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 ↓


Lienard–Wiechert potentials


Electromagnetism occupies a fundamental place in the vast fabric of physics. When the study of charges and currents is combined with the principles of relativity, intriguing and complex formulations emerge. One such formulation in the field of relativistic electrodynamics is the concept of "Liénard-Wiechert potential". These potentials describe the electromagnetic effect of a moving charge, incorporating the principles of causality and the finite speed of light.

Introduction to potential in electrodynamics

In classical electrodynamics, potentials are mathematical structures from which electromagnetic fields can be derived. These potentials – namely, scalar potentials (Φ) and vector potentials (A) – play an important role in facilitating the calculation of electric (E) and magnetic fields (B).

Basic equation

The electric field E and the magnetic field B can be expressed in terms of these potentials as follows:

E = -∇Φ - ∂A/∂t 
B = ∇ × A

Here, ∇ is the gradient operator, ∇ × A denotes the curl of A, and ∂ denotes partial differentiation. These equations describe how changes in potentials affect the resulting fields.

The concept of Liénard–Wiechert potentials

When dealing with moving point charges, one must take into account the delay caused by the finite speed at which electromagnetic information travels – the speed of light, denoted by c. Liénard-Wiechert potentials provide an elegant solution to this problem. Named after Alfred-Marie Liénard and Emil Wiechert, these potentials encompass the effect of a moving point charge on the electromagnetic field at a given point in space and time.

Lienard–Wiechert potential formulation

The expressions for the Lienard–Wiechert potential for a moving charge are as follows:

Φ(r, t) = (q / (4πε₀)) * (1 / (1 - (v·n)/c)) * (1 / |r - r₀|) 
A(r, t) = (q / (4πε₀c)) * (v / (1 - (v·n)/c)) * (1 / |r - r₀|)

In these expressions, several words are important:

  • q: charge of the moving particle.
  • v: velocity of the charge.
  • n: unit vector in the direction of the field point relative to the instantaneous position of the charge.
  • c: speed of light.
  • r and r₀: position vectors of the field point and charge respectively.
  • ε₀: permittivity of free space.

The terms 1/(1 - (v·n)/c) are important because they represent the distortion of the fields due to relativistic effects, in particular the relative motion of the charges.

Delayed time explanation

The Liénard–Wiechert potential is evaluated at a time called the "retarded time", t' ret, which is the time over which the electromagnetic effect propagates from the charge to the field point. This time is explicitly defined as:

t' = t - |r - r₀(t')|/c

This expression indicates that the effect of charge motion on t' ret should be evaluated by taking into account the time delay taken by the electromagnetic signal to travel the distance between the position of the charge and the field point.

Visual example

The geometric relationship between the position of the charge, its velocity and the field point can be seen from the following example:

r₀(t') R |r - r₀(t')|

In this illustration, the blue dot represents the position of the charge at slow time t' ret, while the red dot represents the point in the field. The dashed line represents the path the electromagnetic effect travels.

Application examples

Electric field of a moving charge

The electric field produced by a moving point charge can be obtained from the Liénard-Wiechert potential. This electric field is composed of two main components: the "Coulomb field" (which resembles the field of a stationary charge) and the "radiation field" (which reflects the time-varying nature of the motion of the charge). The electric field can be given as follows:

E = q/(4πε₀) * [(n - n·v/c) / (γ²(1 - n·v/c)³) + n × ((n - v/c) × a) / c²(1 - n·v/c)³]

In this expression:

  • a represents the acceleration of the charge.
  • γ is the Lorentz factor, γ = 1/√(1 - v²/c²).

Radiation pattern

Radiation emitted by moving charges can be analyzed using these expressions. Releasing electromagnetic radiation is a hallmark of accelerating charges. Liénard-Wiechert potentials allow physicists to predict how radiation propagates in space, aiding in understanding natural phenomena and technological applications such as antennas and wireless communication systems.

Conclusion

Liénard-Wiechert potentials epitomize how classical electromagnetism adapts and evolves under the rigorous demands of relativity. Combining vector calculus, relativistic theories, and electromagnetic theory, these potentials form a bridge that spans the realm of moving charges, time-delayed effects, and field interactions.

In a world where relativity figures prominently, such as in the movement of particles near the speed of light and in advanced communication systems, the applicability of Liénard-Wiechert potentials continues to provide important insights and indispensable solutions. The discovery of these potentials not only deepens our understanding of electromagnetic phenomena but also enhances our ability to innovate within the framework of modern physics.


Graduate → 2.3.2


U
username
0%
completed in Graduate

← Prev (2.3.1)
Covariant formulation of electrodynamics
Next (2.3.3) →
Synchrotron radiation and bremsstrahlung

Comments 



Lienard–Wiechert potentials

https://www.physicsflow.com/g/2.3.2?pfal=en