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In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
However, many such phenomena may be explained using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations.
In physics (specifically electromagnetism), Gauss's law, also known as Gauss's flux theorem (or sometimes Gauss's theorem), is one of Maxwell's equations. It is an application of the divergence theorem , and it relates the distribution of electric charge to the resulting electric field .
These forms are equivalent due to the divergence theorem. The name "Gauss's law for magnetism" [1] is not universally used. The law is also called "Absence of free magnetic poles". [2] It is also referred to as the "transversality requirement" [3] because for plane waves it requires that the polarization be transverse to the direction of ...
If electromagnetic energy is not gained from or lost to other forms of energy within some region (e.g., mechanical energy, or heat), then electromagnetic energy is locally conserved within that region, yielding a continuity equation as a special case of Poynting's theorem: = where is the energy density of the electromagnetic field. This ...
By the divergence theorem, ... In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation.
There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density. Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. [1]
In electromagnetic field theory, vector calculus can be used to express the law in terms of charge density ... By the Gauss divergence theorem, this means the rate of ...