Maxwell equations current density

Ionic current density maps can be recorded with the aid of the pulse sequence shown in Figure 2.9.2. The principle of the technique [48-52] is based on Maxwell s fourth equation for stationary electromagnetic fields,... 

E and B are the fundamental force vectors, while P and H are derived vectors associated with the state of matter. J is the vector current density. The Maxwell equations in terms of E and B are... 

By interpreting the term in brackets as the total current density the inhomogeneous Maxwell equation (2) is also written as... 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

Bartlett and Corle [46] proposed modification of Maxwell s equations in the vacuum by assigning a small nonzero electric condictivity to the formalism. As pointed out by Harmuth [47], there was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell s theory. Thus, the equations of the latter fail for waves with nonnegligible relative frequency bandwidth when propagating in a dissipative medium. To resolve this problem, a nonzero electric conductivity ct and a corresponding current density... 

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. 

There is a clear symmetry. In particular there are electrical sources for both fields P, N. There is a simple change of sign in the source, but monopoles do not arise However, there is no such sign difference for the current density J. There are two equations of continuity, one for each field P, N, while there is only one in the unsymmetrized version of Maxwell s equations. 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

We start from the first pair of Maxwell equations written with account of electric charges moving in vacuum. Let vector J be the electric current density produced by these charges. Combining the above-mentioned equations, we get the second-order differential equation for electric field vector E ... 

The remaining three conditions are found using a similar procedure to the other Maxwell equations. K(x) is the surface current density. 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

The media which are of interest for this review do not contain macroscopic charge and current densities, hence p = 0 and 7 = 0, and they are not magnetized, so that M = Q. Then Maxwell s equations and the constitutive relations may be combined to yield the following coupled partial differential equation between the electric field E and the dielectric polarization P. 

Let this new function jo (r) be an extraneous current density for some electromagnetic field. The solutions of the Maxwell s equations with this source are the fields Bq (r) and ho (r) themselves, which are zero outside domain D. Therefore, the currents jo (r) are nonradiating ... 

These four vector relations compactly summarize the experimental laws describing all known electrical and magnetic phenomena. In these expressions, p is the electric charge density, J, the current density, E, the electric field and B, the magnetic induction. Maxwell s equations in free space (in the absence of dielectric or magnetic media) can be written... 

Maxwell (68) calculated the potential distribution for a single spherical particle immersed in a conducting medium and subjected to a uniform electrical field He solved Laplace s equation within the two regions subject to continuity of potential, and continuity of the normal component of the current density, at the surface of the particle. Maxwell then extended his single-sphere solution to dilute mixtures and obtained the following expression for... 

The backing in this model is treated with the Stefan-Maxwell equation to yield the partial pressure of oxygen at the backing/catalyst layer interface from the total pressure Ptot in the gas flow channel, the backing characteristic current density, 7b, and the mole fractions of water vapor and of oxygen, Xws and Xon, respectively ... 

Since Fpy is an antisymmetric tensor in spacetime and since the components of the ordinary affine connection are symmetric in the indices (ap), it follows that the 4-divergence of the current density /, automatically vanishes. In other words, as in the standard formulation, the equation of continuity follows from taking the covariant divergence of Maxwell s equation (31a) ... 

The matter field was originally postulated by Louis de Broglie, and discovered in the electron diffraction studies of Davisson and Germer [30] and of G. P. Thomson [31]. From Schrodinger s understanding of the matter field of, say, an electron, it must be represented in the source terms (charge and current density) of Maxwell s equations, as the moduli of these waves. 

Integration of the local limit of Eq. (50) for the four-current density source of Maxwell s equations, together with the boundary condition that the... 

Here 2 and E are the electric induction and the electric field strength, respectively, and jext, Pext are the external electric current and the charge densities. As is known, the Maxwell equations (4.50) must be complemented by the so-called material equations, which enable the expression of 2 as a function of E. In the linear approximation the relation between T> and E can be taken as... 

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) ... 

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