The electric field

The electric field vector E at a point in space is the negative gradient of the electrostatic potential at that point  

As E is the negative gradient vector of the potential, the electric force is directed downhill and is proportional to the slope of the potential function. The explicit expression for E is obtained by differentiation of the operator r — r T1 in Eq. (8.2) towards x, y, z, and subsequent addition of the vector components. With 3 r — r l/dx = r — r -2 dt /dx, we obtain, for the negative slope of the potential in the x direction, 

In Section 10.4.6, the relation between the held and the potential was shown to be  

If the potential varies linearly with distance, then the held is uniform over that distance, but if the potential varies non-linearly with distance then the held over that distance is non-uniform. 

In the case of conductance studies, the ions move under the influence of the electric field. Because the field is uniform this means that the ions migrate at a constant velocity. This will become relevant in the section on ionic mobilities (Section 11.17), and when discussing the relaxation and electrophoretic effects in the theories of conductance (see Sections 12.1, 12.2 and 12.4). 

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Writing the electromagnetic field in terms of transverse electric (TE) and transverse magnetic (TM) components, the electric field has the form ... 

The equation (1) assumes the knowledge of the incident field E (r) which is the electrical field in the anomalous domain considering the flaw absent. This field must be computed before and one can imagine that small errors in estimation of this field may 2586... 

By using the method of the dyadic Green s function [4] and the adequate boundary conditions [5], the expressions of the electric field in the zone where the transducer is placed can be written as [2]... 

This subject has a long history and important early papers include those by Deijaguin and Landau [29] (see Ref. 30) and Langmuir [31]. As noted by Langmuir in 1938, the total force acting on the planes can be regarded as the sum of a contribution from osmotic pressure, since the ion concentrations differ from those in the bulk, and a force due to the electric field. The total force must be constant across the gap and since the field, d /jdx is zero at the midpoint, the total force is given the net osmotic pressure at this point. If the solution is dilute, then... 

Derive the expression for the electric field around a point dipole, Eq. VI-5, by treating the dipole as two charges separated by a distance d, then moving to distances X d. 

Consider the interaction of a neutral, dipolar molecule A with a neutral, S-state atom B. There are no electrostatic interactions because all the miiltipole moments of the atom are zero. However, the electric field of A distorts the charge distribution of B and induces miiltipole moments in B. The leading induction tenn is the interaction between the pennanent dipole moment of A and the dipole moment induced in B. The latter can be expressed in tenns of the polarizability of B, see equation (Al.S.g). and the dipole-mduced-dipole interaction is given by... 

In this fonn it is clear that k leads to an attenuation of the electric field amplitude with distance (i.e. absorption). 

This expression may be interpreted in a very similar spirit to tliat given above for one-photon processes. Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian H. In general, there is also a second-order interaction with the electric field through which returns a portion of the excited-state amplitude to surface a, with subsequent evolution on surface a. The Feymnan diagram for this second-order interaction is shown in figure Al.6.9. 

Defining EJh + oij, replacing v /(-co) by v r(0), since the difference is only a phase factor, which exactly cancels in the bra and ket, and assuming that the electric field vector is time independent, we find... 

As described at the end of section Al.6.1. in nonlinear spectroscopy a polarization is created in the material which depends in a nonlinear way on the strength of the electric field. As we shall now see, the microscopic description of this nonlinear polarization involves multiple interactions of the material with the electric field. The multiple interactions in principle contain infomiation on both the ground electronic state and excited electronic state dynamics, and for a molecule in the presence of solvent, infomiation on the molecule-solvent interactions. Excellent general introductions to nonlinear spectroscopy may be found in [35, 36 and 37]. Raman spectroscopy, described at the end of the previous section, is also a nonlinear spectroscopy, in the sense that it involves more than one interaction of light with the material, but it is a pathological example since the second interaction is tlirough spontaneous emission and therefore not proportional to a driving field... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The work done increases the energy of the total system and one must now decide how to divide this energy between the field and the specimen. This separation is not measurably significant, so the division can be made arbitrarily several self-consistent systems exist. The first temi on the right-hand side of equation (A2.1.6) is obviously the work of creating the electric field, e.g. charging the plates of a condenser in tlie absence of the specimen, so it appears logical to consider the second temi as the work done on the specimen. 

Any cavity contains an infinite number of electromagnetic modes. For radiation confined to a perfectly conducting cubical cavity of volume V= L, the modes are given by the electric field components of the fomi ... 

An alternative approach is to consider ions of charge z e accelerated by the electric field strengtii, E, being subject to a frictional force, Kj, that increases with velocity, v, and is given, for simple spherical ions of... 

It is important to recognize the approximations made here the electric field is supposed to be sulficiently small so that the equilibrium distribution of velocities of the ions is essentially undisturbed. We are also assuming that the we can use the relaxation approximation, and that the relaxation time r is independent of the ionic concentration and velocity. We shall see below that these approximations break down at higher ionic concentrations a primary reason for this is that ion-ion interactions begin to affect both x and F, as we shall see in more detail below. However, in very dilute solutions, the ion scattering will be dominated by solvent molecules, and in this limiting region A2.4.31 will be an adequate description. 

The experimental data and arguments by Trassatti [25] show that at the PZC, the water dipole contribution to the potential drop across the interface is relatively small, varying from about 0 V for An to about 0.2 V for In and Cd. For transition metals, values as high as 0.4 V are suggested. The basic idea of water clusters on the electrode surface dissociating as the electric field is increased has also been supported by in situ Fourier transfomr infrared (FTIR) studies [26], and this model also underlies more recent statistical mechanical studies [27]. 

In order to evaluate equation B1.2.6, we will consider the electric field to be in the z-direction, and express the interaction Hamiltonian as... 

Flere, is the static polarizability, a is the change in polarizability as a fiinction of the vibrational coordinate, a" is the second derivative of the polarizability with respect to vibration and so on. As is usually the case, it is possible to truncate this series after the second tenn. As before, the electric field is = EQCOslnvQt, where Vq is the frequency of the light field. Thus we have... 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

Flere the Fresnel transfonnation describes the relationship between the electric field /iciin medium 1... 

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

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