PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Electrodynamics ↓


Radiation and scattering


In the field of electrodynamics, the concepts of radiation and scattering are of vital importance due to their wide implications in fields such as astrophysics, telecommunications, and quantum mechanics. Understanding these concepts often means understanding the fundamental interactions between electromagnetic waves and matter.

Electromagnetic radiation

Electromagnetic radiation refers to waves of the electromagnetic field, which propagate in space and carry electromagnetic radiation energy. It includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

Nature of electromagnetic waves

Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields, which are perpendicular to each other and the direction of wave propagation. The general form of an electromagnetic wave traveling in x direction is given as:

    E(x, t) = E₀ sin(kx – ωt + φ)
    b(x, t) = b₀ sin(kx - ωt + φ)

Here, E(x, t) and B(x, t) denote the electric and magnetic fields respectively, E₀ and B₀ are the amplitudes, k is the wave number, ω is the angular frequency, and φ is the phase constant.

Radiation emission

Whenever charges accelerate, radiation is emitted. According to classical electrodynamics, a charge in uniform motion does not radiate. However, if its motion is non-uniform – for example, if the charge is oscillating – it does produce radiation.

For example, consider an oscillating electric dipole, one of the simplest radiating systems. The power radiated by an oscillating dipole is given by Larmor's formula:

    P = (frac{{mu_0 q^2 a^2}}{6 pi c})

where P is the radiation power, μ₀ is the permittivity of free space, q is the charge, a is the acceleration, and c is the speed of light.

Radiation pattern

The radiation pattern of an electromagnetic wave describes how the power emitted by a source is distributed in space. The most common visual representation of this pattern is a polar plot showing how the power varies with angle.

This example shows a simple dipole radiation pattern. The blue semicircular line indicates the radiation intensity in one direction, while the red line indicates the other, perpendicular direction.

Scattering of electromagnetic waves

Scattering refers to the deflection of electromagnetic waves by particles. The interaction between waves and particles can cause dispersion, absorption, or redirection of waves. Scattering processes are classified based on the size of the scatterer relative to the wavelength of the radiation.

Types of scattering

Rayleigh scattering

Rayleigh scattering occurs when the scattering particles are much smaller than the wavelength of the radiation. This is why the sky appears blue; shorter wavelengths of sunlight are scattered more than longer wavelengths when passing through air molecules in the atmosphere. The Rayleigh scattering cross-section is given by:

    (sigma = frac{{2pi^5 (n^2-1)^2 d^6}}{3 lambda^4 (n^2 + 2)^2})

Here, σ is the scattering cross section, n is the refractive index, d is the scatterer diameter, and λ is the wavelength of the incident radiation.

Mie scattering

Mie scattering describes scattering by particles of similar size to the wavelength of the radiation. This type of scattering does not depend strongly on wavelength, which is why clouds (made up of large water droplets) appear white. Mie theory is complex and is usually solved using computational methods.

Thomson scattering

Thomson scattering considers the scattering of electromagnetic waves by free electrons. It is important in many astrophysical applications, especially in understanding the cosmic microwave background radiation. The differential cross-section for Thomson scattering is given by:

    (frac{{dsigma}}{dOmega} = frac{{e^4}}{(4pi varepsilon_0)^2 m^2 c^4}) = (frac{{r_0^2}}{2} (1+cos^2theta)

Here, dσ/dΩ is the differential cross section, r₀ is the classical electron radius, and θ is the scattering angle.

Wave–particle interaction

Radiation and scattering phenomena in electrodynamics often require consideration of the quantum nature of light and its interaction with matter, resulting in phenomena such as absorption and emission spectra.

Absorption and emission

Atoms absorb electromagnetic energy by moving electrons to higher energy levels. When these electrons return to their ground state, they often emit radiation at different wavelengths. This is widely observed in spectroscopy.

Compton scattering

Compton scattering is a quantum mechanical phenomenon in which X-ray or gamma-ray photons scatter off electrons, resulting in a decrease in energy (increase in wavelength) of the photons, which is described by the Compton equation:

    (lambda' - lambda = frac{h}{m_e c} (1 - cos theta))

Where λ' is the wavelength after scattering, λ is the initial wavelength, h is the Planck constant, mₑ is the electron mass, c is the speed of light, and θ is the scattering angle.

Scatter visualization

Observing the scattering of waves helps in understanding their interaction with matter.

In this visual representation, a wave comes from the left, collides with a particle (gray circle), and scatters in multiple directions (blue and red paths represent possible scattering paths).

Theoretical and practical implications

Understanding the theory behind radiation and scattering provides the basis for many practical applications. These principles aid in the design and analysis of antenna systems, medical imaging techniques, and even climate studies through the analysis of light scattering in the atmosphere.

Antennas and radiation patterns

The study of radiation patterns is important in the design of antennas that play a vital role in communication systems. The design of an antenna depends heavily on the desired radiation pattern, which affects how efficiently the antenna can transmit or receive signals over distances.

Medical imaging

Scattering is a key principle behind techniques such as X-ray imaging and MRI, which rely on the scattering of electromagnetic waves to form images of the inside of the body. Understanding the different scattering mechanisms allows for the development of better imaging techniques.

Climatology

The scattering of sunlight in the Earth's atmosphere plays an important role in climate science. Rayleigh and Mie scattering are crucial for understanding how light interacts with air molecules and aerosols, which has a great impact on weather patterns and assessments of climate change.

Conclusion

Radiation and scattering are fundamental in understanding how electromagnetic waves interact with matter. As widely applicable phenomena, they provide insights into both classical and quantum physics and demonstrate how theoretical principles can be applied to real-world challenges. The interactions between electromagnetic waves and particles not only enhance our understanding of the universe but also drive technological advancements that have a lasting impact on modern society.


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