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[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 term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials. These relations are named for the nineteenth-century physicist James Clerk Maxwell .
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.
The inhomogeneous Maxwell equation leads to the continuity equation: =, = implying conservation of charge. Maxwell's laws above can be generalised to curved spacetime by simply replacing partial derivatives with covariant derivatives:
Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity. [1] Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special ...
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]
Maxwell's equations (in partial differential form) are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a cyclic manner: the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated over and over ...