Equipotentials

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

Figure Bl.6.1 Equipotential surfaces have the shape of lenses in tlie field between two cylinders biased at different voltages. The focusing properties of the electron optical lens are specified by focal points located at focal lengthsandy2, measured relative to the principal planes, The two principal rays emanating...

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

An electric potential placed across a needle and a flat (plate) electrode. The lines of equipotential in the resulting electric field are focused around the tip of the needle, where the electric field becomes very large. 

The CHA is used for xps today in most commercial spectrometers. The most common version of the CHA is the 180° device shown in Figure 22. It consists of two concentric hemispherical surfaces of radii R2 and R. These surfaces have a potential difference of AH appHed between them so that the outer surface is negative and the inner surface is positive. The median equipotential surface Rq falls between these surfaces ideally, Rq = The... 

Foi shts placed at from the center of curvature, the electrons passed by this analyzer foUow the equipotential surface described by R. With an acceptance angle 8a shown in Figure 22 and a sht width w, the energy resolution of the CFIA is given by... 

Fig. 3-31 Equipotential lines on the soil surface above a defect in the pipe coating (DN 700, thickness of soil covering 1.5 m).

Installation of grounds, standing surface insulation, or equipotential grounding mats are necessary in the case of pipeline potentials higher than 65 V [2]. In the case of long sections of parallelism for an overhead power line and a pipeline, continuous grounding with respect to the distances and the resistances of the grounds... 

The impressed current, 7, flows from the spherical anode radially in a symmetrical field (i.e., the equipotential lines represent spherical shells). It follows from Eq. (24-1)... 

If the spherical anode is situated at a finite depth, f, the resistance is higher than for t and lower than for t = 0 (hemisphere at the surface of the electrolyte). Its value is obtained by the mirror image of the anode at the surface (f = 0), so that the sectional view gives an equipotential line distribution similar to that shown in Fig. 24-4 for the current distribution around a pipeline. This remains unchanged if the upper half is removed (i.e., only the half space is considered). 

The CHA is shown in schematic cross-section in Fig. 2.5 [2.5]. Two hemispheres of radii ri (inner) and T2 (outer) are positioned concentrically. Potentials -Vi and -V2 are applied to the inner and outer hemispheres, respectively, with V2 greater than Vi. The source S and the focus E are in the same plane as the center of curvature, and Tq is the radius of the equipotential surface between the hemispheres. If electrons of energy E = eVo are injected at S along the equipotential surface, they will be focused at Eif ... 

If electrons are injected not exactly along the equipotential surface, but with an angular spread Aa about the correct direction, then the energy resolution is given by ... 

Aquipartition,/. equipartition. aquipotential, aquipotentiell, a. equipotentiaL equivalent, a. equivalent. 

Thickening of the passive film on all surfaces occurs and decreases a small current now flows from the crevice to the outer surface, but this is so small that the surfaces are practically equipotential. 

It is possible that the stationary-state situations leading to an active ion transport occur only in localized regions of the membrane, i.e., at ATPase molecule units with diameters of about 50 A and a length of 80 A. The vectorial ion currents at locations with a mixed potential and special equipotential lines would appear phenomenologically like ionic channels. If the membrane area where the passive diffusion occurs is large, it may determine the rest potential of the whole cell. 

We suppose that the boundary surface S is equipotential, that is,... 

Inasmuch as the boundary surface is an equipotential surface for both potentials Ui and U2, their difference is also constant on this surface and correspondingly we can write... 

As was pointed out in the Chapter 1, it is very useful to study a scalar field with the help of equipotential or level surfaces. At each point of such a surface the potential is constant, and correspondingly its equation is... 

Inasmuch as this system is not used so often as Cartesian, cylindrical, or spherical coordinates, let us describe it in some detail. First of all, we find a condition when a family of non-intersecting surfaces can be a family of equipotential surfaces. Suppose that the equation of the surfaces is... 

This formula was introduced by the Italian geodesist Somigliana in 1929 and defines the behavior of the gravitational field on the level surface of the spheroid for any distribution of density inside, as long as the outer surface remains equipotential. 

We have derived formulas for the gravitational field outside and at the surface of the rotating spheroid with an arbitrary value of flattening /, provided that this surface is equipotential. Such a distribution of the potential U(p) takes place only for a certain behavior of the density of masses. For instance, as follows from the condition of the hydrostatic equilibrium this may happen if the spheroid is represented as a system of confocal ellipsoidal shells with a constant density inside each of them. 

Here C/ is the potential of the field of attraction. Inasmuch as we assume that the earth s surface is equipotential, the vector lines of the normal gravitational field are perpendicular to this surface. This condition can be represented as... 

This equation describes any level surface of the potential U of the gravitational field y, where x, y are coordinates of a point on the surface, while C is the value of the potential. At the same time, the potential of the attraction field varies on this surface. Our next step is to represent the left hand side of Equation (2.195) in the spherical system of coordinates and then, using Equation (2.192), obtain the equation of the equipotential surface, which coincides with the outer surface of the earth spheroid. As was shown earlier, the potential related to a rotation is... 

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