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 ↓


Boundary conditions and uniqueness theorem


In advanced electrodynamics, understanding boundary conditions and uniqueness theorems is important for solving complex problems involving electric and magnetic fields. These concepts allow physicists to predict and analyze the behavior of electromagnetic fields under various constraints and configurations.

Introduction to boundary conditions

Boundary conditions are necessary relations applying at the interface between different media through which electromagnetic fields pass. These conditions ensure that the solutions to Maxwell's equations are physically realizable and consistent across these boundaries.

Let us consider some common types of boundary conditions in electromagnetism:

1. Continuity of the tangential component of the electric field

At the boundary between two different media, the tangential component of the electric field must remain constant. Mathematically, it is expressed as:

E₁t = E₂t

where (E₁t) and (E₂t) are the tangential components of the electric field in media 1 and 2, respectively.

2. Continuity of the normal component of the electric displacement field

The normal component of the electric displacement field (D) must satisfy:

D₁n - D₂n = sigma_f

where (D₁n) and (D₂n) are the normal components of the electric displacement field in media 1 and 2, and (sigma_f) is the free surface charge density at the boundary.

3. Continuity of the tangential component of the magnetic field

For a magnetic field, the tangential component must satisfy:

H₁t - H₂t = K_f

where (H₁t) and (H₂t) are the tangential components of the magnetic field in media 1 and 2, and (K_f) is the surface current density across the boundary.

4. Continuity of the normal component of the magnetic field

The normal component of the magnetic field (B) must be constant:

B₁n = B₂n

where (B₁n) and (B₂n) are the normal components of the magnetic field in media 1 and 2.

Uniqueness theorem in electrodynamics

Uniqueness theorems are important in electromagnetism because they guarantee that solutions to electromagnetic problems are not only consistent but also unique given specific boundary conditions and sources within the field. Let us look at these theorems in detail:

1. Uniqueness theorem for the electrostatic field

The electrostatic field in the volume (V) enclosed by the surface (S) is completely specified by the charge distribution inside (V) and the potential at (S). If you know the potential and the charge distribution at the boundary, the electric field configuration is unique. Formally, this theorem can be expressed as:

∇²φ = -ρ/ε₀ in V, φ = φ₀ on S

Where (phi) is the electric potential, (rho) is the charge density, and (epsilon_0) is the permittivity of free space.

2. Uniqueness theorem for a magnetic constant field

Similarly, the magnetic field within a volume is completely determined by the current distribution and boundary conditions. Given that the currents are known, the magnetic field configuration will be unique.

Mathematically, this can be represented as the following solution:

∇ × H = J, ∇ ⋅ B = 0

Where (H) is the magnetic field, (J) is the current density, and (B) is the magnetic flux density.

Visual example

Consider a simple example of two dielectric media separated by a boundary. The boundary conditions affect the behavior of electric fields when transitioning from one medium to another. Below is a representation of this scenario.

Medium 1 Medium 2 electric field

This figure shows two regions characterized by different dielectric properties, where boundary conditions apply to the electric field.

Textual examples

Imagine you have a cube with different electric potentials inside, and given boundary conditions on the surface of the cube. According to the uniqueness theorem, for a specified charge distribution inside the cube and a known potential on the surface, the electric field solution is uniquely determined.

Example problem: Suppose you have a metal sphere with radius (R) and charge (Q). Determine the electric field outside and inside the sphere.

Outside the sphere (r > R): E = (1/(4πε₀)) * (Q/r²) Inside the sphere (r ≤ R): E = 0

This example highlights how boundary conditions and charge distributions uniquely determine the electric field configuration.

Conclusion

Boundary conditions and uniqueness principles provide the framework necessary to solve electromagnetic problems in a rigorous and systematic way. By ensuring that solutions of Maxwell's equations are unique and satisfy physical requirements at the boundaries, these principles form the backbone of predictive power in electromagnetism.


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