Average macroscopic field

In Equation (2.183) new surface charges, qex, have been introduced these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of qex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium more details on this equivalence will be given in Section 2.7.4. 

In condensed media consisting of molecules, the intermolecular forces such as permanent and induced dipole interactions are generally small compared to intramolecular chemical binding forces. Therefore, the molecular identities and properties are conserved to a certain extent. They nevertheless differ significantly from those of an isolated molecule in the gas phase. Therefore, both in linear and non-linear optics the question arises of how to relate molecular to macroscopic properties. More specifically, how do the individual permanent and induced dipole moments of the molecules translate into the macroscopic polarization of the medium The main problem is to determine the local electric field acting on a molecule in a medium which differs from the average macroscopic field E (Maxwell field) in this medium. 

Dielectric permeability unit—dimensionless (a value indicating by how much an averaged macroscopic field in a dielectric is less than an external field)... 

By analogy with macroscopic electrodynamics, we consider the currents in the turbulent fluid as molecular ones, the averaged actual field h we denote by B, and we introduce 77 here curl 77 = 0 in the region where there are only unordered, turbulence-dependent currents. From (13) we obtain B = (r/ )77 = 77/Rem. Thus, macroscopically, the turbulent conducting fluid acts like a diamagnet1 with very small permeability n 1/Rem. 

Here, E n = 0 on Sp (Neumann type boundary condition), where n is the unit outward normal from the pore region, and T> is compact. E can be interpreted as the microscopic electric field induced in the pore space when a unit macroscopic field e is applied, assuming insulating solid phase and uniform conductivity in the pore fluid. Its pore volume average is directly related to the tortuosity ax ... 

In addition to the field generated by the adjacent dipoles, there is a macroscopic fieldE due to the presence of charges and of the average polarization P of the medium. In the Lorentz treatment of polarization, for a constant macroscopic field in a linear and homogeneous medium of dielectric constant e (hence satisfying P = e0(e — 1)E), the local field E n(fl at a site of a selected dipole is related to the macroscopic field E via19... 

The effective Eeld E, is the sum between the macroscopic field Eo and the internal field due to the other molecules of the medium. The latter can be calculated by adding the individual contributions of the other molecules close to the selected one, Elncal, and subtracting the contribution from the same molecules treated in an average continuum approximation described by the polarization P ... 

In the traditional double layer theory, the average dipole moment of a water molecule is proportional to the macroscopic field (the derivative of the electrical potential)... 

The idea is that only the field from molecules near to the solute has to be considered, so that the complete system can be separated into microscopic and macroscopic regions. The molecules in the latter can be described by the average macroscopic properties. Lorentz derived a simple relation between the local field and the macroscopic polarization, which is still in use [113,133,134,135]. 

We note firstly that the energy dissipation depends on Eq and so will be particularly strong where the field is locally enhanced, e.g. at electrodes and where space charges are present. Secondly, the appropriate value of the loss compliance will be that for the craze region in the amorphous phase, and this is likely to be greater than the average macroscopic value, which includes the crystalline phase. 

Consider a suspension of spherical soft particles in a general electrolyte solution of volume V. We define the macroscopic average electric field (E) and current (/) in... 

On performing in (187a) an averaging with the statistical function (100) for the perturbation (188), we obtain by equation (187) and the general expression for the mean macroscopic field Et within a dielectric sphere of permittivity placed in a strong external field E, viz. 

Having derived the microscopic expression for polarization, the focus is now on the macroscopic formulation of the dielectric constant for a cubic crystal. The relative dielectric constant r (the ratio of absolute dielectric constant eabs with 0) can be introduced through the average electric field E acting on a crystal unit cell as ... 

Let us consider a macroscopic volume Vt containing N molecules. Let E0 be the mean macroscopic field due to electric charges situated outside the volume under consideration. The average moment of the volume, in the direction of the field is... 

After having accomplished formally the derivation of the Maxwell equations and having defined the macroscopic field quantities in terms of ensemble averages, it is perhaps useful to add a few remarks which may help to clarify the relation between the earlier Lorentz treatment and the present one. 

For a multidomain ferromagnetic sample in zero external field, the magnetization is homogeneous within a domain, but the directions of magnetizations differ for different domains. Thus the macroscopic magnetization M(Hq = 0) and the average demagnetizing field vanish, and... 

Here y is average molecular polarizability (generally, yij is a tensor), Eioc is a local electric field acting on each molecule and Pe is a field induced electric dipole in a molecule. So, if we find Eioc we could calculate P and then the value of s, using Eq. 7.12, and known macroscopic field E in the sample, see Fig. 7.4a. 

---