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In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
Faraday's law of induction (or simply Faraday's law) is a law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf). This phenomenon, known as electromagnetic induction , is the fundamental operating principle of transformers , inductors , and many types of electric ...
Faraday's law describes two different phenomena: the motional emf generated by a magnetic force on a moving wire (see Lorentz force), and the transformer emf that is generated by an electric force due to a changing magnetic field (due to the differential form of the Maxwell–Faraday equation).
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 ...
The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical fluid dynamics : the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.
Using the differential form of Faraday's law, ∇ × E = − ∂B / ∂t , this gives =. By definition, B = μ 0 ( H + M ) , where M is the magnetization of the material and μ 0 is the vacuum permeability .
The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form: [1] [2] = . Therefore, F is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
Faraday's law of induction was suggestive to Einstein when he wrote in 1905 about the "reciprocal electrodynamic action of a magnet and a conductor". [15] Nevertheless, the aspiration, reflected in references for this article, is for an analytic geometry of spacetime and charges providing a deductive route to forces and currents in practice.