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 → Special relativity ↓


Relativistic electrodynamics


Relativistic electrodynamics is a fascinating field of physics that combines Einstein's theory of special relativity with the principles of electrodynamics. At the core of this theory is the concept that the laws of physics, including those governing electric and magnetic fields, should be the same for all observers, whether or not they are at constant velocity relative to one another. This simple premise leads to profound consequences for the way we understand electromagnetic interactions and the nature of space and time.

Fundamentals of special relativity

Special relativity is a theory proposed by Albert Einstein in 1905. It revolutionized our understanding of space, time, and motion. There are two principles at the base of this theory:

  1. The laws of physics are the same for all observers in uniform motion relative to each other (theory of relativity).
  2. The speed of light in a vacuum is the same for all observers, regardless of the speed of the light source or the observer.

From these assumptions, several counterintuitive results follow, such as time dilation and length contraction. For example, if you travel at a significant fraction of the speed of light, time appears to slow down for you relative to a stationary observer. Similarly, objects moving at high speeds will appear shorter in the direction of motion.

Variation of electric and magnetic fields

In nonrelativistic physics, electric and magnetic fields are often treated as separate entities. However, in relativistic electrodynamics, they are components of a unified entity known as the electromagnetic field tensor. Let us consider how the electric and magnetic fields change when moving from one inertial frame to another in special relativity.

Suppose we have two inertial frames, S and S', where S' is moving at a constant velocity v relative to S along the x-axis. The transformation equations for the electric field ( E ) and the magnetic field ( B ) are given by the following Lorentz transformations:

E'_x = E_x
E'_y = γ(E_y - vB_z)
E'_z = γ(E_z + vB_y)
B'_x = B_x
B'_y = γ(B_y + (v/c²)E_z)
B'_z = γ(B_z - (v/c²)E_y)

Here, c is the speed of light, and γ = 1/√(1 - v²/c²) is the Lorentz factor. These equations show how electric and magnetic fields are interrelated; a magnetic field in one reference frame can appear as a mixture of electric and magnetic fields in another.

Visual example: region change

Below is a visual representation demonstrating the transformation of fields between two reference frames:

E' (electric field) B' (magnetic field)

Maxwell's equations in relativistic form

Maxwell's equations describe how electric and magnetic fields are generated and transformed by each other and by charges and currents. In their traditional form, they reflect experiments and observations made at non-relativistic speeds. In a relativistic framework, Maxwell's equations can be expressed more compactly using four-vector and tensor notation. This form makes the equations explicitly invariant under Lorentz transformations.

∂_μ F^μν = μ₀ J^ν
∂_σ F_μν + ∂_μ F_νσ + ∂_ν F_σμ = 0

Here, F^μν is the electromagnetic field tensor, and J^ν is the four-current density. The first equation relates the electromagnetic field tensor to the current density, while the second equation is a mathematical statement of the absence of magnetic monopoles and the fact that the electromagnetic field is a closed two-form.

Electromagnetic field tensor

The electromagnetic field tensor F^μν is an antisymmetric 4x4 matrix that elegantly sums up the electric and magnetic fields:

F^μν = | 0    Ex  Ey  Ez  |
        |-Ex   0  Bz -By |
        |-Ey -Bz   0  Bx |
        |-Ez  By -Bx   0 |

This tensor notation shows that the electric and magnetic fields are two sides of the same coin and transform into each other under Lorentz transformations.

Four-vector in electrodynamics

Four-vectors are a crucial ingredient in the language of relativity, because they make the equations explicitly covariant. This means that they have the same form in all inertial frames, making it easy to ensure that the laws of physics are consistent under Lorentz transformations.

Some important four-vectors in electrodynamics include the four-potential A^μ and the four-current J^μ:

A^μ = (φ/c, A_x, A_y, A_z)
J^μ = (cρ, J_x, J_y, J_z)

Here, φ is the electric scalar potential, A is the magnetic vector potential, ρ is the charge density, and J is the current density.

Visual example: four-vector representation

Imagine a four-vector as an arrow in four-dimensional space:

Aμ (space-time)

Covariance of electromagnetic theory

Special relativity requires that electromagnetic theories be consistent in all inertial frames. The covariance of Maxwell's equations maintains this consistency. For this reason, formulas using the electromagnetic field tensor and four-vectors form a natural language for expressing electromagnetic phenomena in relativistic terms.

Lorentz invariant quantities, such as the magnitude of a four-vector or the action principle derived from the Lagrangian, remain unchanged in different frames, which reinforces this covariance.

Lorentz force in relativity

The Lorentz force describes the force on a charged particle due to electric and magnetic fields. In relativistic terms, it is expressed using a four-vector:

F^μ = q(E + v × B)^μ

Here, q is the charge, E is the electric field vector, and B is the magnetic field vector. The expression v × B represents the vector cross product responsible for the magnetic component of the force.

Practical implications

Understanding relativistic electrodynamics has profound implications in fields such as particle physics, astrophysics, and even engineering disciplines dealing with high-speed electronic components. Relativistic effects become important in systems with high velocities or strong electromagnetic fields, making it essential to consider these theories for accurate predictions and technologies.

Example: synchrotron radiation

Synchrotron radiation occurs when charged particles moving at relativistic speeds are deflected by magnetic fields, producing radiation. This phenomenon is important in the design and operation of synchrotrons, a type of particle accelerator widely used in research.

Conclusion

Relativistic electrodynamics beautifully unifies electric and magnetic phenomena with relativistic concepts of space and time. Using mathematical tools such as tensors and four-vectors, the theory provides an elegant and comprehensive framework for understanding electromagnetic interactions at high velocities. This unified approach has practical applications in physics, from experimental particle physics to cutting-edge technological innovations.


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Relativistic electrodynamics

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