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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.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
A continuity equation or transport equation is an equation that describes the transport of ... although charge conservation is more fundamental than Maxwell's equations.
Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials.
Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.
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:
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]
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor in three dimensions that is used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum.