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The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.
Maxwell's equations are the basis of theoretical and computational methods describing light scattering, but since exact solutions to Maxwell's equations are only known for selected particle geometries (such as spherical), light scattering by particles is a branch of computational electromagnetics dealing with electromagnetic radiation ...
The first equation listed above corresponds to both Gauss's Law (for β = 0) and the Ampère-Maxwell Law (for β = 1, 2, 3). The second equation corresponds to the two remaining equations, Gauss's law for magnetism (for β = 0) and Faraday's Law (for β = 1, 2, 3).
Eighteen of Maxwell's twenty original equations can be vectorized into six equations, labeled to below, each of which represents a group of three original equations in component form. The 19th and 20th of Maxwell's component equations appear as and below, making a total of eight vector equations. These are listed below in Maxwell's original ...
Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
[24] [25] Maxwell deals with the motion-related aspect of electromagnetic induction, v × B, in equation (77), which is the same as equation (D) in Maxwell's original equations as listed below. It is expressed today as the force law equation, F = q ( E + v × B ) , which sits adjacent to Maxwell's equations and bears the name Lorentz force ...
Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case: = , = + . It must be recognized that the symbols in the equations of this article represent the complex multipliers of e j ω t {\displaystyle e^{j\omega t}} , giving the in-phase and out-of-phase parts with respect ...