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 ↓


Relativistic electrodynamics


Relativistic electrodynamics is a theory that combines the principles of electromagnetism with the fundamental notions of special relativity. It provides a framework for understanding how electric and magnetic fields behave in various inertial reference frames that move at speeds close to the speed of light.

Foundations of electrodynamics

At the core of electrodynamics are Maxwell's equations, which govern the behavior of electric and magnetic fields. In their traditional form, these equations are:

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

Here, E and B are the electric and magnetic fields respectively, ρ is the electric charge density, J is the current density, ε₀ is the permittivity of free space, and μ₀ is the permittivity of free space.

Special relativity and its impact on physics

Albert Einstein's theory of special relativity introduced new concepts in physics, notably the ideas that the laws of physics are the same in all inertial frames, and that the speed of light is constant regardless of the motion of the observer.

Basically, relativistic electrodynamics is an extension of the notion that electric and magnetic fields are not separate entities, but two aspects of the same physical phenomenon. They transform into each other when observed from different reference frames.

Lorentz transformations

Central to special relativity – and hence to relativistic electrodynamics – are the Lorentz transformations, which relate the space and time coordinates of events measured in different inertial frames.

x' = γ(x - vt) y' = y z' = z t' = γ(t - vx/c²) γ = 1/√(1 - v²/c²)

Here, v is the relative velocity between the two frames, c is the speed of light, and γ is the Lorentz factor. Lorentz transformations show how measurements of time and space by two observers are interrelated by their relative motion.

Relativity of simultaneity

A consequence of special relativity is the relativity of simultaneity, which holds that two events that appear simultaneous in one inertial frame cannot be simultaneous in another frame moving relative to the first.

Example using the Lorentz transformation

Let two events occur simultaneously at positions x₁ and x₂ in frame S, so that t₁ = t₂. In frame S' moving with velocity v relative to S, their time coordinates are:

t₁' = γ(t₁ - vx₁/c²) t₂' = γ(t₂ - vx₂/c²)

Although t₁ = t₂, more generally t₁' ≠ t₂'. This shows that simultaneity is not absolute, but depends on the observer's frame of reference.

Four-vector and covariant formulation

In relativity, it is often useful to describe physical quantities using four-vectors, which integrate space and time components. For example, the four-position is defined as:

X = (ct, x, y, z)

Accordingly, the four-velocity U is:

U = dX/dτ = γ(c, vₓ, vᵧ, v_z)

where τ is the proper time. This framework extends to the electromagnetic field, which is described using the electromagnetic field tensor F

Electromagnetic field tensor

The electromagnetic field tensor is a mathematical structure that represents the electric and magnetic fields concisely. It is an antisymmetric tensor whose components are defined as follows:

F^μν = | 0 -Eₓ -Eᵧ -E_z | | Eₓ 0 -B_z Bᵧ | | Eᵧ B_z 0 -Bₓ | | E_z -Bᵧ Bₓ 0 |

Using this tensor, Maxwell's equations can be written explicitly in covariant form, giving explicit information about their invariance under Lorentz transformations.

Covariant form of Maxwell's equations

Using the field tensor, Maxwell's equations can be written as two tensor equations:

∂_ν F^μν = μ₀J^μ ∂_σ F^μν + ∂_μ F^νσ + ∂_ν F^σμ = 0

Here, J^μ is the four-current density, which includes both charge and current densities. These equations summarize the behavior of electromagnetic fields in a form that remains valid in different inertial frames.

Electromagnetic force in relativity

The force acting on a charged particle moving in an electromagnetic field is given by the relativistic form of the Lorentz force law:

f^μ = q F^μν U_ν

where f^μ is the quaternion, q is the charge of the particle, F^μν is the electromagnetic field tensor, and U_ν is the quaternion velocity of the particle.

Variation of electric and magnetic fields

To understand how the electric and magnetic fields transform between different frames, consider two inertial frames S and S', where S' moves with velocity v relative to S along x axis.

Eₓ' = Eₓ Eᵧ' = γ(Eᵧ - vB_z) E_z' = γ(E_z + vBᵧ) Bₓ' = Bₓ Bᵧ' = γ(Bᵧ + (v/c²)E_z) B_z' = γ(B_z - (v/c²)Eᵧ)

It shows how electric and magnetic fields mix under the Lorentz transformation and emphasizes the unified nature of electromagnetic fields.

Visual representation of electric and magnetic field changes

E-Region B-region

In the diagram above, you see concentric circles representing the electric field lines and a perpendicular line representing the magnetic field. This helps to visualize how the direction and intensity of the fields change when transformed into another inertial frame.

Field from a moving charge

The electric and magnetic fields produced by a point charge q moving with a constant velocity v are described as:

E(r, t) = (q/4πε₀) (1 - v²/c²) / [(r - vt)² (1 - (v²/c²)sin²θ)^(3/2)] B = (1/c²) v × E

Here, θ is the angle between the position vector r and the velocity vector v. These equations shed light on how motion affects the size and strength of the fields.

Conclusion

Relativistic electrodynamics is a cornerstone of modern physics, combining classical electromagnetism and special relativity. This amalgamation not only resolves conceptual dilemmas that exist at high velocities, but also lays the groundwork for later theories such as quantum electrodynamics.


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Relativistic electrodynamics

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