Capacitor plates

For this purpose we compare a parallel plate capacitor under vacuum and one containing a dielectric, as shown in Figs. 10.4a and b, respectively. The plates of the capacitor carry equal but opposite charges Q which can be described as aA, where o is the surface charge density and A is the area of the plates. In this case, the field between the plates is given by... 

Figure 10.4 Parallel-plate capacitor with surface charge density a. (a) The field is Eo with no dielectric present, (b) The field is reduced to E by a dielectric which acquires a surface charge of its own,...

Consider the leaky parallel plate capacitor shown in Figure A-4-1.3. If the capacitor is momentarily charged and allowed to discharge through resistor / L, so that the charging current Iq = 0, the leakage current... 

Erom C-1.1 the equation for capacitance of a parallel plate capacitor (plate area A, separation d) s ... 

A simple model of the e.d.l. was first suggested by Helmholz in which the charges at the interface were regarded as the two plates constituting a parallel plate capacitor, e.g. a plate of metal with excess electrons (the inner Helmholz plane I.H.P.) and a plate of excess positively charged ions (the outer Helmholz plane O.H.P.) in the solution adjacent to the metal the... 

The electrical double layer resembles an ordinary (parallel-plate) capacitor. For an ideal capacitor, the charge (q) is directly proportional to the potential difference ... 

Figure 1-13 displays the experimental dependence of the double-layer capacitance upon the applied potential and electrolyte concentration. As expected for the parallel-plate model, the capacitance is nearly independent of the potential or concentration over several hundreds of millivolts. Nevertheless, a sharp dip in the capacitance is observed (around —0.5 V i.e., the Ep/C) with dilute solutions, reflecting the contribution of the diffuse layer. Comparison of the double layer witii die parallel-plate capacitor is dius most appropriate at high electrolyte concentrations (i.e., when C CH). 

The charge density (2s)t the zeta potential, and the thickness 5q are interrelated by the plate-capacitor relation... 

Whatever the most acceptable model may be and as we need only a rough estimate of the amount of ions discharged, we start from the Helmholtz model of a simple parallel-plate capacitor, whose potential difference is... 

The electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g (ion) = qMd/4ire. Itincludes a contribution from the polarizability of the electrolyte, since the dielectric constant is included in the expression. The distance d between the layers of charge is often taken to be from the outer Helmholtz plane (distance of closest approach of ions in solution to the metal in the absence of specific adsorption) to the position of the image charge in the metal a model for the metal is required to define this position properly. The capacitance per unit area of the ideal capacitor is a constant, e/Aird, often written as Klon. The contribution to 1/C is 1 /Klon this term is much less important in the sum (larger capacitance) than the other two contributions.2... 

The growth of an anodic alumina film, at a constant current, is characterized by a virtually linear increase of the electrode potential with time, exemplified by Fig. 10, with a more or less notable curvature (or an intercept of the extrapolated straight line) at the beginning of anodization.73 This reflects the constant rate of increase of the film thickness. Indeed, a linear relationship was found experimentally between the potential and the inverse capacitance78 (the latter reflecting the thickness in a model of a parallel-plate capacitor under the assumption of a constant dielectric permittivity). This is foreseen by applying Eq. (38) to Eq. (35). It is a consequence of the need for a constant electric field on the film in order to transport constant ionic current, as required by Eqs. (39)-(43). 

So the double-layer capacity is the same as that of a parallel-plate capacitor with the plate separation given by the Debye length. Since for high concentrations the latter are of the order of a few Angstroms, these capacities can be quite high. 

Method involves placing a specimen between parallel plate capacitors and applying a sinusoidal voltage (frequencies ranging from 1 mHz to 1 MHz) to one of the plates to establish an electric field in the specimen. In response to this field, a specimen becomes electrically polarized and can conduct a small charge from one plate to the other. Through measurement of the resultant current, the dielectric constant and dielectric loss constant for a specimen can be measured. The sharp increases in both the dielectric constant and the dielectric loss constant during a temperature scan are correlated with the occurrence of Tg... 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

The region between the surface plane and the IHP, and the region between the IHP and the OHP are considered to behave electrostatically as parallel plate capacitors, with charge related to potential by the capacitances C- and C2 ... 

The relationship between charge and potential are derived by assuming that the planes can be treated as plates of two parallel plate capacitors in series (18) with... 

Fig. 9.18 (a) Schematic of the device, which was designed for simultaneous measurement of the SWNT network capacitance and conductance, (b) Dependence of the network capacitance (red) and conductance (green) on the substrate voltage, FS. The network capacitance is approximately 1/4 the value of the capacitance for a parallel-plate capacitor with an equivalent area and oxide thickness (Kong et al., 2003. With the permission from American Chemical Society) (See Color Plates)... 

As DDTC adsorbs on jamesonite electrode chemically, the double electric charge layer is treated as a plate capacitor, the capacitance C of the tight layer as a constant, and the change of the capacitance of the double electric charge layer is designated to the capacitance Ct of the diffusion layer. Thereby, the tight layer and the diffusion layer are looked upon as two series capacitances according to the method from Cooper and Harrison, then ... 

The double-layer consists of ions juxtaposed with the electrode, so it resembles a plate capacitor (e.g. see Figure (5.3)) - we will describe this in terms of its double-layer capacitance, Cji (cf Chapter 5). 

However, although the WO3 surface is filled , ions still move from the electrolyte reservoir toward the WO3-electrolyte interface. Such a situation results in the accumulation of charge at this interface. In effect, we have a structure which is physically very similar to that of a typical plate capacitor (see Figure 5.3). For this reason, the equivalent circuit (see Figure 8.12(b)) also contains a capacitor Q (where the subscript denotes surface ). 

Thus, according to this model, the interphase consists of two equal and opposite layers of charges, one on the metal ( m) the other in solution (q ). This pair of charged layers, called the double layer, is equivalent to a parallel-plate capacitor (Fig. 4.5). The variation of potential in the double layer with distance from the electrode is linear (Fig. 4.4). A parallel-plate condenser has capacitance per unit area given by the equation... 

Figure 4.5. Electrical equivalent of the Helmholtz double layer a parallel-plate capacitor.

In Section 4.3 it was shown that the electrical equivalent of the Helmholtz double layer is a parallel-plate capacitor (Fig. 4.5). In Section 4.5 (Fig. 4.9) it was shown that... 

Equation 6.3 is identical to the equation that relates the charge density, voltage difference, and distance of separation of a parallel-plate capacitor. This result indicates that a diffuse double layer at low potentials behaves like a parallel capacitor in which the separation distance between the plates is given by k. This explains why k is called the double layer thickness. 

For our example, this means that the benzenesulfonate ions are adsorbed in a surface layer and that the sodium ions form an atmosphere of counter ions in which their concentration gradually decreases with the distance from the benzenesulfonate layer. A small separation between the negative and the positive charges has been created at the interface between the mobile and the stationary phases. As in a plate capacitor, this charge separation creates a difference in electrostatic potential between the charged stationary phase and the surrounding mobile phase. By convention, the surface has a negative electrostatic potential in this case. 

LASL, Los Alamos, NM. The transducer described in his paper and shown in Fig 31 was in the form of an uncharged parallel-plate capacitor which had an explosive as a dielectric. One plate was connected to the signal input terminal of an oscilloscope, while the other plate was grounded and acted as part of the attenuator in the boosting system. When the shock wave in the grounded attenuator plate hir the explosive, a voltage appeared across the capacitor and a pulse appeared on the oscilloscope. Two oscilloscopes were used to record the waveform of the current in the transducer circuit which consisted of a small capacitance shunted by the small resistance of the signal cable. 

Figure 6.22 A parallel-plate capacitor (a) in vacuum and (b) with a dielectric material between the plates. Reprinted, by permission, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 640. Copyright 2000 by John Wiley Sons, Inc.

Since Do is the surface charge density on the plate in vacuum, it can be related to the capacitance of a parallel plate capacitor in vacuum, Co, which is defined as... 

A voltage of 10 V is applied across a parallel-plate capacitor with a plate separation of 2 X 10 m and plate area of 6.45 x 10 m, both in vacnnm and then with a dielectric material placed between the plates. The polarization, P, dne to the presence of the dielectric material is 2.22 x 10 C/m. ... 

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