Dielectric

Dielectric

The previous example highlights the difficulty of storing a large amount of charge in capacitors. If \(d\) is made smaller to produce a larger capacitance, then the maximum voltage must be reduced proportionally to avoid breakdown (since \(E=V/d\)). An important solution to this difficulty is to put an insulating material, called a dielectric, between the plates of a capacitor and allow \(d\) to be as small as possible. Not only does the smaller \(d\) 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 \(C={\epsilon }_{0}\cfrac{A}{d}\) by a factor \(\kappa \), called the dielectric constant. A parallel plate capacitor with a dielectric between its plates has a capacitance given by

\(C={\text{κε}}_{0}\phantom{\rule{0.10em}{0ex}}\cfrac{A}{d}\phantom{\rule{0.25em}{0ex}}\text{(parallel plate capacitor with dielectric)}.\)

Values of the dielectric constant \(\kappa \) for various materials are given in this table. Note that \(\kappa \) 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 this example, then the capacitance is greater by the factor \(\kappa \), which for Teflon is 2.1.

Dielectric Constants and Dielectric Strengths for Various Materials at 20ºC

Material	Dielectric constant \(\kappa \)	Dielectric strength (V/m)
Vacuum	1.00000	—
Air	1.00059	\(3×{\text{10}}^{6}\)
Bakelite	4.9	\(\text{24}×{\text{10}}^{6}\)
Fused quartz	3.78	\(8×{\text{10}}^{6}\)
Neoprene rubber	6.7	\(\text{12}×{\text{10}}^{6}\)
Nylon	3.4	\(\text{14}×{\text{10}}^{6}\)
Paper	3.7	\(\text{16}×{\text{10}}^{6}\)
Polystyrene	2.56	\(\text{24}×{\text{10}}^{6}\)
Pyrex glass	5.6	\(\text{14}×{\text{10}}^{6}\)
Silicon oil	2.5	\(\text{15}×{\text{10}}^{6}\)
Strontium titanate	233	\(8×{\text{10}}^{6}\)
Teflon	2.1	\(\text{60}×{\text{10}}^{6}\)
Water	80	—

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 \(E=V/d\) for a parallel plate capacitor.) Also shown in this table 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 this example, the separation is 1.00 mm, and so the voltage limit for air is

\(\begin{array}{lll}V& =& E\cdot d\\ & =& (3×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m})(1\text{.}\text{00}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m})\\ & =& \text{3000 V.}\end{array}\)

However, the limit for a 1.00 mm separation filled with Teflon is 60,000 V, since the dielectric strength of Teflon is \(\text{60}×{\text{10}}^{6}\) V/m. 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

\(\begin{array}{lll}Q& =& \mathit{\text{CV}}\\ & =& {\mathit{\kappa C}}_{\text{air}}V\\ & =& (2.1)(8.85 nF)(6.0×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{V})\\ & =& 1.1 mC.\end{array}\)

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 \(\kappa \). 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.

This figure 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 \(d\) away.

Another way to understand how a dielectric increases capacitance is to consider its effect on the electric field inside the capacitor. This figure (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 \(V=\text{Ed}\), so it too is reduced by the dielectric. Thus there is a smaller voltage \(V\) for the same charge \(Q\); since \(C=Q/V\), the capacitance \(C\) is greater.

The dielectric constant is generally defined to be \(\kappa ={E}_{0}/E\), 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 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. This figure illustrates the separation of charge in a water molecule, which has two hydrogen atoms and one oxygen atom \(({\text{H}}_{2}\text{O})\). 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 tutorial, polar water provides a shield or screening of the electric fields in the highly charged molecules of interest in biological systems.