Electric Fields Have Fluxes

Chapter 17 describes the flux of a flowing fluid. You can also define the flux for electric field vectors. Why is it useful to define a flux for electric fields The electric field flux has the important general property that it is independent of 

Two profound implications of Equation (20.15) lead to a general method for calculating electrostatic forces and fields. First, the right-hand side of Equation (20.15) does not depend on r. For the electric flux through a spherical container, the r dependence of E cancels with the r- dependence of the spherical surface area. Because of this cancelation, the electric held flux out of a sphere is independent of its radius. It doesn t matter how large you choose your imaginary balloon to be, the flux is the same through spherical balloons of any radius. Second, we show now that the shape of the balloon doesn t matter either. 

The Electric Field Flux is Independent of Balloon Shape 

To relate Corner to dinner, compute the field through the outer element, 

The flux is the same through the outer balloon area element as through the inner balloon element. 

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When dislocations move in a piezoelectric crystal, not only mechanical but also electrical fields are produced around them. For a continuous distribution of moving dislocations, these mechanical and electrical fields have been given in terms of the dislocation-density and flux tensors by the aid of convolution integrals with respect to the whole region where there exist dislocations [1]. Further computations have been made to estimate the fields produced by a uniformly moving infinite straight dislocation by application of Cauchy s theorem and Jordan s lemma (see [2] to [4]). The present paper is devoted to a short summary of these investigations with some additional remarks. 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

Henry et al (23) have collected experimental data on cross-flow electro-filtration of Kaolin clay suspensions and oil-water emulsions. Since both the Kaolin particles and the oil droplets are negatively charged in aqueous suspensions, a direct electric field will always give higher filtration rates than cross-flow filtration alone. The level of improvement depends on the intensity of the fluid shear and the electric-field strength. Figures 47 and 48 present data for the Increase in flux with electric field strength for the oil-water emulsion and the clay suspension. 

Cabral and coworkers [253] have investigated the batch mode synthesis of a dipeptide acetyl phenylalanine leucinamide (AcPhe-Leu-NH2) catalyzed by a-chymotrypsin in a ceramic ultrafiltration membrane reactor using a TTAB/oc-tanol/heptane reverse micellar system. Separation of the dipeptide was achieved by selective precipitation. Later on the same group successfully synthesized the same dipeptide in the same reactor system in a continuous mode [254] with high yields (70-80%) and recovery (75-90%). The volumetric production was as high as 4.3 mmol peptide/l/day with a purity of 92%. The reactor was operated for seven days continuously without any loss of enzyme activity. Hakoda et al. [255] proposed an electro-ultrafiltration bioreactor for separation of RMs containing enzyme from the product stream. A ceramic membrane module was used to separate AOT-RMs containing lipase from isooctane. Application of an electric field enhanced the ultrafiltration efficiency (flux) and it further improved when the anode and cathode were placed in the permeate and the reten-tate side respectively. 

Although attractive isotope effects may be expected especially in the Ca2H selective solvent polymeric membranes mentioned [see B. E. Jepson and R. DeWitt, J. Inorg. Nucl. Chem., 38, 1175 (1976)], we have not so far studied such effects. As compared to the flux of ions in the elec-trodialytic transport experiments we carried out on solvent polymeric membranes, the flux of ions is negligible in the absence of an electric field (other parameters kept constant). 

When neither an external fluid flow nor an electrical field is imposed, the diffuse double layer is at equilibrium and the net ion fluxes due to the electrical migration and Brownian diffusion should vanish. Thus, the ions have Boltzmann distributions,... 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

This is the fundamental electric field equation that applies at any point in an isotropic medium. In this context the quantity e0e is the absolute permittivity of the material, and the ratio e, which we have called the dielectric constant of the material, is more properly termed the relative permittivity (with respect to the absolute permittivity of free space e0) and we shall use this term. The flux of dielectric displacement begins and ends on free charge and otherwise is continuous, even at an interface between two media. Electric field, on the other hand, is discontinuous at an interface between two different materials as a result of the different degrees of polarisation. 

Thus, even though two electrolytic conductors have the same geometry, they need not necessarily have the same specific conductivity (Fig. 4.52 and Table 4.8) the number of charge carriers in that normalized geometry may be different, in which case their fluxes under an applied electric field will be different. Since the specific conductivity of an electrolytic solution varies as the concentration, one can write... 

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