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 → Radiation and scattering ↓


Multipole expansion


In advanced electrodynamics it is important to understand how electromagnetic fields, such as dipoles, tripoles and higher order, behave and propagate in space. A powerful analytical technique used to simplify complex field interactions is the "multipole expansion". This method allows us to express the electromagnetic field as a series of simpler, often more manageable, contributions called multipoles.

Introduction to electromagnetic radiation and scattering

When a charge distribution is made to oscillate or accelerate, it emits electromagnetic radiation. Similarly, when an incoming electromagnetic wave hits an obstacle, it is scattered, producing a complex pattern of emitted fields. Both processes involve interactions that can be described mathematically using the multipole expansion.

Basics of multipole expansion

At its core, the multipole expansion is a way of expressing a complex electromagnetic field as a sum of simpler fields. We begin by considering the potential due to a charge distribution:

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

Here, [Phi] denotes the electric potential at the position [mathbf{r}], while [rho(mathbf{r'})] is the charge density at the position [mathbf{r'}]. The challenge lies in evaluating this integral, since the term [|mathbf{r} - mathbf{r'}|] is often cumbersome to handle directly.

Breaking down the possibilities

Using mathematical techniques including Taylor expansions, we can express the potential [Phi] as a sum of different contributions, each of which corresponds to a different "multipole":

- Monopole: Represents a point charge. This is the simplest order. - Dipole: Consists of two equal and opposite charges that are some distance apart. - Quadrupole: Uses a configuration of four charges. - Higher orders such as octupoles follow, but these are generally small and can often be neglected in approximations.

Let's imagine a simple dipole using a geometric illustration:





-Why
+Q

In the figure, the red circle represents a negative charge, while the blue circle represents a positive charge. The line has a dipole moment vector, which shows the direction and magnitude of the moment.

Mathematical representation of polypoles

Each term in the multipole expansion is related to certain symmetries and associated with distances or powers of separation:

[Phi(mathbf{r}) approx frac{1}{4piepsilon_0} left( frac{Q}{r} + frac{mathbf{p} cdot mathbf{r}}{r^3} + frac{1}{2} frac{Q_{ij} r_i r_j}{r^5} + ldots right)]

- Q is the total charge (monopole term). - [mathbf{p}] represents the dipole moment. - [Q_{ij}] is the quadrupole moment tensor.

At large distances, the importance of the higher-order terms quickly diminishes because they fall off faster than the lower-order terms. In many applications, this property allows us to shorten the series and consider only the first few terms.

Electric and magnetic polypoles

The multipole expansion is not limited to electric fields only. Magnetic fields can also be expanded using a similar approach. In practice, magnetic multipoles arise from current loops or distributions and depend on currents rather than charge distributions. For example:

[mathbf{A}(mathbf{r}) approx frac{mu_0}{4pi} left( frac{mathbf{m} times mathbf{r}}{r^3} + ldots right)]

Here, [mathbf{m}] denotes the magnetic dipole moment. Scattering processes, especially in complex media, benefit greatly from analyzing these multipolar contributions.

Applications in radiation and scattering

1. Antenna design: Engineers use multipole expansion to optimize radiation patterns. The far-field region of antennas often transforms complex interactions into simpler dipole or quadrupole patterns, helping to understand and improve design features.

2. Molecular interactions: In quantum chemistry, the molecular potential energies between atoms or molecules can be deconvoluted into multipolar contributions, giving better information about bond strengths and angles.

3. Cosmic phenomena: Multipole expansion helps in understanding the complex magnetic fields around celestial bodies, where considering a planet as an ideal magnetic dipole simplifies calculations and predictions.

Challenges and limitations

While the multipole expansion simplifies many scenarios, it is not universally applicable. Its efficacy is reduced when: - The observation point is not far enough from the charge or current distribution. - The charge distribution undergoes rapid changes, introducing significant higher-order terms.

Despite these limitations, its utility remains important in systems where symmetry dominates or far-field interactions are substantial.

Conclusion

The multipole expansion, though inherently a mathematical technique, goes beyond mathematics, providing physical intuition about natural and engineered electromagnetic systems. Its reach remains vast and varied, from engineering solutions to unravelling cosmic mysteries.

Understanding and applying multipole expansions in electrodynamics provides both elegant simplicity and powerful insights into the nature of electromagnetic interactions. With continued advances in computational techniques, multipole methods hold the potential to unlock deeper understanding and more sophisticated applications in the future.


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Multipole expansion

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