Parallel Plate Capacitor

The parallel plate capacitor shown in Figure 4 has two identical conducting plates, each having a surface area [latex]\boldsymbol{A}[/latex], separated by a distance [latex]\boldsymbol{d}[/latex] (with no material between the plates). When a voltage [latex]\boldsymbol{V}[/latex] is applied to the capacitor, it stores a charge [latex]\boldsymbol{Q}[/latex], as shown. We can see how its capacitance depends on [latex]\boldsymbol{A}[/latex] and [latex]\boldsymbol{d}[/latex] by considering the characteristics of the Coulomb force. We know that like charges repel, unlike charges attract, and the force between charges decreases with distance. So it seems quite reasonable that the bigger the plates are, the more charge they can store—because the charges can spread out more. Thus [latex]\boldsymbol{C}[/latex] should be greater for larger [latex]\boldsymbol{A}[/latex]. Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. So [latex]\boldsymbol{C}[/latex] should be greater for smaller [latex]\boldsymbol{d}[/latex].

It can be shown that for a parallel plate capacitor there are only two factors ([latex]\boldsymbol{A}[/latex] and [latex]\boldsymbol{d}[/latex]) that affect its capacitance [latex]\boldsymbol{C}[/latex]. The capacitance of a parallel plate capacitor in equation form is given by

[latex]\boldsymbol{A}[/latex] is the area of one plate in square meters, and [latex]\boldsymbol{d}[/latex] is the distance between the plates in meters. The constant [latex]\boldsymbol{{\varepsilon}_0}[/latex] is the permittivity of free space; its numerical value in SI units is [latex]\boldsymbol{{\varepsilon}_0 = 8.85 \times 10^{-12} \;\textbf{F} / \textbf{m}}[/latex]. The units of F/m are equivalent to [latex]\boldsymbol{\textbf{C}^2 / \textbf{N} \cdot \textbf{m}^2}[/latex]. The small numerical value of [latex]\boldsymbol{{\varepsilon}_0}[/latex] is related to the large size of the farad. A parallel plate capacitor must have a large area to have a capacitance approaching a farad. (Note that the above equation is valid when the parallel plates are separated by air or free space. When another material is placed between the plates, the equation is modified, as discussed below.)

Another interesting biological example dealing with electric potential is found in the cell’s plasma membrane. The membrane sets a cell off from its surroundings and also allows ions to selectively pass in and out of the cell. There is a potential difference across the membrane of about –70 mV. This is due to the mainly negatively charged ions in the cell and the predominance of positively charged sodium (Na⁺) ions outside. Things change when a nerve cell is stimulated. Na⁺ ions are allowed to pass through the membrane into the cell, producing a positive membrane potential—the nerve signal. The cell membrane is about 7 to 10 nm thick. An approximate value of the electric field across it is given by

This electric field is enough to cause a breakdown in air.

Dielectric

The previous example highlights the difficulty of storing a large amount of charge in capacitors. If [latex]\boldsymbol{d}[/latex] is made smaller to produce a larger capacitance, then the maximum voltage must be reduced proportionally to avoid breakdown (since [latex]\boldsymbol{E = V/d}[/latex]). An important solution to this difficulty is to put an insulating material, called a dielectric, between the plates of a capacitor and allow [latex]\boldsymbol{d}[/latex] to be as small as possible. Not only does the smaller [latex]\boldsymbol{d}[/latex] make the capacitance greater, but many insulators can withstand greater electric fields than air before breaking down.

There is another benefit to using a dielectric in a capacitor. Depending on the material used, the capacitance is greater than that given by the equation [latex]\boldsymbol{C = {\varepsilon}_0 \frac{A}{d}}[/latex] by a factor [latex]\boldsymbol{\kappa}[/latex] , called the dielectric constant. A parallel plate capacitor with a dielectric between its plates has a capacitance given by

Values of the dielectric constant [latex]\boldsymbol{\kappa}[/latex] for various materials are given in Table 1. Note that [latex]\boldsymbol{\kappa}[/latex] for vacuum is exactly 1, and so the above equation is valid in that case, too. If a dielectric is used, perhaps by placing Teflon between the plates of the capacitor in Example 1, then the capacitance is greater by the factor [latex]\boldsymbol{\kappa}[/latex], which for Teflon is 2.1.

Note also that the dielectric constant for air is very close to 1, so that air-filled capacitors act much like those with vacuum between their plates except that the air can become conductive if the electric field strength becomes too great. (Recall that [latex]\boldsymbol{E = V/d}[/latex] for a parallel plate capacitor.) Also shown in Table 1 are maximum electric field strengths in V/m, called dielectric strengths, for several materials. These are the fields above which the material begins to break down and conduct. The dielectric strength imposes a limit on the voltage that can be applied for a given plate separation. For instance, in Example 1, the separation is 1.00 mm, and so the voltage limit for air is

However, the limit for a 1.00 mm separation filled with Teflon is 60,000 V, since the dielectric strength of Teflon is [latex]\boldsymbol{60 \times 10^6 \;\textbf{V} / \textbf{m}}[/latex]. So the same capacitor filled with Teflon has a greater capacitance and can be subjected to a much greater voltage. Using the capacitance we calculated in the above example for the air-filled parallel plate capacitor, we find that the Teflon-filled capacitor can store a maximum charge of

This is 42 times the charge of the same air-filled capacitor.

Microscopically, how does a dielectric increase capacitance? Polarization of the insulator is responsible. The more easily it is polarized, the greater its dielectric constant [latex]\boldsymbol{\kappa}[/latex]. Water, for example, is a polar molecule because one end of the molecule has a slight positive charge and the other end has a slight negative charge. The polarity of water causes it to have a relatively large dielectric constant of 80. The effect of polarization can be best explained in terms of the characteristics of the Coulomb force. Figure 5 shows the separation of charge schematically in the molecules of a dielectric material placed between the charged plates of a capacitor. The Coulomb force between the closest ends of the molecules and the charge on the plates is attractive and very strong, since they are very close together. This attracts more charge onto the plates than if the space were empty and the opposite charges were a distance [latex]\boldsymbol{d}[/latex] away.

Another way to understand how a dielectric increases capacitance is to consider its effect on the electric field inside the capacitor. Figure 5(b) shows the electric field lines with a dielectric in place. Since the field lines end on charges in the dielectric, there are fewer of them going from one side of the capacitor to the other. So the electric field strength is less than if there were a vacuum between the plates, even though the same charge is on the plates. The voltage between the plates is [latex]\boldsymbol{V = Ed}[/latex], so it too is reduced by the dielectric. Thus there is a smaller voltage [latex]\boldsymbol{V}[/latex] for the same charge [latex]\boldsymbol{Q}[/latex]; since [latex]\boldsymbol{C = Q/V}[/latex], the capacitance [latex]\boldsymbol{C}[/latex] is greater.

The dielectric constant is generally defined to be [latex]\boldsymbol{\kappa = E_0/E}[/latex], or the ratio of the electric field in a vacuum to that in the dielectric material, and is intimately related to the polarizability of the material.

We will find in Chapter 30 Atomic Physics that the orbits of electrons are more properly viewed as electron clouds with the density of the cloud related to the probability of finding an electron in that location (as opposed to the definite locations and paths of planets in their orbits around the Sun). This cloud is shifted by the Coulomb force so that the atom on average has a separation of charge. Although the atom remains neutral, it can now be the source of a Coulomb force, since a charge brought near the atom will be closer to one type of charge than the other.

Some molecules, such as those of water, have an inherent separation of charge and are thus called polar molecules. Figure 7 illustrates the separation of charge in a water molecule, which has two hydrogen atoms and one oxygen atom [latex]\boldsymbol{\textbf{H}_2 \textbf{O}}[/latex]. The water molecule is not symmetric—the hydrogen atoms are repelled to one side, giving the molecule a boomerang shape. The electrons in a water molecule are more concentrated around the more highly charged oxygen nucleus than around the hydrogen nuclei. This makes the oxygen end of the molecule slightly negative and leaves the hydrogen ends slightly positive. The inherent separation of charge in polar molecules makes it easier to align them with external fields and charges. Polar molecules therefore exhibit greater polarization effects and have greater dielectric constants. Those who study chemistry will find that the polar nature of water has many effects. For example, water molecules gather ions much more effectively because they have an electric field and a separation of charge to attract charges of both signs. Also, as brought out in the previous chapter, polar water provides a shield or screening of the electric fields in the highly charged molecules of interest in biological systems.