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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.
An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or a principal U(1)-bundle, on the fibers of which U(1) acts regularly. The principal U(1)- connection ∇ on the line bundle has a curvature F = ∇ 2 , which is a two-form that automatically satisfies d F = 0 and can be interpreted as a field strength.
The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. In other words, the medium must be continuous[no need to be continuous][This paragraph need to be revised, the wrong concept of "continuous" need to be ...
One of the early uses of the matrix forms of the Maxwell's equations was to study certain symmetries, and the similarities with the Dirac equation. The matrix form of the Maxwell's equations is used as a candidate for the Photon Wavefunction. [8] Historically, the geometrical optics is based on the Fermat's principle of least time. Geometrical ...
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 ...
The requirement that the equations remain consistent when viewed from various moving observers led to special relativity, a geometric theory of 4-space where intermediation is by light and radiation. [2] The spacetime geometry provided a context for technical description of electric technology, especially generators, motors, and lighting at first.
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.
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 ...