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Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299 792 458 m/s [2]). Known as electromagnetic radiation , these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays .
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]
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.
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.
The Riemann–Silberstein vector is used as a point of reference in the geometric algebra formulation of electromagnetism.Maxwell's four equations in vector calculus reduce to one equation in the algebra of physical space:
[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 magnetic field in the magnet frame and the electric field in the conductor frame must generate consistent results in the conductor. At the time of Einstein in 1905, the field equations as represented by Maxwell's equations were properly consistent. Newton's law of motion, however, had to be modified to provide consistent particle trajectories.