Electric field equipotential surfaces

Templating by electric fields equipotential and tangential field surfaces... 

Figure 3. Vertical cross-section showing equipotential contours inside a conductive cylindrical silo containing a symmetric conical heap of uniformly charged solids. The electrostatic potential maximum exists on the center line somewhat below the powder surface, while the maximum electric field intensity occurs near the wall just above the powder.

In this case the surfaces fi = constant correspond to the equipotentials and the surfaces a = constant correspond to the surfaces on which the electric field lines are fixed. The distance from the origin to the focus, a, is related to the distance between the apex and the origin, d, and the radius of curvature of the tip, rt, by a = [d(d + rt) m. 

Figure 2. Schematic diagram of the imaging apparatus with ion lens. The detector is a dual microchannel plate/phosphor screen assembly (40 mm active diameter) coupled with a CCD camera. Electric field lines are shown to illustrate the ion lens. Equipotential surfaces in the repeller/extractor region are also included.

In 1949 Herzog and Viehbock reported a novel ion source for mass spec-trography (Fig. 4.2) [9]. This source provided separate accelerating fields for the primary and secondary ions and thus became the first modem instrument designed specifically for SIMS. The design included acceleration of the positive secondary ions from an equipotential surface through an electric field acting as an electron-optic lens. 

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). 

Since the reaction chamber is surrounded by an equipotential surface, we may assume that there is no electric field acting on the ions produced by electron impact. Therefore, those ions have only the kinetic energy of the molecules, and they may diffuse across the magnetic field at much higher rates than the electrons. However, the component of their velocity perpendicular to the magnetic field is less than the velocity... 

Principle. Corroding and passive rebars in concrete show a difference in electrical potential of up to 0.5 V, thus a macrocell generates and current flows between these areas (Chapter 8). The electric field coupled with the corrosion current between corroding and passive areas of the rebars (Figure 16.4) can be measured experimentally with a suitable reference electrode (half-cell) placed on the concrete surface, resulting in equipotential lines (potential field) that allow the location of corroding rebars at the most negative values [5-8]. 

AC Electro-Osmotic Flow, Fig. 2 The basic mechanism of AC electroosmosis electrochemical relaxation top) and induced-charge electroosmotic flow bottom) in response to a suddenly applied voltage across an electrode pair, (a) At first the electric field has no tangential component on the electrodes, since they are equipotential surfaces, and thus there is no electroosmotic flow, (b) Capacitive double-layer charging begins near the gap... 

Fig. 2 Physical mechanism for induced-charge electroosmosis around an ideally polarizable metal cylinder in a suddenly applied electric field (From Bazant and Squires [3]). (a) When the field is turned on, electronic charges relax to make the surface an equipotential, but the normal current drives double-layer charging, (b) After charging,...

A careful account of the problem can be found in Ref. [95]. Ohshima et al. [96] first found a numerical solution of the problem, valid for arbitrary values of the zeta potential or the product Ka. In the same paper, they dealt with the problem of finding the sedimentation potential and the DC conductivity of a suspension of mercury drops. The problems are solved following the lines of the electrophoresis theory of rigid particles previously derived by O Brien and White [18]. The liquid drop is assumed to behave as an ideal conductor, so that electric fields and currents inside the drop are zero, and its surface is equipotential. The main difference between the treatment of the electrophoresis of rigid particles and that of drops is that there is a velocity distribution of the fluid inside the drop, Vj, governed by the Navier-Stokes equation with zero body force (in the case of electrophoresis), and related to the velocity outside the drop, v, by the boundary conditions ... 

Thunderstorm currents totaling 1500 A flow to the ionosphere, which is an approximately equipotential layer in the upper atmosphere, and return to earth in fair-weather regions. The resistance between the ionosphere and earth is 200 f2, giving rise to an ionospheric potential of 300 kV. The vertical electric field near the earth s surface is of the order of 100 V/m, and the associated current density is typically several picoamperes/m. The ac component of the fair-weather field is relatively much smaller than the ac component of individual thunderstorm currents. 

The constant electric field is generated by the pair of parallel electrodes of infinite length. The equipotential surfaces of the electric field run parallel to the electrodes. The current density vector, a gradient of the electric potential, is vertical to the equipotential surface. Since the electrodes were placed parallel to the x-axis, the current density vector is written by ... 

The development of the wave-shape pattern is described as follows. The electric field activates and drives the surfactant molecules, which are adsorbed on the gel and deform it. As the adsorption progresses, the deformation occurs in such a way that the surface normal of the gel approaches parallel to the equipotential surface of the electric field. Fig. 7.22 illustrates the geometry of the gel and the electric field. Horizontal lines are the equipotential surfaces of the electric field. Arrows on the gel surface are normal vectors of the gel. Prom equation (2.7), the effect of the electric field to the gel disappears when the surface normal of the gel and the equipotential surface of the electric field become parallel (Fig. 7.22(a)). The angle of the tip of the gel 4> reaches maximum when the deformation speed near the root and one near the tip balance (Fig. 7.22(b)). The gel deformation works to deactivate the adsorption reaction and causes oscillatory motion. 

The distribution of the primary phase was studied in the same manner under an external electric field. The external field was imposed vertically by pladng electrodes onto the AI2O3 insulating plate parallel to the sample s free surface. Apphcation of the external field horizontally was ineffective since the wall and bottom of the Pt-Rh container diminished the potential difference and led to an equipotential state. The field was generated by the power supply set at a frequency of 500 Hz since the effect of the electric field on the chemical potential is given by as defined in Eq. (15.7) and the directionality of the electric field did not have to be considered. Thus, the fluid flow driven by the electric field could be neglected on the macroscopic scale. 

Figure 8.2.7 Schematic diagram of electron dipole scattering in front of a surface. The long-range electric field of the electron (electric field lines marked by dashed lines) interacts with the oscillating electric dipole field (lines of equipotential are marked dot-dashed) because of a surface vibration.

Electric fields are formed around solid surfaces that have a potential on them. The locations in space that have the same potential with respect to the surface are called equipotential surfaces. When the surface is fiat or nearly so, the equipotential surfaces will be conformal with the solid surface. When the solid surface has a complex morphology, the equipotential surfaces will not be able to conform to the solid surface configuration and will smooth out the irregularities. Surfaces with closely spaced features, such as an open mesh (high transmission) grid, appear as a solid surface to the electric field. The separation between the equipotential surfaces establishes the electric field gradient. Electrons and ions are accelerated normal to the equipotential surfaces. Figure 5.4 shows some equipotential surfaces and the effects of curvature on the equipotential surfaces. The variation of field over a non-smooth surface leads to variations in the bombardment of the surfaces by ions. 

This correspondence between equipotential and tangential field surfaces leads to a simple construction for the likely diffusion trajectories of ions within fast-ion conductors, also called "solid electrolytes". These solids (typically binary salts) are electrically highly conducting (the electrical conductivity of... 

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