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 → Advanced Electrodynamics ↓


Green's function and potential theory


In advanced electrodynamics, Green's functions and potential theory play a vital role in understanding and solving various electromagnetic problems. These concepts are deeply rooted in mathematical physics and allow us to effectively address and solve boundary value problems. Here, we will explore these topics with an emphasis on clarity using simple language and illustrative examples.

Understanding Green's functions

Green's functions are mathematical tools used to solve differential equations. In the context of electromagnetism, they help solve heterogeneous differential equations describing electric and magnetic fields.

Basically, Green's function, G(x, x'), represents the response of a system to a point source. In electrostatics, it helps solve Poisson's equation:

        ∇²Φ(x) = -ρ(x)/ε₀

In this equation, Φ(x) is the electric potential, ρ(x) is the charge density, and ε₀ is the electric permittivity of free space. The Green's function satisfies:

        ∇²G(x, x') = -δ(x - x')

where δ(x - x') is the Dirac delta function. The power of the Green's function is evident when expressing the solution of the incompressible equation:

        Φ(x) = ∫ G(x, x')ρ(x') dx'

Visual example: point source effect

X'g(x, x')

In this visualization, a point source located at x' produces an impact spread represented by the green function G(x, x'). The red circle shows the impact radius at another point x.

Use of Green's functions in electrodynamics

Green's function is widely used to solve electrostatic problems, especially those with complex boundary conditions. For example, in a bounded region, Green's function must also satisfy the boundary conditions of the problem.

Suppose a problem in a cavity requires that the potential at its surface be zero. Here, you would use the Green function corresponding to this condition.

Example: Solve ∇²Φ(x) = 0 for the region inside a spherical shell with radius R and boundary condition Φ(x) = 0 on the surface.

Potential theory in electrodynamics

Potential theory deals with the properties and behavior of potentials. In electrodynamics, these potentials are central to understanding electric and magnetic fields. The primary potentials used in electrodynamics are the scalar potential, Φ, and the vector potential, A

The scalar potential is related to the electric field E as follows:

        e = -∇Φ

The vector potential is related to the magnetic field B as follows:

        b = ∇ × a

Together, Φ and A can describe any electromagnetic field configuration.

Visual example: Prospects in an area

IΦ path

In this illustration, the blue line represents the electric field lines from the scalar potential, while the curved green path represents the path where Φ changes.

Solving electromagnetic problems using Green's functions

Solving electromagnetic problems often involves calculating the potential of a given charge or current distribution. When using Green's functions, you turn the problem into an integration one, allowing you to apply known boundary conditions directly.

Example: Given the charge distribution in a cube, find the electric potential everywhere inside it using a suitable Green's function.

The general process includes the following:

  • Identify the differential equation governing the system, usually the Poisson or Laplace equation.
  • Choose or derive a Green's function that accommodates the boundary conditions of the system.
  • Use the Green function to fully formulate the problem.
  • Perform integration to find the capacity.

Conclusion

In summary, Green's functions and potential theory provide powerful frameworks for tackling complex problems in electromagnetism. By understanding the interplay of sources and fields and using appropriate mathematical tools, we can obtain solutions in a wide range of scenarios. As you delve deeper into these topics, you will appreciate their applicability and beauty in solving complex physical problems.


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Green's function and potential theory

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