Displacement Current Maxwells Equations

To end this chapter it is useful to generalize all the aspects discussed they compose a system of equations of classical Maxwellian electrodynamics from which all the laws of electricity and magnetism can be derived, including electromagnetic radiation. Nearly all the equations are already known to the readers, so we can concentrate mainly on the physical conclusions. 

Let us start with those equations that describe stationary phenomena. One of the equations is the Gauss law (Section 4.1.3). Its physical sense concerns the statement the sources 

The Ampere law from which the nonpotential character of the magnetic field follows is the next Maxwell equation. It follows from this law that the source of the magnetic field is the electric current and that it is of a nonpotential character  

The potential character of an electrostatic field follows from equality to zero of circulation of the strength vector E of the electrostatic field. 

Since there are no magnetic charges (monopoles) in nature, the flux of the magnetic induction through the closed surface is zero AO). 

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Chubykalo and Smirnov-Rueda [2,56] have presented a renovated version of Hertz theory, that is in accordance with Einstein s relativity principle. For a single point-shaped charged particle moving at the velocity v, the displacement current in Maxwell s equation is modified into a convection displacement current ... 

These laws are useful but represent cause without effect, that is, fields propagating without sources, and the Maxwell displacement current is an empirical construct, one that happens to be very useful. These two laws can be classified as U(l) invariant because they are derived from a locally invariant U(l) Lagrangian as discussed already. Majorana [114] put these two laws into the form of a Dirac-Weyl equation (Dirac equation without mass)... 

The definition of electric charge density in Eq. (76) agrees with our opinion that 0 in Maxwell s equations represents charge neutrality (see Section HI) the simplest case is 5+ + S = 0. Also note that X/ defined by Eq. (74) is independent of pe thus allowing for the existence of a displacement current in the absence of electric charge, as also discussed in Section HI. 

It should be mentioned that in the approach with nonzero electric divergence, the photon mass is also related to the space charges in vacuo. Now, in the approach with a / 0, we have j = ctE but jeff = 0. Let us now assume j = aE and j 7 0, which means fs 0. In such a case, jo is assumed to be associated with p, where p is the charge density in vacuo. So, in such an approach one can think of the existence of a kind of space charge in vacuo that is to be considered to be associated to nonzero electric field divergence. This will result in a displacement current in vacuum similar to that measured by Bartlett and Corle [43]. The assumption of the existence of space charge in vacuo makes our theory not only fully relativistic but also helps us to understand gauge condition. In the conventional framework of Maxwell s equations... 

Indeed, it was Maxwell s generalization of the laws of electrodynamics that revealed that the radiation solutions of these equations, which would not have appeared in the earlier version (without the displacement current term) predicted all the known optical phenomena. After Maxwell s investigation of these optical implications of electrodynamics, other portions of the spectrum of radiation solutions were predicted and discovered empirically radiowaves, X- rays, infrared radiation, and gamma rays. Thus, it was Maxwell s intuitive feeling for the need of symmetry in his laws of electrodynamics that led to the full unification of electrodynamics and optics in the expression of Maxwell s equations. 

Their results are based on three approximations (i) The conduetivity of the eell membrane was eonsidered to be very small eompared with the cytoplasm and the suspending medium conductivity ( m and cr,). (ii) The membrane thickness is small when compared with the cell inner radius d R). (iii) The displacement current in the suspending medium and the cytoplasm is negligible compared with the conduction current s = e, = 0). Taken together, these three approximations make r/ independent of the permittivity of the suspending medium and the cytoplasm, and independent of the permittivity and conductivity of the membrane. Recently we used Laplace transforms of Maxwell s mixture equation (Eqs. (1) and (2)) to derive complete analytical expressions for the two characteristic relaxation time constants [39]. 

Numerous experiments have shown the appropriateness of selecting the vector X in this form. This quantity was called a displacement current. As follows from the second Maxwell equation, there are two sources for the magnetic field conduction currents and displacement currents. Applying Stoke s theorem, we obtain the integral form of the second Maxwell equation ... 

Maxwell equations neglecting the displacement current because of relatively low velocities ... 

From Maxwell s equations, it follows that only the sum of direct and displacement currents... 

The province of conventional dielectric measurements is here taken to be the determination of the relations of the polarization E and current density J. to the electric field in the macroscopic Maxwell equations. Proper theory should account for these relations in condensed phases as a function of state variables time dependence of applied fields and molecular parameters by appropriate statistical averaging over molecular displacements determined by the equations of motion in terms of molecular forces and fields. Simplifying assumptions and approximations are of course necessary. One kind often made and debated is use of an effective or mean local field at a molecule rather than the sum of microscopic... 

The Maxwell s equations link the four macroscopic Helds D, E, H and B with the external charge density pe and the external current density Jg. The term (9/9t)D is called displacement current and it is added to the external current density term in Eq. (1.4) to satisfy the charge conservation, i.e. the continuity equation ... 

We will deal with electromagnetic phenomena in the electrostatic regime, that is, we disregard any magnetic and radiative effects. In accordance with the continuum hypothesis, the governing equations for continuous media are Maxwell equation. Here, the eleetric field E, the electric displacement field D, the magnetic field B, the polarization field P, the electrical current density and the electrical potential (p are averaged locally over their microscopic counterparts. The fundamental equations are... 

A Wo is the steady state value of AW. In analogy to the displacement current in Maxwell s equations [Eq. (1.1.1)] the variable component of AW, called AWi,has to be a time derivative. Thus... 

It is important to note that equations (2.17) and (2.18) predict that every electric wave will be accompanied by an associated magnetic wave, and vice versa. The waves are therefore more properly termed electromagnetic waves. In 1888, following an experimental search for the magnetic effects of Maxwell s displacement current. Hertz discovered waves which had exactly this electromagnetic character and possessed all the other properties which can be predicted from Maxwell s equations. Our present sophisticated radio and telecommunication systems have all developed from these first primitive experiments. 

Comparison of eqs. (5.4.1) and (5.4.2) allows us to find an infringement of symmetry in the magnetic and electric laws. In fact, if the source of a vertex electric field is an alternating magnetic field, it can be expected that the alternating electric field should cause the occurrence of the magnetic field. From the equations presented above, this does not follow. So Maxwell came analytically to the symmetry brake phenomenon, which later originated the new notion of displacement current. 

The conditions of propagation of an electromagnetic wave in the ionospheric medium are determined by the Maxwell and Lorentz equations. These equations determine the properties of the electric (E) and magnetic (H) fields, as well as the displacement (D) and the induction (B) as a function of the electric charge and current densities (J) (see, for example, Budden, 1961 Davies, 1965) ... 

Since one of Maxwell s equations leads directly to the condition V i = 0, the total current (including both conduction and displacement terms) is space invariant at all times even in dynamic situations. This is often a useful condition in the onedimensional case since it allows one to write... 

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