Maxwell’s structure

Maxwell s structure is often used for modelling parallel processes. As seen in Figure 4.21, Maxwell s structure is basically RC circuits in parallel. The concept of admittance is useful because the overall admittance of parallel elements is simply the sum of the individual admittances, i.e.,... 

A simple example of Maxwell s structure with two parallel processes is shown in Figure 4.22a. 

We shall assume light propagation along z-axis and electromagnetic field distribution independent of y coordinate. Solution of Maxwell s equations for such a structure can be assumed in the form ... 

Figure 1. Schematic illustration of waveguiding structures, a ray-picture b modal intensity distribution obtained with the aid of Maxwell s equation c mode coupling c whispering gallery mode.

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elementary solid, which reflected the vibrational energy of a three-dimensional solid, should be equal to 3RJK-1 mol-1. The anomaly that the free electron theory of metals described a metal as having a three-dimensional structure of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add another (3/2)R to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas the quantum theory of free electrons shows that these quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

Maxwell s equations, 16 513, 514 24 652 Maxwell-Stefan equations, 1 43-46, 598 Maxzide, molecular formula and structure, 5 165t... 

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

MODELING OF ONE-DIMENSIONAL NONLINEAR PERIODIC STRUCTURES BY DIRECT INTEGRATION OF MAXWELL S EQUATIONS IN THE FREQUENCY DOMAIN... 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

Here I describe a simple numerical method for solving Maxwell s equation in the frequency domain. As the structure to be analysed is onedimensional, Maxwell s equations turn into a system of two coupled ordinary differential equations that can be solved with standard numerical routines. 

Consider a one-dimensional optical structure that consists of nonlinear layers. The structure is illustrated in Fig. 1. Assuming no field variation in the X and z directions and the convention exp(icot) for the time dependence of the fields, Maxwell s equations take the form... 

A Gibbs angel, like Maxwell s demon, unfortunately does not exist. Neither is there at present an experimental technique to achieve what could be accomplished by a Gibbs angel. Hence, one has to try to build up a picture of the structure of the interface by letting the mind play with the other types of cruder information that are available. 

To summarize this subsection, the description of the dynamics of the force lines as the level curves of two maps. S 3i -rS2, given by two complex functions topological structure, in such a way that the mere existence of a pair of such functions guarantee that the corresponding pullbacks of the area 2-form in S2 automatically obey the Maxwell s equations in empty space. 

A three-layer, planar optical waveguide consists basically of a high-RI film with RI np that is sandwiched between two lower-refractive-index materials referred to as the substrate and the cover medium with RIs ns and c> respectively Fig. 2. If Maxwell s equations are solved for such a layered structure, the following solutions for the electromagnetic field [Pg.281]

Beyond the simple thin surfactant monolayer, the reflectivity can be interpreted in terms of the internal structure of the layer, and can be used to determine thicker layers and more complex surface structures, and this can be done in two different ways. The first of these uses the optical matrix method [18, 19] developed for thin optical films, and relies on a model of the surface structure being described by a series or stack of thin layers. This assumes that in optical terms, an application of Maxwell s equations and the relationship between the electric vectors in successive layer leads to a characteristic matrix per layer, such that... 

A convenient way of comparing many unit systems used in electromagnetism is to rewrite Maxwell s equations, by introducing six constants (k to k6) plus the dimensionless Sommerfeld79 fine-structure constant a, and remembering that for SI i 0 = 107/47ic2 and /i0 = in x 10 7 ... 

The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

The reader should examine not only the individual steps in the derivation, but also the internal structure of every presentation, the exploitation of Maxwell s equations, and the relation between commonly used thermodynamic functions- and experimental information, such as equations of state. 

A basic waveguide structure, which is sketched in Fig. 1, is composed of a guiding layer surrounded by two semi-infinite media of lower refractive indices. The optical properties of the stmcture are described by the waveguiding layer refractive index Hsf, and thickness t, and by the refractive indices of the two surrounding semi-infinite media, here called (for cover) and (for substrate). Application of Maxwell s equations and boundary conditions leads to the well-known waveguide dispersion equation [6] ... 

Our assumption here is that the conception of the C.C. and their actual form will one day be as commonplace to structural biologists as Maxwell s equations are now to physicists. The C.C. are out there in nature, just as much as Maxwell s equations. What can be said of them today ... 

Analysis of motions of a structure using a and n representations has many applications beyond chemistry. For example, some interesting results come from applications of these ideas to macroscopic, engineering-scale structures. In stmctural mechanics. Maxwell s rule gives a necessary though not sufficient condition for rigidity of a 3D bar-and-joint assembly. It is... 

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