Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Electromagnetism ↓


Electromagnetic wave propagation


Electromagnetic wave propagation is a fundamental concept in electromagnetism and involves the movement of electromagnetic waves through various mediums. These waves carry energy and information from one place to another without the need for a physical medium. This means they can travel through the vacuum of space as well as through air, water, and other materials. Understanding electromagnetic wave propagation is important to many technologies, including radio communications, microwave devices, and even the way light travels.

Basic concepts of electromagnetic waves

Electromagnetic waves are composed of oscillating electric and magnetic fields that are perpendicular to each other and in line with the direction of wave propagation. Electric fields ((E)) and magnetic fields ((B)) carry energy through space or a medium.

The relationship between electric fields, magnetic fields, and wave vectors is governed by Maxwell's equations. These equations describe how electric and magnetic fields interact, causing waves to propagate.

1. ∇ • E = ρ/ε₀ 
2. ∇ • B = 0 
3. ∇ × E = -∂B/∂t 
4. ∇ × B = μ₀(J + ε₀∂E/∂t)

In the absence of free charges and currents, the equations become simple and form the basis of electromagnetic wave propagation.

Wave equation

The propagation of electromagnetic waves can be described by the wave equation. In free space, it is given as:

∂²E/∂t² = c²∇²E 
∂²B/∂t² = c²∇²B

Where (c) is the speed of light ((c ≈ 3 times 10^8 , text{m/s})). These equations show that both electric and magnetic fields propagate as waves with speed (c).

Visual example: Wave components

Direction of propagation E-Region B-region

In this diagram, the blue wave represents the electric field ((E)) oscillating around an axis, and the red wave represents the magnetic field ((B)) oscillating perpendicular to the electric field. The line represents the direction in which the wave is propagating.

Polarization

Polarization refers to the orientation of the oscillations of the electric field component of an electromagnetic wave. Polarization can be linear, circular, or elliptical, depending on how the electric field vector changes with time.

Generally, linear polarization is encountered when the electric field oscillates in a single plane. Circular polarization occurs when the electric field rotates in a circular motion while maintaining a constant amplitude. It can be right-handed or left-handed depending on the direction of rotation. Elliptical polarization is a more common form where the electric field describes an ellipse.

Reflection and transmittance

When electromagnetic waves hit the boundary between two different media, some part of the wave gets reflected while the rest gets transmitted through the medium. This can be explained by the concepts of reflection ((R)) and transmittance ((T)). They are defined as:

R = (frac{|E_{text{reflected}}|^2}{|E_{text{incident}}|^2}) 
T = (frac{|E_{text{transmitted}}|^2}{|E_{text{incident}}|^2})

Where (E_{text{incident}}), (E_{text{reflected}}), and (E_{text{transmitted}}) are the amplitudes of the incident, reflected, and transmitted waves, respectively. The sum of reflection and transmission is equal to one ((R + T = 1)), which suggests conservation of energy.

The Fresnel equations provide a detailed description of reflection and transmission at the boundary of two media, taking into account the angle of incidence and the refractive index of the two media.

Practical applications

Electromagnetic wave propagation is an integral concept in a variety of fields, including telecommunications, radar technology, and medical imaging.

  • Telecommunication: Radio waves are used to send signals over long distances. Understanding their propagation helps design efficient antennas and improve signal quality.
  • Radar technology: Radar systems use microwave radiation to detect objects. Understanding wave reflection is important for determining the distance and speed of a target.
  • Medical imaging: Techniques such as MRI (magnetic resonance imaging) rely on electromagnetic waves to create detailed images of the human body.

Conclusion

Electromagnetic wave propagation is fundamental to understanding how energy and information travel through space and various mediums. By studying how electric and magnetic fields interact, particularly under Maxwell's equations, we gain insight into a vast range of phenomena in physics and many practical applications in our technological world. Through diverse methods such as wave equations, understanding reflections, and different polarization states, the propagation of electromagnetic waves continues to be an important area of physics.


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Electromagnetic wave propagation

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