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 ↓


Electromagnetic waves


Electromagnetic waves are fascinating phenomena that are fundamental to many technologies and scientific fields. From the light we see to the radio waves used in communication, electromagnetic waves pervade our lives. In the PhD-level field of electrodynamics, understanding these waves requires an understanding of Maxwell's equations, wave propagation theories, and the interaction of electric and magnetic fields.

Maxwell's equations: the basis

To understand electromagnetic waves, we need to start with James Clerk Maxwell's four key equations:

∇·E = ρ/ε₀ ∇·B = 0 ∇×E = -∂B/∂t ∇×B = μ₀J + μ₀ε₀∂E/∂t
∇·E = ρ/ε₀ ∇·B = 0 ∇×E = -∂B/∂t ∇×B = μ₀J + μ₀ε₀∂E/∂t

These equations describe how electric fields (E) and magnetic fields (B) interact with charges (ρ) and currents (J). The constants ε₀ and μ₀ are the permittivity and permeability of free space, respectively. The last equation includes the concept of displacement current, which was Maxwell's genius addition, enabling the prediction of electromagnetic waves.

Deriving the wave equation

The wave equation for electromagnetic waves can be derived by studying Maxwell's equations in depth. Consider the curl equations:

∇×E = -∂B/∂t ∇×B = μ₀ε₀∂E/∂t
∇×E = -∂B/∂t ∇×B = μ₀ε₀∂E/∂t

Taking the curl of both equations and applying vector calculus identities, we can show:

∇²E = μ₀ε₀∂²E/∂t² ∇²B = μ₀ε₀∂²B/∂t²
∇²E = μ₀ε₀∂²E/∂t² ∇²B = μ₀ε₀∂²B/∂t²

These are the wave equations for the E and B fields, which describe how electromagnetic waves propagate in a vacuum at the speed of light (c), where:

c = 1/√(μ₀ε₀)
c = 1/√(μ₀ε₀)

This shows that electromagnetic waves are transverse waves; their electric and magnetic fields oscillate perpendicular to the direction of wave propagation.

Visualization of electromagnetic waves

Imagine an electromagnetic wave traveling in space. The electric field vector oscillates in a plane, while the magnetic field vector oscillates in a plane perpendicular to it. Here is a representation:


    
    
    
    
    E-Region
    B-region

In this diagram, the blue wave represents electric field oscillations, while the red wave represents magnetic field oscillations. Both fields are perpendicular to the direction of propagation, which is along the x-axis.

Polarization of electromagnetic waves

Polarization describes the orientation of the oscillations of the electric field vector. If the fields oscillate in the same direction, the wave is linearly polarized. For example, in the SVG diagram above, the electric field is linearly polarized. However, electromagnetic waves can also be circularly or elliptically polarized, which describes more complex oscillation patterns. Polarization is important in many technologies, such as antennas and optical devices.

Wave interaction with matter

When electromagnetic waves strike materials, several interactions can occur: reflection, refraction, diffraction, and absorption. Each interaction depends on the properties of the material and the characteristics of the wave, such as wavelength and polarization.

n = c/v
n = c/v

This is the formula for the refractive index (n) of a material, which shows how much the speed of an electromagnetic wave is reduced inside a medium compared to its speed in a vacuum (v is the speed of the wave in the material).

As an example of refraction, consider visible light passing through a glass prism. Different wavelengths of light are bent at different angles, resulting in a spectrum of colors.

Electromagnetic spectrum

Electromagnetic waves are classified based on their wavelength and frequency. This classification is known as the electromagnetic spectrum, which ranges from low-energy radio waves to high-energy gamma rays. Widely recognized parts of the spectrum include:

  • Radio waves: Used for communications, such as television and radio.
  • Microwave: Used for radar technology and heating food.
  • Infrared: Emitted by hot objects and used in remote controls.
  • Visible light: The narrow band of wavelengths that our eyes can see.
  • Ultraviolet: Responsible for sunburn and used to sterilize equipment.
  • X-rays: Used in medical imaging to see inside the human body.
  • Gamma rays: Emitted during nuclear reactions and used in the treatment of cancer.
Here is a visual representation of the electromagnetic spectrum: 


    
    
    
    
    
    
    
    radio
    Microwave
    Infrared
    Visible
    Ultraviolet
    X-rays
    Gamma Ray

The role of electromagnetic waves in modern technology

The study and application of electromagnetic waves has led to significant technological advances. Examples include:

  • Wireless communications: Radio, Wi-Fi, and Bluetooth rely on electromagnetic waves to transmit data.
  • Medical imaging: Techniques such as MRI use radio waves, while X-rays aid in diagnosis.
  • Optical instruments: Cameras, telescopes, and microscopes use lenses that modulate electromagnetic waves.
  • Remote sensing: Satellites use electromagnetic waves to collect information about Earth's climate and landscape.

Challenges and research limitations

Electromagnetic waves remain a rich area of research. Leading topics include:

  • Metamaterials: Engineered materials with unique properties can control electromagnetic waves in new ways, opening up possibilities for invisibility cloaks and super lenses.
  • High frequency waves: Terahertz and gamma waves offer new possibilities in imaging and communications, but their production and control remain challenging.
  • Quantum Electrodynamics: The intersection of quantum mechanics and electromagnetic theory is crucial to developing our understanding of fundamental processes.

Conclusion

Electromagnetic waves are not just theoretical constructs; they form the basis of our understanding of the universe and the advancement of technology. From making phone calls to observing distant galaxies, electromagnetic waves are central to both modern convenience and scientific exploration.


PHD → 2.2


U
username
0%
completed in PHD

← Prev (2.1.5)
Boundary conditions in Maxwell's equations
Next (2.2.1) →
Wave equation

Comments 



Electromagnetic waves

https://www.physicsflow.com/phd/2.2?pfal=en