Intensity vector

For weakly guiding structures, the second term can be neglected, and we obtain the standard Helmholtz equation in which individual components of the electric field intensity vector E remain uncoupled. For high contrast waveguides this is clearly not the case. The second term in Eq. (2) in which the transversal electric field components are mutually coupled must be retained. 

The biorthogonality relations (11.15) make clear the far-reaching symmetry between intensive vectors R ) and their conjugates R ) in the geometrical formalism. The formal symmetry is also seen in relations of the form... 

Taken together, (11.22) and (11.23) lead to various thermodynamic identities between measured response functions, as will be illustrated below. Equation (11.23) shows that the inverse metric matrix M-1 plays a role for conjugate vectors R/) that is highly analogous to the role played by M itself for the intensive vectors R,). In view of this far-reaching relationship, we can define the conjugate metric M,... 

As previously remarked (Sidebar 10.1), intensive vectors have privileged status in Ms The asymmetry between intensive and extensive variables can already be recognized in the U-based (or S-based) fundamental equation of Gibbs... 

The thermodynamic vector a) for the coexistence coordinate cr is therefore composed from the old intensive vectors T), — P) by the equation... 

Arbitrarily chosen intensive vectors Rai) are in general neither orthogonal nor normalized. It is therefore of interest to identify a particular choice of A for which these vectors become an orthonormal set. Such unit-like response vectors, denoted as E ), can be expressed as usual as linear transformations of standard reference intensities... 

The coexistence conditions (12.68) that relate each excess intensive vector RK) to the chosen axis intensities R ) can also be written in terms of the conjugate extensive vectors R ) = X/). With the usual metric relationship between intensive and extensive vectors,... 

Equations (12.68) establish how any /+ 1 intensive vectors must be related to one another in the heterogeneous equilibrium of p phases. Let us now assume for definiteness that these are labeled... 

The chemical species set of a state is the set of chemical constituents that are associated with the system description of that state. The values of the attributes chemical-species-set, operating-conditions, and system-volume provide the (n + 2) independent variable quantities that are necessary to define a thermodynamic state. An important feature of this representation is that each state is described by a vector of intensive and extensive variables. The intensive vector defines the operational state of the process, while the extensive vector defines the maximum accumulation of mass and energy that can occur. This is bounded by flowrate, reaction rate and physical size of the process equipment. The values of these variables are accessed through the attributes interval flowrate vector, interval accumu-... 

The most usual situation that leads to low power factors is the use of inductive motors that, as the name implies, can introduce very large inductive reactance in the line. The electric load introduced by an inductive motor can be represented as a resistance and an inductance in series. This combination will have an intensity vector that will be delayed with respect to the voltage. 

If we add a capacitor in parallel to the motor, we are imposing an intensity vector that will be advanced 90° with respect to the voltage. By using the right size capacitor we can decrease the angle by which the overall intensity is delayed with respect to the voltage and, therefore, increase coscp. A capacitor with only a few micro-farads (pF) usually suffices to increase the power factor from, say, 75% to approximately 85%i or higher. 

In Fig. 1, / is the initial current in amperes, equivalent to the vector length oa. Addition of a capacitor in parallel will create an intensity vector Ic advanced 90°... 

The distribution of electric and magnetic intensity vectors of a surface plasmon can be obtained from Eqs. 31-33 ... 

In the space coordinate system, a, b and c the exciting light is within direction a. polariz.cd in the b-c plane (Fig. 5.1) (Albrecht, 1961). Fluorophore molecules which possess a transition moment parallel to the a direction will be excited. These molecules arc shown in Figure 5.1 with shaded planes. An intensity vector can be prcgected into the three axis a, b and c. One intensity will be parallel to the a direcum and thus will be called lu. The other two intensities oriented parallel to the h and c axis and thus oriented perpendicular to the a axis. Tliese two intensities are called l, Therefore, the global inten ty I is equal to... 

The presence of dispersed fillers in the polymer material in low amounts may intensify electrization, increase the residual charge and change the friction coefficient. Introduction of the filler in the electret state exerts a still stronger effect on polymer electrization on frictional interaction with metals. Depending on the direction of the field intensity vector formed by the filler particles, the field generated by triboelectrization can be attenuated or intensified. This means that the principle of the electret-triboelectrization superposition is realized [49], which can be used to regulate the parameters of frictional interactions. Thus, by the introduction of the electret filler, e.g. mechanically activated F-3 powder, it is possible to decrease the friction force (Fig. 4.9). 

The acoustic energy density and intensity vector for a plane wave are given by... 

The intensity vector of the radiated plane wave in the fluid, which is a measure of the acoustic energy flux carried by the waves, is given by... 

This action may corresponds with the application of an electromagnetic field (or an electric plus a magnetic field) on a direction around which the intensities vectors of the electric fields (for example) oscillate as non-polarized, as shown in Figure 4.25. 

This last optical effect, is also called as the optical rotation power, referring to the ability of (an)isotropic medium to rotate (to the left or to the right) the transmission polarization plane (of oscillation) of the electric intensity vector respecting the incident direction of polarization. 

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