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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.
[24] [25] Maxwell deals with the motion-related aspect of electromagnetic induction, v × B, in equation (77), which is the same as equation (D) in Maxwell's original equations as listed below. It is expressed today as the force law equation, F = q ( E + v × B ) , which sits adjacent to Maxwell's equations and bears the name Lorentz force ...
The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector F. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with r -forms and there are corresponding operations.
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".
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.
Oliver Heaviside reduced the complexity of Maxwell's theory down to four partial differential equations, [118] known now collectively as Maxwell's Laws or Maxwell's equations. Although potentials became much less popular in the nineteenth century, [119] the use of scalar and vector potentials is now standard in the solution of Maxwell's ...
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.
This process culminated in the 1860s with the discovery of Maxwell's equations, a set of four partial differential equations which provide a complete description of classical electromagnetic fields. Maxwell's equations provided a sound mathematical basis for the relationships between electricity and magnetism that scientists had been exploring ...