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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.
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]
In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations [1] which were to become known as Maxwell's equations, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in his 1861 paper "On Physical Lines of Force".
Lorentz force on a charged particle (of charge q) in motion (velocity v), used as the definition of the E field and B field.. Here subscripts e and m are used to differ between electric and magnetic charges.
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 ...
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector , + = or , + =, which expresses the conservation of linear momentum and energy by electromagnetic interactions.
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The electromagnetic field admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not rectilinear. For example ...