Equipotential plane

Polanyi s potential theory of adsorption views the area immediately above an adsorbent s surface as containing equipotential lines which follow the contour of the surface potential. When a molecule is adsorbed it is considered trapped between the surface and the limiting equipotential plane at which the adsorption potential has fallen to zero. Figure 9.1 illustrates these equipotential planes. In the diagram, Y represents a pore and X depicts some surface impurity. 

According to the potential theory the volume V, defined by the adsorbent s surface and the equipotential plane , can contain adsorbate in three different conditions depending upon the temperature. Above the critical temperature the adsorbate can not be liquified and the gas in the adsorption volume V simply becomes more dense near the surface. At temperatures near, but less than the critical temperature, the adsorbate is viewed as a liquid near the surface and a vapor of decreasing density away from the surface. Substantially below the critical temperature... 

In corrosion systems, a salt film may cover an electrode that is itself covered by a porous oxide layer. If two different layers are superimposed, the geometrical analysis shows that the equivalent circuit corresponds to that described in Section 9.3.1 with an additional series Rti a. circuit to take into account the effect of the second porous layer. The circuit shown in Figure 9.5 is approximate because it assumes that the botmdary between the inner and outer layers can be considered to be an equipotential plane. This plane will, however, be influenced by the presence of pores. The circuit shown in Figure 9.5 will provide a good representation for systems with an outer layer that is much thicker than the inner layer and with an inner layer that has relatively few pores. 

Of course, in order to obtain an equipotential plane in this case, the reaction taking place at the counter-electrode must have a large reaction constant. 

Fig. la. The physical design of Soref s analog bar graph is shown. The long substrate has a high resistivity transparent conductor so that a potential drop Vi is produced down the substrate from X=0 to X=L. The narrow substrate has a low resistivity transparent conductor so that it is an equipotential plane at zero potential. 

For an ideahzed, energetically homogeneous surface, aU the points equidistant from the surface will have the same potential s and wiU, therefore, form an equipo-tential plane. Thus, the broken lines in the figure represent planes connecting points of equal potential. The value of the adsorption potential s of the parallel equipotential planes decreases as their distance from the surface increases and falls to zero at the maximum distance. Each equipotential surface encloses between itself and the surface of the adsorbent a volume W. The maximum volume is enclosed between the adsorbent surface and the limiting equipotential plane at which the potential has decreased to zero. Thus, the volumes enclosed between the adsorbent and the equipotential surfaces s= s . .. are W2 W3... The quantity represents the volume of the entire adsorption space. As W increases from zero to... 

Equipotential plane. A mass of conducting material that offers a negligible impedance to current flow, thus producing zero volts (equipotential) between points on the plane. 

FIGURE 10.280 The equipotential plane (a) individual signal sources, (b) net effect on the signal reference plane. (Courtesy of Intertec Publishing. After [5].)... 

Practical versions of an equipotential plane include the following ... 

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...

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 ... 

Figure 9.12 Three-dimensional map of the calculated electrostatic potential at 0.25 nm above the symmetry plane in a hexagonally ordered network of dipoles with a dipole-dipole distance of 1.61 nm and a dipole moment of 10 D. The dipoles are positioned at the minima. Note that the potential is lowered at every position on the surface. Equipotential lines for -1.05, -0.84, -0.63 and -0.42 V are indicated in the bottom plane. The contours are circular at short distances from a potassium atom, indicating that at these sites the nearest potassium atom largely dominates the potential. The equipotential tine for -0.42 V, however, has hexagonal symmetry due to the influence of the dipoles further away (from Janssens et al. [40]).

Equation (16.13) implies that all the points in the x-O-y plane, for which y — x define the asymptotes of the equilateral hyperbola, are at the same potential. The field lines are orthogonal to equipotential curves, which implies that for each point A/Xyz inside the quadrupole, the value of the electrical field will be ... 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

Fig. 9.9 (a),(b) Calculated closed trajectories of electron motion at energy W = 0, corresponding to the first seven resonances shown in Fig. 9.8. The final state is in all cases m = 0. Projections onto the p,z plane are depicted and the equipotential W = 0 is shown, (c) Closed trajectory of type 3 at W = 0 shown in projection onto the x,y plane. The final state is again m = 0 (from ref. 23).

FIGURE 5. Electrostatic potentials of vinylamine with respect to a positive point charge. Equipotential curves through planes parallel to the CCN plane at a distance d— 1.220 A for non-planar equilibrium configuration (a) and d = 1.292 A for the all planar structure (b). All energies in kcalmol-1. Reproduced by permission of Helvetica Chimica Acta from Reference 23... 

B) Electrostatic equipotential contours for the water molecule, in the plane perpendicular to the molecular plane [13]. The contour spacing is 2000 cm1 (for a unit test charge), and the distance units for the horizontal and vertical axes are atomic units (au). 

Fig. 4.45. A geometric representation of the electric field In an electrochemical system in which plane-parallel electrodes are immersed in an electrolyte so that they extend up to the walls of the rectangular insulating container. The equipotential surfaces are parallel to the electrodes.

The imposition of a potential difference between two electrodes thus makes an electrolytic solution the scene of operation of an electric field (i.e., an electric force) acting upon the charges present. This field can be mapped by drawing equipotential surfaces (all points associated with the same potential lie on the same surface). The potential map yields a geometric representation of the field. In the case of plane-parallel electrodes extending to the walls of a rectangular cell, the equipotential surfaces are parallel to the electrodes (Fig. 4.45). 

Figure 4.3 Schematics of equipotential curves in the Q Qg (or 6) plane of the vibrational potential U Qs, Qg, Gp) that governs intermonomer modes of a single H-bond, (or (p) keeping a constant value.

Expression 16.13 implies that in all xy-planes the points having the same potential are situated on the branches of an equilateral hyperbola of which the asymptotes are the straight lines y = x. The field lines are orthogonal to the equipotential curves for each point of the space inside the quadrupole. This field has the value ... 

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