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 ↓


Multipole expansion


In advanced electrodynamics, multipole expansion is a powerful mathematical technique used to describe the spatial distribution of charges or currents and their effects on electric and magnetic fields. This method involves expressing the potential generated by a given distribution in terms of simpler components, called multipole moments. This simplifies the study of fields generated by complex distributions, especially when observations are made at large distances. This technique is not only applicable in electromagnetism but is also useful in gravitational fields and quantum mechanics.

Understanding the basics

First, let us consider the potential V(mathbf{r}) due to the charge distribution (rho(mathbf{r'})) The expression for the electric potential is given as:

V(mathbf{r}) = frac{1}{4piepsilon_0} int frac{rho(mathbf{r'})}{|mathbf{r} - mathbf{r'}|} d^3r'

Here, (mathbf{r}) is the position vector of the point where the potential is being calculated, and (mathbf{r'}) is the position vector within the charge distribution.

Distance and approximation

In many practical scenarios, we are interested in evaluating the potential at a point far away from the charge distribution. In such cases, |mathbf{r}' ll |mathbf{r}| and |mathbf{r} - mathbf{r'}| in powers of |mathbf{r'}| / |mathbf{r}|:

|mathbf{r} - mathbf{r'}| = sqrt{r^2 - 2mathbf{r} cdot mathbf{r'} + r'^2}

Using the Taylor series expansion, we can estimate:

|mathbf{r} - mathbf{r'}| approx r left(1 - frac{mathbf{r} cdot mathbf{r'}}{r^2} + cdots right)

In this expansion, each term corresponds to a different polypole order.

Multipole moment

1. Monopole

The simplest term in the expansion is the monopole term, which corresponds to treating all charges as if they were concentrated at a single point. The monopole moment is simply the total charge Q of the distribution.

Q = int rho(mathbf{r'}) d^3r'

The potential due to the monopole moment at a distance r is given by:

V_{mon}(mathbf{r}) = frac{Q}{4piepsilon_0 r}

2. Dipole

The next term is the dipole moment which is responsible for the first-order deviation from the point charge. It is defined as:

mathbf{p} = int mathbf{r'} rho(mathbf{r'}) d^3r'

The potential due to dipole moment is:

V_{dip}(mathbf{r}) = frac{mathbf{p} cdot mathbf{r}}{4piepsilon_0 r^3}

The dipole moment is a vector that points from the negative charge to the positive charge and its magnitude depends on the amount of separation.

Visual representation

Unipolar

Why Potential Area

Dipole

-Why +Q D Dipole

In the diagram of a dipole, the negative charge is at one end while the positive charge is at the opposite end. The dipole moment vector lies along the line joining the charges, which points toward the positive charge.

Quadrupole moment

The quadrupole moment is the next term in the series. It describes second-order deviation and involves two points with opposite charges. The quadrupole tensor Q_{ij} is defined as:

Q_{ij} = int (3x'_i x'_j - r'^2 delta_{ij}) rho(mathbf{r'}) d^3r'

where delta_{ij} is the Kronecker delta. When the distribution is symmetric or when the observation point is very far away, the quadrupole term becomes more important than the dipole.

Quadripolar representation

-Why +Q +Q -Why Quadrupole

This diagram shows a quadrupole setup where there are equal negative and positive charges in two perpendicular arrangements. The effects of quadrupole become essential in fields such as nuclear physics and astrophysical bodies.

Higher multipole moment

Higher-order moments at the octupole and beyond can also be calculated using similar strategies, involving more complex terms and higher derivatives. These terms diminish rapidly with distance, but they can be important for high-precision calculations in some scientific discoveries.

Applications of multipole expansion

Multipole expansions are widely used in areas where the spatial separation between objects allows simplification:

  • Gravitational fields: Similar extensions can be used to describe gravitational potentials in celestial mechanics, especially in analyzing the stability of satellite orbits and the structure of galaxies.
  • Molecular physics: The electron cloud in a molecule and the resulting electric and magnetic fields can be described using multipoles to simplify calculations in quantum chemistry.
  • Antenna theory: In the design of antennas, the radiation pattern can be described as a multipole expansion to optimize directional emission properties.

Conclusion

Multipole expansion provides a systematically elegant approach to tackling complex problems in electromagnetism by breaking down possible contributions into manageable components. Its utility across a variety of disciplines underscores its importance in the understanding of physical phenomena at many scales. As science progresses, mastering these fundamental techniques remains crucial to advancing technology and furthering our understanding of the universe.


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Multipole expansion

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