Continuity equation electromagnetic

Essentially what has enabled us to rewrite the matrix element in this form is the fact that the sources of the electromagnetic potential, that is, the currents of the charged particles obey the continuity equation d j x) — 0. 

Then what is the source of /iM in electromagnetic theory Are there restrictions on A that should also apply to /iM The answer is yes —it is the restriction of gauge invariance in order to yield a unique representation for the electric and magnetic field variables. Additionally, gauge invariance is the necessary and sufficient condition for the existence of conservation laws in the formalism—in this case the requirement of the conservation of electrical charge [13]. The latter follows from the continuity equation,... 

The governing equations for MHD have two components classical fluid dynamics and electromagnetics. The former includes mass continuity equation and N-S equation. The latter includes Maxwell s equation, current continuity equation, and constitutive equations. For an incompressible electrolyte solution of density, p, and viscosity, p, the continuity and N-S equations are, respectively, described as... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The electromagnetic field is required to satisfy the Maxwell equations at points where e and ju, are continuous. However, as one crosses the boundary between particle and medium, there is, in general, a sudden change in these properties. This change occurs over a transition region with thickness of the order of atomic dimensions. From a macroscopic point of view, therefore, there is a discontinuity at the boundary. At such boundary points we impose the following conditions on the fields ... 

As noted elsewhere [67], Eq. (14) means that the continuity condition does not prohibit the existence of an electromagnetic current density J in free space. It is stressed that Eq. (14) is a mathematical prediction of Maxwell s equations, completely independent of any interpretation. 

In elecfrodynamics the coupling between matter and the electromagnetic field is mediated by the charge density, p(x,t), and the current density, j(x,t), which together satisfy an equation of continuity,... 

We employ matrix methods in order to obtain the reflection and transmission coefficient of the electromagnetic field within the device. Stratified structures with isotropic and homogeneous media and parallel-plane interfaces can be described by 2 x 2 matrices because the equations governing the propagation of the electric field are linear and the tangential component of the electric field is continuous [15,16], We consider a plane wave incident from the... 

Electrcnnagnetic Field in Dielectrics.—MaxweWs Equations. icompassing the laws of Ampere, Faraday, and Gauss, Maxwell in 1864 proposed in final form the macroscopic theory of the electromagnetic field in unbounded space filled with matter. The complete set of Maxwell s equations for a continuous material medium is, when expressed in vector symbolism, of the form ... 

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