Equipotential line

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).

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).

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). 

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. 

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]).

Figure 5.13 (a) Potential drop across a SAM and the adjacent double layer (DL) in a region free of defects shown in top part. Solid and dashed lines are two examples of different positions of the Fermi level Ep. Dash-dot line indicates the course of an equipotential line representing the potential at the outer surface of the SAM including defects, (b) Illustration of processes involved in electrochemical metal deposition. For details, see text. 

The charge-transport equation includes the electrochemical kinetics for both anode and cathode catalyst layers. If we assume an infinitely large electric conductivity of the electronic phase, the electrode becomes an equipotential line, such that... 

Fig. 1. Electrical dipole development around a conductor crossing redox equipotential lines. Positive current in the country rock (not shown) travels perpendicular to electrical field lines from the anode to the cathode (modified after Govett 1976 and Hamilton 1998).

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. 

While a proper aiming of the atom-probe can be experimentally determined, information on field lines and on equipotential lines is difficult to derive with an experimental method because of the small size of the tip. Yet this information is needed for interpreting quantitatively many experiments in field emission and in field ion emission. We describe here a highly idealized tip-counter electrode configuration which may be useful for describing field lines at a short distance away from the tip surface but far enough removed from the lattice steps of the surface. The electrode is assumed to consist of a hyperboloidal tip and a planar counter-electrode.30 In the prolate spheroidal coordinates, the boundary surfaces correspond to coordinate surfaces and Laplace s equation is separable, so that the boundary conditions can be easily satisfied. 

To account for these conclusions, we propose the following hypothesis The spots are due to molecules which rest or on protrude above the first two layers and which share electrons with the atoms in these layers. On this basis we can treat these molecules as if they were small metallic spheres or parts of spheres on a metal surface. We can then draw equipotential lines above the surface and above these spheres and deduce differences in field strength. We can also sketch probable paths for electrons from those parts of the spheres where the field strength is greatest. 

Figure 24 is a schematic diagram of equipotential lines and lines of electron flow for a doublet which is clearly resolved. If, as we have postulated, the atoms share or interchange electrons with the substrate, it should be permissible to treat them as conductors and to draw equi-... 

This new approach employed two-dimensional equipotential lines, instead of three-dimensional equipotential surfaces, using an interpretation of islands of compressed gas or liquid that grow in places of lower potential on the adsorbent surface. 

As the specific resistance of the solution increases, the geometry becomes more important, as has been shown in the previous sections. The uncompensated resistance will remain large unless the tip of the reference electrode is located very close to the working-electrode surface. This tip must be quite small otherwise the current density will be nonuniform over the electrode surface because of distortion of the equipotential lines by the top of the reference electrode. 

When polarization occurs at an electrode with nonideal geometry (e.g., when the current is limited by rate of electron transfer or by mass transport), there is a gradient in potential in the solution adjacement to the electrode, and associated with this is a tangential as well as normal component of the current at the electrode surface.13 This causes the equipotential lines to intersect the electrode and the current lines to enter the electrode at angles other than 90°. (In the absence of polarization, or in a polarized electrode with ideal geometiy, the equipotential lines would be parallel to the electrode surface, and the current lines would intersect the electrode at an angle of 90°.)... 

Hg/HgO is often the electrode of choice in alkaline aqueous medium, silver/silver acetate in many nonaqueous medium such as acetic acid, etc. When the experiment requires large currents to flow between the working and the counterelectrodes, a particular attention must be paid to place the reference electrode at an equipotential line close to the working electrode or to make an appropriate ohmic drop correction. 

Figures 8(a) and 8(b) show potential energy diagrams for such a system [70], The potential energy is plotted as equipotential lines in a coordinate system...

Water table maps consist of equipotential lines—contours of equal water table altitudes. The contours are drawn to fit measured water table altitudes, as shown in Fig. 4.4. An immediate outcome is the deduction of the dominant direction of groundwater flow (Fig. 4.4b). 

Fig. 4.4 Drawing equipotential lines (a) a map with well locations and water tables in masl (b) equipotential lines based on the well data the arrows show deduced main directions of groundwater flow.

Shown in Fig. 7.2 are the relevant structural parameters, schematic representations of the extreme, intermediate and transition state conformations of the chelate ring, and the calculated energy profile. Heavy equipotential lines are spaced by... 

Current lines are perpendicular to isopotential lines otherwise known as equipotential lines, which are lines of constant potential in two dimensions or surfaces of constant potential in three dimensions. Current lines do not terminate at insulating surfaces. 

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