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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.
General purpose, includes 2D and 3D magnetics solvers, both static and harmonic. 3D solver is based on the Whitney AV formulation of Maxwell's equations. VSimEM: Commercial Yes Yes Yes Yes Yes Automatic, variable mesh FDTD, PIC, finite volume: Simulating electromagnetics, and electrostatics in complex dielectric and metallic environments.
The resulting linear system of equations is commonly solved using conjugate gradient iterations. The discretization matrix has symmetries (the integral form of Maxwell equations has form of convolution) enabling fast Fourier transform to multiply matrix times vector during conjugate gradient iterations.
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 .
English: Figure 1-4 Electromagnetic spectrum diagram from The Army Institute for Professional Development, Principles of Radio Wave Propagation. February 2005, Number SS0130 Edition B February 2005, Number SS0130 Edition B
The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions.
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave [6] (electromagnetism). However, Jefimenko's equations show an alternative point of view. [7]
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation .