16.2 Plane Electromagnetic Waves

Electromagnetic waves are fascinating phenomena that combine electric and magnetic fields. They propagate through space at the speed of light, carrying energy and information across vast distances. Understanding their properties is crucial for grasping how modern technology works.

Maxwell's equations describe the behavior of electromagnetic waves, revealing their nature as self-sustaining oscillations of electric and magnetic fields. These waves can be produced by accelerating charges and detected through various methods, enabling a wide range of applications from radio communication to medical imaging.

Plane Electromagnetic Waves

Electromagnetic wave components and propagation

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• In a plane electromagnetic wave, the electric field ($\vec{E}$), magnetic field ($\vec{B}$), and propagation direction ($\vec{k}$) are mutually perpendicular to each other
  • $\vec{E}$ and $\vec{B}$ oscillate in phase with each other, meaning their peaks and troughs align at the same points in space and time
  • The direction of propagation is given by the cross product of $\vec{E}$ and $\vec{B}$: $\vec{k} = \frac{\vec{E} \times \vec{B}}{|\vec{E}||\vec{B}|}$, which is a vector perpendicular to both $\vec{E}$ and $\vec{B}$
    • This relationship is often represented by the "right-hand rule" (point fingers of right hand in direction of $\vec{E}$, curl fingers toward $\vec{B}$, thumb points in direction of $\vec{k}$)
  • The orientation of the electric field oscillations determines the wave's polarization

Maxwell's equations and light speed

• Maxwell's equations describe the relationships between $\vec{E}$, $\vec{B}$, and $\vec{k}$ in electromagnetic waves
  • Faraday's law: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ states that a changing magnetic field induces an electric field (electric generator)
  • Ampère's law with Maxwell's correction: $\nabla \times \vec{B} = \mu_0\left(\vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t}\right)$ states that a changing electric field and electric current induce a magnetic field (electromagnet)
  • Gauss's law for electric fields: $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$ states that electric fields originate from electric charges (point charges)
  • Gauss's law for magnetic fields: $\nabla \cdot \vec{B} = 0$ states that magnetic fields have no divergence, meaning there are no magnetic monopoles (bar magnets always have north and south poles)
• In free space, where there are no charges ($\rho = 0$) or currents ($\vec{J} = 0$), Maxwell's equations simplify and lead to wave equations for $\vec{E}$ and $\vec{B}$
  • The speed of the electromagnetic wave in both equations is given by: $v = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$, where $\mu_0$ is the permeability and $\varepsilon_0$ is the permittivity of free space
  • Substituting known values yields: $v \approx 3 \times 10^8 \text{ m/s}$, equal to the speed of light in vacuum, $c$
    • This result showed that light is an electromagnetic wave and unified electricity, magnetism, and optics

Electric to magnetic field ratio

• In a plane electromagnetic wave, the ratio of the electric field magnitude to the magnetic field magnitude is constant and equal to the speed of light in free space
  • The relationship is given by: $\frac{|\vec{E}|}{|\vec{B}|} = c$, where $c \approx 3 \times 10^8 \text{ m/s}$
• To calculate the ratio, divide the amplitude of the electric field ($E_0$) by the amplitude of the magnetic field ($B_0$): $\frac{|\vec{E}|}{|\vec{B}|} = \frac{E_0}{B_0}$
  • For example, if $E_0 = 100 \text{ V/m}$ and $B_0 = 3.33 \times 10^{-7} \text{ T}$, then $\frac{|\vec{E}|}{|\vec{B}|} = \frac{100 \text{ V/m}}{3.33 \times 10^{-7} \text{ T}} \approx 3 \times 10^8 \text{ m/s} = c$

Principles of EM wave production

• Electromagnetic wave production:
  • Accelerating charges produce electromagnetic waves
    • Oscillating electric dipoles (antennas) and alternating currents (AC circuits) are common sources
  • The frequency of the produced wave matches the frequency of the charge oscillation or acceleration
    • For example, a 100 MHz oscillation produces a 100 MHz electromagnetic wave
  • The wavelength of the produced wave is related to the frequency by: $\lambda = \frac{c}{f}$, where $\lambda$ is wavelength and $f$ is frequency
    • Higher frequencies correspond to shorter wavelengths (gamma rays), while lower frequencies correspond to longer wavelengths (radio waves)
• Electromagnetic wave detection:
  • Electromagnetic waves are detected through their interaction with matter
    • Photoelectric effect (solar cells), Compton scattering (X-ray diffraction), and pair production (PET scans) are examples
  • Antennas detect electromagnetic waves by converting oscillating electric and magnetic fields into alternating currents at the same frequency as the incident wave
    • This is the reverse process of electromagnetic wave production
  • The type of detector depends on the frequency or wavelength of the electromagnetic wave:
    1. Radio waves: antennas and receivers (AM/FM radio)
    2. Microwaves: antennas, waveguides, and bolometers (microwave ovens, radar)
    3. Infrared: thermocouples, bolometers, and photoconductors (night vision, thermal imaging)
    4. Visible light: photomultiplier tubes, photodiodes, and CCD sensors (cameras, telescopes)
    5. Ultraviolet, X-rays, and gamma rays: scintillation detectors, proportional counters, and semiconductor detectors (medical imaging, astronomy)

Wave properties and energy transfer

• The wave equation describes the propagation of electromagnetic waves in space and time
• The Poynting vector represents the energy flux density of an electromagnetic wave, indicating the direction and magnitude of energy flow
• Dispersion occurs when different wavelengths of light travel at different speeds in a medium, causing the wave to spread out
• The phase velocity is the speed at which the phase of a wave propagates in a medium