Energy of a dipole

In addition to the nonelectrostatic adsorptive force, there is an image force between a dipole and a metal, which will be present whenever charged or dipolar particles in a medium of one dielectric constant are near a region of another dielectric constant. If the metal is treated as an ideal conductor, the image-force contribution to the energy of a dipole in the electrolyte is proportional to p2j z3, where z is the distance of the dipole from the plane boundary of the metal (considered ideal, with no surface structure), and to 1 + cos2 0. This ideal term is, of course, the same for all metals. If... 

The energy of a dipole in a constant electric field in SI units is given by ... 

From the studies made by I.R. and Raman spectra on 1,2-dihaloethanes, the conclusion has been that the gauche form is more important in polar than in non polar solvents. This is because that the gauche form has a considerable dipole moment while the anti has nearly none. Solvation by polar solvents reduces the potential energy of a dipole and so makes the gauche form more stable relative to the anti. 

The potential energy of a dipole with charges e and — e at a distance d from one another, and an ion with charge e at a distance r from the dipole, Figure 33> will be... 

On pages 159 and 160 formulae have been derived for the energy of a dipole pi and a particle with polarizability a in the field of an ion a at a distance r. If the field of the ion is called F, these expressions become —piF and —- aF2. It was noted before that the factor [jl in the latter formula is due to the fact that the energy-required to polarize a particle by a field F is equal to half the energy of the particle in that field. 

Form factor of the hat-curved model Normalized concentration of molecules Kirkwood correlation factor Steady-state energy (Hamiltonian) of a dipole Dimensionless energy of a dipole Moment of inertia of a molecule Longitudinal and transverse components of the spectral function Complex propagation constant Elasticity constant (in Section IX)... 

The trajectories of the major portion part of dipoles-rotators occupy in this case almost all the sphere (it is seen in Figs. 7 and 9). Consequently, a nonhomogeneity of the potential (94) could be neglected also in this case. Thus again we may consider free rotation of a dipole, but now in a zero potential, unlike the small-(3 case considered above. Hence, the kinetic energy of a dipole... 

The corresponding spectral functions, denoted L(z) and L(z), are derived, as well as the SF L for the rotators, in Section V.E in the form of simple integrals from elementary functions over a full energy of a dipole (or over some function of this energy). The total spectral function is thus represented as... 

How would such a dipole respond to an applied electric field The energy of a dipole Mdipoie pointing at an angle 6 to the direction of an electric field E is —E //.dipoie = -Mdipoie E cosO. (Remember that an electric field pushes the positive charge and pulls the negative.)... 

Two charges interact according to Coulomb law, with a strength directly proportional to the product of their magnitude and inversely to their distance. The interaction energy of a dipole with a charge, a dipole, or a generic multipole of rank /, is the scalar product between the dipole and the electric... 

Other ions in the solution. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager s theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechaiucs from which the adsorption isotherm is derived. 

The Electric-Field-Induced Second-Harmonic Generation (EFISHG) technique makes it possible to measure the molecular hyperpolarizability, p, on liquids or molecular solutions. The centrosymmetry of tire solution is broken by applying a DC electric field to induce an average orientation of the molecules due to interactions of the permanent dipoles of the molecules and the electric field. The energy of a dipole with a permanent dipole fi in an electric field E is given by ... 

The potential energy of a dipole depends on the orientation of the dipole moment /z ip with respect to the applied field E or... 

Let us consider the case where dipole 1 is fixed and dipole 2 is free to rotate in the field of dipole 1. The potential energy of a dipole in an electric field (J5) is given by ... 

The energy of a dipole is a state function otherwise, the system could not store energy. The variation of this inductive (electromagnetic) energy is necessarily an exact differential... 

Our main goal in this chapter is to investigate how these classical results change when the dipoles belong to molecules that are governed by quantum mechanics. In Section 2.3 we evaluate the energy of a dipole exposed to an external field. In Section 2.4 we consider the field produced by a quantum-mechanical dipole. In Section 2.5 we address the interaction between two dipolar molecules. 

Whereas the classical energy of a dipole in a field can take a continuum of values between its minimum and maximum, this is no longer the case for a quantum-mechanical molecule. In this section we will establish the energy spectrum of a polar molecule in an electric field, building fi om a set of simple examples. To start, we will define the laboratory z-axis to coincide with the direction of an externally applied electric field, so that = z. In this case, the projection of the total angular momentum on the z-axis is a conserved quantity. 

This simple idea can be generalised to give other useful formulae. The electric field E is minus the derivative of the potential, but it is not most efficiently obtained by differentiating Eq. (18). Instead we recall that the energy of a dipole ii in an electric field is —/i E, so the component of the field is minus the coefficient of the dipole... 

EXAMPLE 21.4 The energy of a dipole in an electric field. Figures 21.9 and 21.10 show a dipole with its center fixed in space. The dipole is subject to an orienting force from an electric field E. Compute the work w that the field performs in rotating the dipole from 0 (parallel to the field) to an angle 0,... 

The potential energy of a dipole /r, in the presence of a charge Q2 is calculated by taking into account the interaction of the charge with the two partial charges of the dipole, one resulting in a repulsion and the other an attraction. The result for the arrangement shown in (7) is... 

To arrive at the mean field description, we stiU need an expression for P. Knowing that an energy of a dipole in an external field is (-p-E), the mean potential for a dipole of a species i reads... 

If it is taken into consideration that the potential energy of a dipole in an electric field is much smaller than the thermal energy, then the time variation of the orientation polarization P of N dipoles having dipole moment p is found to follow a linear differential equation ... 

The potential energy of a dipole in an electric field is given by U = —p E. Therefore, an attractive potential exists between two electric dipoles that are aligned nose to tail that goes as and the force between them goes as —pipijr. Thus a dipole-dipole bond is... 

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