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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.
These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and ...
Some observed electromagnetic phenomena cannot be explained with Maxwell's equations if the source of the electromagnetic fields are the classical distributions of charge and current. These include photon–photon scattering and many other phenomena related to photons or virtual photons , " nonclassical light " and quantum entanglement of ...
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.
The electromagnetic field is a covariant antisymmetric tensor of degree 2, which can be defined in terms of the electromagnetic potential by =.. To see that this equation is invariant, we transform the coordinates as described in the classical treatment of tensors: ¯ = ¯ ¯ ¯ ¯ = ¯ (¯) ¯ (¯) = ¯ ¯ + ¯ ¯ ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯ = ¯ ¯ = ¯ ¯.
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay (retarded time) of the fields due to the finite speed of light and relativistic effects.
The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that emf (electromagnetic work done on a unit charge when ...
The modern theoretical treatment of electromagnetism is as a quantum field in quantum electrodynamics. In many situations of interest to electrical engineering, it is not necessary to apply quantum theory to get correct results. Classical physics is still an accurate approximation in most situations involving macroscopic objects.