Induction vector

According to the macroscopic Maxwell approach, matter is treated as a continuum, and the field in the matter in this case is the direct result of the electric displacement (electric induction) vector D, which is the electric field corrected for polarization [7] ... 

In the foregoing, B is the magnetic induction vector. For a more elementary derivation of the above results, based on a rather specialized case, see Exercise 1.6.3, while a more elegant derivation is furnished in the Appendix, Section 1.7. Again, Eq. (1.6.15a) is awkward in actual use because H and B are local fields which reflect the reaction of the medium to the applied field, and because the integration extends over all space. For practical applications it is more convenient to use Heine s (1956) formulation... 

E and H being the electric and magnetic field strength vectors, D the electric displacement vector, B the magnetic induction vector, J the electric current density, and p, the electric charge density. 

One considers a composite material made of a mixture of conductors and insulators. As above, for p < Pc, the material is insulating, and conducting for p > Pc- We consider first the case with p smaller than Pc- The equations for the induction vector D and the field E are ... 

The long-wavelength field can be easily found if we take into account that in a medium without external charges the longitudinal component of the induction vector T> vanishes, and the macroscopic electric field is longitudinal, if the retardation, as assumed in the theory of Coulomb excitons, is not taken into account. From this considerations we obtain... 

To find the operator EH(r), we consider only the long-wavelength limit of the longitudinal field, where some well-known relations from phenomenological theory can be applied. In particular, in virtue of the solenoidal character of the induction vector (divD = 0), we have kD(w, k) = 0 for plane waves. Simultaneously D = E + 4-7tP (see also Section 4.4) so that the longitudinal parts of the vectors E and P are related by... 

In determining the quantities Coulomb interaction has been completely taken into account. In this case, linear response theory determines only the so-called (see (44) and the next Ch. 7) transverse dielectric tensor ej y(w, k). This tensor relates the induction vector T> to the transverse part of the macrofield E Di(ui, k) = see also eqn (7.3). 

Nonlinear optical effects can be described within the framework of macroscopic electrodynamics (see, e.g. Bloembergen (49)), by applying the nonlinear relation between the induction vector T> and the strength E of the macroscopic electric field. When the value of E in the light wave is small compared to the intra-atomic electric fields, this nonlinear relation can be written in the form of the expansion... 

The theory of nonlinear optical processes in crystals is based on the phenomenological Maxwell equations, supplemented by nonlinear material equations. The latter connect the electric induction vector D(r,t) with the electric field vector E(r, t). In general, the relations are both nonlocal and nonlinear. The property of nonlocality leads to the so-called spatial dispersion of the dielectric tensor. The presence of nonlinearity leads to the interaction between normal electromagnetic waves in crystals, i.e. makes conditions for the appearance of nonlinear optical effects. 

If the thickness of the surface layer in which a surface exciton-polariton is localized considerably exceeds the lattice constant of a crystal, the electric and magnetic field strength vectors, i.e. vectors E and H of a wave with energy hw in both media (in vacuum and in the crystal the crystal is assumed to be nonmagnetic so that the magnetic induction vector B = H), satisfy Maxwell s equations... 

Early investigators of electric and magnetic fields observed that when the magnetic induction vector B changes with time throughout a surface S bounded by a contour L, an electromotive force S exists along that contour with an intensity ... 

Respectively, the flux of the magnetic induction vector piercing the receiver is ... 

In the Maxwell approach, in which matter is treated as a continuum, we must in many cases ascribe a dipole density to matter. Let us compare the vector fields D and E for the case in which only a dipole density is present. Differences between the values of the field vectors arise from differences in flieir sources. Both the external charges and the dipole density of the sample act as sources of these vectors. The external charges contribute to D and E in the same maimer (2). The electric displacement (electric induction) vector D is defined as... 

Integration in Eqn (10.2) is performed over V area occupied by the current sources. As (r, (p, z) one can mark cylindrical coordinates of a free point in space, and as (p,6,Q—cylindrical coordinates of a point that belongs to V area. The magnetic induction vector is determined from the equality B = rotA.. Magnetic carriers (separate nanoparticles or their aggregates) can be considered as magnetic dipoles. In general, the dipole is influenced by the force F and mechanical moment M which are defined by the known formulas ... 

In the electrodynamics of dielectrics, and electric induction vector D(r) and a dielectric polarization vector P(r) are introduced for describing the fields acting in the medium. Correspondences between these objects and those coming from the GSCRF theory are established below. 

B magnetic induction vector 9 wave vector of magnons or para-... 

Poisson equation, i.e., the electroneutrality equation is basically the Gauss law, that is to say the Maxwell s equation giving the dependence between the electric induction vector and charge density... 

We introduce here the notation Ob = and ObPh = q K2lm Ob is effective conductivity, i.e., the semiconductor conductivity in the direction of electric field when there is an influence of magnetic field. In a general case its value depends on the intensity of magnetic induction vector, is Hall mobility, representing effective drift mobility under the influence of transversal electric field with an... 

Vector B = Bn is referred to as a magnetic field induction vector. 

Solution In order to find the magnetic induction at the specified point A we should define the directions of induction vectors Bj and CTeated by each conductor separately and then combine two vectors B = Bj + Bj. We can find the module of total induction according to the cosine theorem... 

Solution Since the z-axis is perpendicular to the ring plane and passes the center of the ring, it is a symmetry axis L. This means that the induction vector certainly must be codirectional to the z-axis and only the B component gives contribution to the field induction B. First, derive the general expression for B(z). 

Expression (5.1.20) is the essence of Ampere s law circulation of the induction vector along a closed contour L is equal to the current multiplied by /Iq comprised by this contour. 

If the particle velocity is directed at an angle a to a vector B (Figure 5.16, where B is directed along the z-axis), vector o has to be projected in two directions perpendicular and parallel n to the induction vector B. Accordingly, the component defines the circular motion of the particle and another component On determines its uniform motion along axis z since the Lorentz force for this component is zero. This results in the particle s spiral movement. The basic spiral increment h is defined as... 

The concept of a vector flux d through a surface dS has been given in Sections (2.8.3) and (4.1.3). Being a particular case of a more general theory of a vector field, the same concepts can be applied to a magnetic field as well. An elementary flux d of a magnetic field induction vector B through the surface dS is equal to a scalar product of B and dS... 

Depending on the angle between a normal n to the surface dS (Figure 2.20) and the induction vector B, a flux dd> can vary in limits BdS. In general, flux through surface S is defined by the integration... 

Figure 5.43 Electromagnetic wave (a) schematic representation of electromagnetic bundles and (b) oscillation of the field inductions vectors E and B in an electromagnetic wave.

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