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The electromagnetic wave equation is a second-order partial differential ... The multipole expansion of the electromagnetic field finds application in a ...
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal nĚ‚, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299 792 458 m/s [2]). Known as electromagnetic radiation , these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays .
The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function. In the case of EM fields the required expansion of the field is the Fourier expansion.
Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three ...
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
We introduce the polarization density P, which has the following relation to E and D: = + and the following relation to the bound charge: = Now, consider the three equations: = = = The key insight is that the sum of the first two equations is the third equation.
The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the electric field. [1] The equation is named after Hermann von Helmholtz, who studied it in 1860. [2]