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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.
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.
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such ...
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. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
A Dynamical Theory of the Electromagnetic Field. " A Dynamical Theory of the Electromagnetic Field " is a paper by James Clerk Maxwell on electromagnetism, published in 1865. [1] In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that ...
A linearly polarized electromagnetic plane wave propagating parallel to the z-axis is a possible solution for the electromagnetic wave equations in free space. The electric field, E, and the magnetic field, B, are perpendicular to each other and the direction of propagation. Maxwell's equations can be combined to derive wave equations.
Classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields. In a vacuum, electromagnetic waves travel at the speed of light, commonly denoted c. There, depending on the frequency of oscillation, different wavelengths of electromagnetic spectrum are produced.
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]