Electric exponential decay

The second (real) term accounts for the exponential decay of the electric field intensity in the direction normal to the interface. The reflected beam combines with the incident beam, forming a standing electromagnetic wave at the interface (Fig. 9.9). The electric field that penetrates to the optically rarer medium of refractive index n, the evanescent field, plays a critical role in many optical sensors based on the waveguiding principle. Its depth of penetration dv is defined as the distance at which the initial intensity Eq decays to 1/e of its value. Thus from (9.18), dv is... 

The absorption index, k, introduces an exponential decay in E (and E ) with increasing z (see eq. [3]). Recalling that the intensity, I, is related to the electric field amplitude E, according to I lE 2 it follows from eqs. [3] and [4] that... 

Here, Eq is the permittivity of free space. For a simple Debye-type relaxation process, Eq. (4), and owing to the incorrect representation of the high-frequency limit inherent in any expression for K (to) consistent with an exponential decay function for the electric moment, one obtains for the high-frequency limit of a(to) from Eqs. (4) and (6) (30) ... 

As previously discussed, neighboring nephrons can also influence one another through vascular propagated electrical (electrochemical) signals. To account for this mechanism, the total activation potential for k-th nephron is assumed to be the sum of contributions from all other nephrons in the tree. Moreover, the electrical activation potentials are assumed to propagate along the vascular wall with an exponential decay. In this way, the vascular propagated interaction is delivered to each nephron as an additional part of its activation potential xjr. 

Gruen and Marcelja considered that the electric and polarization fields are not proportional in the vicinity of a surface and that while the electric field has the ion concentrations as its source, the source of the polarization field is provided by the Bjerrum defects. The coupled equations for the electric and polarization fields were derived through a variational method. Attard et al.14 contested the Gruen—Marcelja model because, to obtain an exponential decay of the repulsion, the nonlocal dielectric function was assumed to have a simple monotonic dependence upon the wavelength (eq 33 in ref 13). This was found to be inconsistent with the exact expression for multipolar models.14 In addition, the characteristic decay length for polarization (denoted in eq 18, ref 13) is inversely proportional to the square of the (unknown) concentration of Bjerrum defects in ice. While at large concentrations of Bjerrum defects the disordered ice becomes similar to water and the traditional Poisson—... 

Using Equation 4.16 to estimate g in region I at first appears to be a strange procedure because the case Od = 0 corresponds to no overlap of the double layers. The variation of electric potential with distance is then as sketched by the thin line in Figure 4.2. The thicker line in Figure 4.2 represents an exponentially decaying potential function (see method (b) below). Let us first pursue the mathematics of the Od = 0 case. The crude approximation... 

FIGURE 4.2 Schematic illustration of the behavior of the electric potential function in the two models. The thin line is a sketch of < >(x) for the approximation thick line is a plot of an exponentially decaying function, for which d = 0.135s. The potential dimensionless surface potential 4>s and the Debye screening length 1/k, respectively. The dashed line at x = 2/k designates the midplane. 

We would expect intuitively that tan 0 emd the Deborah number De are related, since both refer to the ratio between the rates of an imposed process and that (or those) of the system. The exact shape of this relationship depends on the number and nature(s) of the releixation process(es). So let us anticipate [3.6.4 la] for the loss tangent of a monolayer in oscillatory motion, which describes a special case of [3.6,12], namely -tan0 = t]°(o/K°. Here, (o is the imposed frequency, equal to the reciprocal time of observation, t(obs) =< . The quotient K° /t]° also has the dimensions of a time in fact it is the surface rheological equivalent of the Maxwell-Wagner relaxation time in electricity, (Recall from sec. 1.6c that for the electrostatic case relaxation is exponential ith T = e/K where e e is the dielectric permittivity and K the conductivity of the relaxing system. In other words, T is the quotient between the storage and the dissipative part.) For the surface rheological case T therefore becomes The exponential decay that is required for such a... 

We have carried out time-resolved EL measurements of polymer-based light-emitting devices. The transient EL spikes exhibit bi-exponential decay pattern. Temperature independence of the decay pattern allows us to rule out polarization due to polymer matrix relaxation as an EL governing process. According to our interpretation, the phenomenon of double light spikes under pulsed electrical... 

Zhang, Q., Monsalve-Gonzalez, A., Qin, B.L., Barbosa-Canovas, G.V., and Swanson, B.G. 1994a. Inactivation of Saccharomyces cerevisiae in apple juice by square-wave and exponential-decay pulsed electric fields. Journal of Food Process Engineering 17 469 78. 

Because these lipids are electrically neutral, the dominant interactions between bilayers are algebraically decaying, attractive van der Waals forces and an exponentially decaying hydration repulsion. These interactions lead to... 

Figure Bl.4.6. Left an experimental optical THz pump-probe set-up using sub-picosecond THz pulse generation and detection by the electro-optic effect. Right the application of such pulses to the relaxation of optically excited TBNC in toluene. The THz electric field used for these experiments is shown in the upper-right inset. Three exponential decay terms, of order 2, 50 and 700 ps, are required to fit the observed temporal relaxation of the solvent [51].

Each rate constant determination represents the average of three decay curves which were read from the photographs and converted into digital form by an Oscar K77 curve reader. The rate constants and confidence limits were calculated from these data by making a regression fit to an exponential decay with the assistance of a General Electric Data Net computer. 

Figure 1. Left Schematic of the coordinate system at a totally reflecting interface separating tux) media of refractive index n and n2. Right Standing wave pattern and exponential decay of the electric field vector into the less dense medium, 2.

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