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 ↓


Plasma physics


Plasma physics is an important part of electromagnetism, primarily because plasma, often considered the fourth state of matter, is composed of charged particles. These particles move quite differently from neutral atoms in a solid, liquid, or gas. Plasma consists of ions and electrons, which behave collectively and exhibit complex dynamics governed by electromagnetic forces.

When discussing plasma physics, it is important to understand that the principles and equations of electromagnetism play a vital role. In this exploration, we will uncover various aspects of plasma physics, delving into concepts such as Debye shielding, plasma oscillations, and magnetic confinement.

Basic properties of plasma

Plasma is created by imparting sufficient energy to a gas, stripping electrons from atoms and producing a mixture of electrons and ions. This state is highly sensitive to electromagnetic fields because of its charged components.

Debye shielding

A remarkable feature of plasma is its ability to shield electrical charges at a distance through a process called Debye shielding. In a plasma, moving electrons surround a positive test charge, reducing the effective range of its influence. The characteristic length over which this shielding occurs is called the Debye length, denoted by λ_D. This length can be calculated using the formula:

λ_D = sqrt((ε₀ k_B T_e) / (n_e e²))

Where:

  • ε₀ is the permittivity of free space.
  • k_B is the Boltzmann constant.
  • T_e is the electron temperature.
  • n_e is the electron number density.
  • e is the elementary charge.
+ Test Charging - Electron

Plasma frequency and oscillation

The plasma frequency is the natural frequency at which the electrons in a plasma oscillate when disturbed from their equilibrium position. This phenomenon is important in understanding how plasmas interact with electromagnetic waves. The plasma frequency ω_p is given by:

ω_p = sqrt((n_e e²) / (ε₀ m_e))

Where:

  • n_e is the electron number density.
  • e is the elementary charge.
  • ε₀ is the permittivity of free space.
  • m_e is the electron mass.
Vibrations

Electric and magnetic fields in plasma

Understanding how plasma behaves in electric and magnetic fields is paramount to controlling and harnessing its properties. The intrinsic nature of plasma allows it to conduct electricity and respond dynamically to magnetic fields.

Magnetic confinement and tokamaks

Magnetic confinement is a technique used to control plasma in a laboratory setting, often applied in nuclear fusion research. The tokamak is one of the most well-known devices using magnetic confinement. It uses a powerful magnetic field to control and stabilize plasma, creating suitable conditions for nuclear fusion.

tokamak

Mathematical description of plasma

The behavior of plasma is typically described using a set of equations derived from fluid dynamics and electromagnetism, often called magnetohydrodynamics (MHD). These equations depict the macroscopic behavior of plasma as a fluid influenced by magnetic and electric fields.

One of the fundamental equations in MHD is the continuity equation:

∂n/∂t + ∇·(nv) = 0

Where:

  • n is the plasma density.
  • v is the flow velocity vector.

Another important equation is the momentum equation, which accounts for the forces acting on the plasma:

m (∂v/∂t + v·∇v) = -∇p + J x B + F_ext

Where:

  • m is the mass of the particle.
  • p is the pressure.
  • J is the current density.
  • B is the magnetic field.
  • F_ext is any external force.

Waves and instabilities in plasma

In plasma physics, waves and instabilities are important phenomena. Plasmas can support various wave modes, such as ion-acoustic waves, Alfvén waves, and magneto-acoustic waves. Understanding these waves is essential for controlling plasmas in various environments.

Alfvén waves

Alfvén waves are low-frequency oscillations of ions and magnetic fields within a plasma. They play an important role in space plasma, such as the solar wind and in the interstellar medium. These waves travel along magnetic field lines and are described by the following dispersion relation:

ω = k v_A

Where:

  • ω is the wave frequency.
  • k is the wave number.
  • v_A is the Alfvén velocity, defined by v_A = B / sqrt(μ₀ n m_i).
  • B is the strength of the magnetic field.
  • μ₀ is the permittivity of free space.
  • n is the ion density.
  • m_i is the ion mass.
Alfvén Wave

Applications and importance of plasma physics

Plasma physics has wide applications in many fields, including astrophysics, nuclear fusion, and space exploration. By understanding the behavior of plasma, scientists can develop technologies such as ion thrusters for spacecraft and improve communication systems.

Nuclear fusion, for example, attempts to replicate the processes that occur in stars to produce energy. This requires controlling plasma at extremely high temperatures and pressures, which can be achieved through magnetic confinement.

Conclusion

Plasma physics is a complex but fascinating field of study in graduate level electromagnetism, involving a wide variety of phenomena and theories governing charged particles in various environments. Foundations and mathematical frameworks such as Debye shielding and magnetohydrodynamics provide deep insights into the behavior and applications of plasma.

By exploring the collective dynamics of charged particles, we not only advance our scientific understanding, but also foster technological advancements crucial for future scientific breakthroughs and innovations.


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Electromagnetic field tensor and Lorentz transformations
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Plasma physics

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