Parallel plate capacitor, charging

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... 

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 ... 

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... 

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... 

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... 

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. 

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... 

In the preceding section we discussed the problem of the variation of potential with distance from an interface from the highly artificial perspective of a parallel plate capacitor. The variation of potential with distance from a charged surface of arbitrary shape is a classical electrostatic problem. The general problem is described by the Poisson equation,... 

Figure 3.8 Schematic parallel-plate capacitor, showing plate area A, separation 5, and net charge Q on each plate surrounding the dielectric medium.

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

Suppose that the radius of the sphere would increase to infinity, this would result in the parallel-plate capacitor as shown in figure 11.4.17. When this parallel-plate capacitor is connected to a voltage source, the source will send electrons to the capacitor which is thus charged. 

Permittivity e is the most basic property of a dielectric material. To understand the concept of permittivity, consider first a parallel-plate capacitor with two metallic plates. When a voltage is applied across the metallic plates, charges are generated on the surface of each plate. The charge Q on the plate surface is given by... 

Important relationships can be developed by considering the effect of filling the space between the plates of a parallel-plate capacitor with a dielectric material, as shown in fig. 2.27. From Gauss s theorem the electric field E between and normal to two parallel plates carrying surface charge density a and separated by a vacuum is... 

The methodology for the calculation of the complex relative permittivity for the dipolar relaxation mechanism is founded on the calculation of the dielectric response function, f(t), for a depolarization produced by the discharge of a previously charged capacitor. In Figure 1.29a, a circuit is shown where a capacitor is inserted in which a dipolar dielectric material is enclosed in the parallel plate capacitor of area, A, and thickness, d, with empty capacitance C0 = Q0/U0 = 0(A/d), and E0 = U0ld. In Figure 1.29b, the corresponding depolarization process is shown. 

The magnitude of the work, W, done by an externally applied force in overcoming the force q E due to the internal electric field E in the parallel-plate capacitor, while moving a charge q from one plate to the other along a line within the capacitor perpendicular to the plates is given by... 

Fig. 6. Net electric field produced by a plane of positive charge to the left and a plane of negative charge to the right, as in the case of a parallel-plate capacitor which is charged to a voltage VB.

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