solar constant). This value determines Earth's climate and the power available to photovoltaic panels.

Radiation pressure

EM waves also carry linear momentum. When a wave is absorbed or reflected by a surface, it exerts radiation pressure: $$P_{rad} = \frac{I}{c}\quad\text{(absorption)},\qquad P_{rad} = \frac{2I}{c}\quad\text{(total reflection)}$$

Applications:

• Solar sails: Space missions (e.g., The Planetary Society's LightSail) use enormous sails propelled by solar radiation pressure, requiring no fuel.
• Optical tweezers: Focused lasers use radiation pressure to trap and manipulate microscopic particles (Nobel 2018: Ashkin, Mourou, Strickland).

When an EM wave propagates in a material medium, its speed is reduced relative to vacuum. The interaction with the medium is described by the refractive index.

Speed and refractive index

In a medium with relative permittivity $\varepsilon_r$ and relative permeability $\mu_r$: $$v = \frac{c}{n},\qquad n = \sqrt{\varepsilon_r\mu_r}$$ For most transparent materials, $\mu_r \approx 1$ and $n \approx \sqrt{\varepsilon_r}$. The refractive index depends weakly on frequency (chromatic dispersion), causing white light to separate into colors in a prism.

Laws of reflection and refraction

When a wave encounters the interface between two media, part is reflected and part is transmitted (refracted):

• Reflection: $\theta_i = \theta_r$ — the angle of incidence equals the angle of reflection.
• Refraction (Snell's law): $n_1\sin\theta_i = n_2\sin\theta_t$ — the ray bends toward the normal when entering a denser medium ($n_2 > n_1$).

Intuition: Refraction is analogous to a cart moving from asphalt onto sand: one wheel slows before the other, causing the cart to turn. Similarly, the wavefront rotates when the speed changes at the boundary between two media.

Critical angle and total internal reflection

When light travels from a denser to a rarer medium ($n_1 > n_2$), there is a critical angle beyond which no refraction occurs: $$\sin\theta_c = \frac{n_2}{n_1}$$ Beyond $\theta_c$, all energy is reflected internally — this is total internal reflection, the principle of optical fibers.

Fresnel equations

The Fresnel equations determine the amplitudes of reflected and transmitted waves as functions of incidence angle and polarization. For normal incidence, the reflection coefficient is: $$R = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2$$ Brewster's angle $\theta_B = \arctan(n_2/n_1)$ is the angle at which reflected light is completely polarized (zero component parallel to the incidence plane).

Applications

• Optical fibers: Ultra-thin glass strands (~10 $\mu$m diameter) guide light via total internal reflection, transmitting data at enormous speeds (fiber up to 100 Gbps and beyond).
• Lenses and optical instruments: Eyeglasses, microscopes, telescopes use refraction to focus light.
• Anti-reflective coatings: Thin layers with intermediate index reduce reflections (Fresnel equations).
• Polarized sunglasses: Exploit Brewster's angle to reduce annoying glare from water or asphalt.

EM waves span an enormous spectrum of frequencies and wavelengths, from radio to gamma rays. The speed in vacuum is constant: $c = \lambda f$, so $\lambda \propto 1/f$.

Electromagnetic spectrum

Here are the main bands in order of increasing frequency:

• Radio waves ($f \sim 10^4$–$10^9\,Hz$, $\lambda \sim 1\,km$–$1\,m$): Used for AM/FM communications, TV, Wi-Fi, Bluetooth.
• Microwaves ($f \sim 10^9$–$10^{11}\,Hz$, $\lambda \sim 30\,cm$–$1\,mm$): Microwave ovens, radar, Wi-Fi, satellite communications. The ~2.45 GHz frequency resonates with water molecules, heating food.
• Infrared ($f \sim 10^{11}$–$4\times10^{14}\,Hz$, $\lambda \sim 1\,mm$–$700\,nm$): Thermal radiation emitted by hot bodies. Thermal cameras, remote controls, infrared heating.
• Visible light ($f \sim 4$–$7.5\times10^{14}\,Hz$, $\lambda \sim 700$–$400\,nm$): The only EM radiation our eyes can detect. Colors from red (longest wavelength) to violet.
• Ultraviolet ($f \sim 7.5\times10^{14}$–$10^{16}\,Hz$, $\lambda \sim 400$–$10\,nm$): Causes sunburn and vitamin D synthesis. Used for sterilization and mineral identification.
• X-rays ($f \sim 10^{16}$–$10^{19}\,Hz$, $\lambda \sim 10$–$0.01\,nm$): Penetrate soft tissue but not bone; essential in medical radiology (diagnostic imaging).
• Gamma rays ($f > 10^{19}\,Hz$, $\lambda < 0.01\,nm$): Produced by nuclear decays and astrophysical phenomena (supernovae, cosmic rays). Used in radiotherapy and sterilization.

Note: There is no sharp boundary between bands; the spectrum is continuous. The fundamental difference between bands is only frequency (and hence photon energy: $E = hf$).

EM waves in conductors and skin effect

When an EM wave penetrates a conductor, induced currents attenuate it exponentially. The skin depth measures how far the wave penetrates before decaying to $1/e$ of its surface value: $$\delta = \sqrt{\frac{2}{\mu_0\mu_r\sigma\omega}}$$ where $\sigma$ is the material's conductivity.

Examples:

• Copper at 60 Hz (mains frequency): $\delta \approx 8.5\,mm$. Power cables use solid conductors.
• Copper at 1 GHz (microwaves): $\delta \approx 2\,\mu m$. For this reason, high-frequency circuits use thin traces and metallic shielding.

Applications of the skin effect:

• Electromagnetic shielding: A thin metal layer blocks EM waves (Faraday cage). Cell phones, microwave ovens, shielded rooms for sensitive equipment.
• Coaxial cables: Use a shielding mesh to confine the signal and reduce interference.
• Induction heating: High-frequency currents flow on metal surfaces, heating them for surface hardening.