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The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
The electromagnetism uniqueness theorem states the uniqueness (but not necessarily the existence) of a solution to Maxwell's equations, if the boundary conditions provided satisfy the following requirements: [1] [2] At =, the initial values of all fields (E, H, B and D) everywhere (in the entire volume considered) is specified;
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 ...
In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation. For example, consider a conductor moving in the field of a magnet. [8]
The Maxwell equations; Principle of least action; Solutions of Maxwell's equations in free space; Solutions of Maxwell's equations with currents and charges; AC circuits; Cavity resonators; Waveguides; Electrodynamics in relativistic notation; Lorentz transformations of the fields; Field energy and field momentum
These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy / matter in general) induce curvature in spacetime, [ 1 ] Maxwell's equations in flat ...
Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS). But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations.
Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. [1] Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇ 2 X (The last term in the right side is the vector Laplacian, not Laplacian applied on ...