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Capacitors and Dielectrics
OpenStax College
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
Abstract
- Describe the action of a capacitor and de ne capacitance.
- Explain parallel plate capacitors and their capacitances.
- Discuss the process of increasing the capacitance of a dielectric.
- Determine capacitance given charge and voltage. A capacitor is a device used to store electric charge. Capacitors have applications ranging from ltering static out of radio reception to energy storage in heart de brillators. Typically, commercial capacitors have two conducting parts close to one another, but not touching, such as those in Figure 1. (Most of the time an insulator is used between the two plates to provide separationsee the discussion on dielectrics below.) When battery terminals are connected to an initially uncharged capacitor, equal amounts of positive and negative charge, +Q and − − Q, are separated into its two plates. The capacitor remains neutral overall, but we refer to it as storing a charge Q in this circumstance.
: A capacitor is a device used to store electric charge. ∗Version 1.3: Jun 10, 2012 11:02 pm GMT- †http://creativecommons.org/licenses/by/3.0/
Figure 1: Both capacitors shown here were initially uncharged before being connected to a battery. They now have separated charges of +Q and − − Q on their two halves. (a) A parallel plate capacitor. (b) A rolled capacitor with an insulating material between its two conducting sheets.
The amount of charge Q a capacitor can store depends on two major factorsthe voltage applied and the capacitor's physical characteristics, such as its size.
: The amount of charge Q a capacitor can store depends on two major factorsthe voltage applied and the capacitor's physical characteristics, such as its size.
A system composed of two identical, parallel conducting plates separated by a distance, as in Figure 2, is called a parallel plate capacitor. It is easy to see the relationship between the voltage and the stored charge for a parallel plate capacitor, as shown in Figure 2. Each electric eld line starts on an individual positive charge and ends on a negative one, so that there will be more eld lines if there is more charge. (Drawing a single eld line per charge is a convenience, only. We can draw many eld lines for each charge,
Field^1 , we know that the voltage across parallel plates is V = Ed. Thus,
V ∝ E. (2) It follows, then, that V ∝ Q, and conversely,
Q ∝ V. (3) This is true in general: The greater the voltage applied to any capacitor, the greater the charge stored in it. Dierent capacitors will store dierent amounts of charge for the same applied voltage, depending on their physical characteristics. We de ne their capacitanceC to be such that the charge Q stored in a capacitor is proportional to C. The charge stored in a capacitor is given by
Q = CV. (4) This equation expresses the two major factors aecting the amount of charge stored. Those factors are the physical characteristics of the capacitor, C, and the voltage, V. Rearranging the equation, we see that capacitance C is the amount of charge stored per volt, or
C =
Q
V
: Capacitance C is the amount of charge stored per volt, or
C =
Q
V
The unit of capacitance is the farad (F), named for Michael Faraday (17911867), an English scientist who contributed to the elds of electromagnetism and electrochemistry. Since capacitance is charge per unit voltage, we see that a farad is a coulomb per volt, or
1 F =
1 C
1 V
A 1-farad capacitor would be able to store 1 coulomb (a very large amount of charge) with the application of only 1 volt. One farad is, thus, a very large capacitance. Typical capacitors range from fractions of a picofarad
1 pF = 10 12^ F
to millifarads
1 mF = 10 −−^3 F
Figure 3 shows some common capacitors. Capacitors are primarily made of ceramic, glass, or plastic, depending upon purpose and size. Insulating materials, called dielectrics, are commonly used in their construction, as discussed below.
(^1) "Electric Potential in a Uniform Electric Field" http://cnx.org/content/m42326/latest/
Figure 3: Some typical capacitors. Size and value of capacitance are not necessarily related. (credit: Windell Oskay)
1 Parallel Plate Capacitor
The parallel plate capacitor shown in Figure 4 has two identical conducting plates, each having a surface area A, separated by a distance d (with no material between the plates). When a voltage V is applied to the capacitor, it stores a charge Q, as shown. We can see how its capacitance depends on A and d 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 storebecause the charges can spread out more. Thus C should be greater for larger A. Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. So C should be greater for smaller d.
Finding the capacitance C is a straightforward application of the equation C = 0 A/d. Once C is found, the charge stored can be found using the equation Q = CV. Solution for (a) Entering the given values into the equation for the capacitance of a parallel plate capacitor yields
C = 0 Ad =
- 85 × 10 12^ mF
) (^1). 00 m 2
- 00 × 10 −−^3 m = 8. 85 × 10 −−^9 F = 8. 85 nF.
Discussion for (a) This small value for the capacitance indicates how di cult it is to make a device with a large capacitance. Special techniques help, such as using very large area thin foils placed close together. Solution for (b) The charge stored in any capacitor is given by the equation Q = CV. Entering the known values into this equation gives
Q = CV =
8. 85 × 10 −−^9 F
3. 00 × 103 V
= 26.6 μC.
Discussion for (b) This charge is only slightly greater than those found in typical static electricity. Since air breaks down at about 3 .00 × 106 V/m, more charge cannot be stored on this capacitor by increasing the voltage.
Another interesting biological example dealing with electric potential is found in the cell's plasma membrane. The membrane sets a cell o from its surroundings and also allows ions to selectively pass in and out of the cell. There is a potential dierence 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 potentialthe nerve signal. The cell membrane is about 7 to 10 nm thick. An approximate value of the electric eld across it is given by
E =
V
d
70 × 10 −−^3 V
8 × 10 −−^9 m
= − − 9 × 106 V/m. (12)
This electric eld is enough to cause a breakdown in air.
2 Dielectric
The previous example highlights the di culty 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 di culty 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 elds than air before breaking down. There is another bene t to using a dielectric in a capacitor. Depending on the material used, the capacitance is greater than that given by the equation C = 0 Ad by a factor κ, called the dielectric constant. A parallel plate capacitor with a dielectric between its plates has a capacitance given by
C = κε 0
A
d
(parallel plate capacitor with dielectric). (13)
Values of the dielectric constant κ for various materials are given in Table 1: Dielectric Constants and Dielectric Strengths for Various Materials at 20ºC. Note that κ for vacuum is exactly 1, and so the above
equation is valid in that case, too. If a dielectric is used, perhaps by placing Te on between the plates of the capacitor in Example 1 (Capacitance and Charge Stored in a Parallel Plate Capacitor), then the capacitance is greater by the factor κ, which for Te on is 2.1.
: How large a capacitor can you make using a chewing gum wrapper? The plates will be the aluminum foil, and the separation (dielectric) in between will be the paper.
Dielectric Constants and Dielectric Strengths for Various Materials at 20ºC
Material Dielectric constant κ Dielectric strength (V/m) Vacuum 1.00000 Air 1.00059 3 × 106 Bakelite 4.9 24 × 106 Fused quartz 3.78 8 × 106 Neoprene rubber 6.7 12 × 106 Nylon 3.4 14 × 106 Paper 3.7 16 × 106 Polystyrene 2.56 24 × 106 Pyrex glass 5.6 14 × 106 Silicon oil 2.5 15 × 106 Strontium titanate 233 8 × 106 Te on 2.1 60 × 106 Water 80
Table 1
Note also that the dielectric constant for air is very close to 1, so that air- lled capacitors act much like those with vacuum between their plates except that the air can become conductive if the electric eld strength becomes too great. (Recall that E = V /d for a parallel plate capacitor.) Also shown in Table 1: Dielectric Constants and Dielectric Strengths for Various Materials at 20ºC are maximum electric eld strengths in V/m, called dielectric strengths, for several materials. These are the elds 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 (Capacitance and Charge Stored in a Parallel Plate Capacitor), the separation is 1.00 mm, and so the voltage limit for air is
V = E · d
3 × 10 6 V/m
1. 00 × 10 −
= 3000 V.
However, the limit for a 1.00 mm separation lled with Te on is 60,000 V, since the dielectric strength of Te on is 60 × 106 V/m. So the same capacitor lled with Te on has a greater capacitance and can be subjected to a much greater voltage. Using the capacitance we calculated in the above example for the
Figure 5: (a) The molecules in the insulating material between the plates of a capacitor are polarized by the charged plates. This produces a layer of opposite charge on the surface of the dielectric that attracts more charge onto the plate, increasing its capacitance. (b) The dielectric reduces the electric eld strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge. The capacitor stores the same charge for a smaller voltage, implying that it has a larger capacitance because of the dielectric.
Another way to understand how a dielectric increases capacitance is to consider its eect on the electric eld inside the capacitor. Figure 5(b) shows the electric eld lines with a dielectric in place. Since the eld 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 eld 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 = 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 de ned to be κ = E 0 /E, or the ratio of the electric eld in a vacuum to that in the dielectric material, and is intimately related to the polarizability of the material.
: The Submicroscopic Origin of Polarization Polarization is a separation of charge within an atom or molecule. As has been noted, the planetary model of the atom pictures it as having a positive nucleus orbited by negative electrons, analogous to the planets orbiting the Sun. Although this model is not completely accurate, it is very helpful in explaining a vast range of phenomena and will be re ned elsewhere, such as in Atomic Physics^2. The submicroscopic origin of polarization can be modeled as shown in Figure 6.
Figure 6: Artist's conception of a polarized atom. The orbits of electrons around the nucleus are shifted slightly by the external charges (shown exaggerated). The resulting separation of charge within the atom means that it is polarized. Note that the unlike charge is now closer to the external charges, causing the polarization.
We will nd in Atomic Physics^3 that the orbits of electrons are more properly viewed as electron clouds with the density of the cloud related to the probability of nding an electron in that location (as opposed to the de nite 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 (H 2 O). The water molecule is not symmetricthe hydrogen atoms are repelled to one side, giving the molecule a boomerang shape. The electrons in a water molecule are more concentrated around
(^2) "Introduction to Atomic Physics" http://cnx.org/content/m42585/latest/ (^3) "Introduction to Atomic Physics" http://cnx.org/content/m42585/latest/
3 Section Summary
- A capacitor is a device used to store charge.
- The amount of charge Q a capacitor can store depends on two major factorsthe voltage applied and the capacitor's physical characteristics, such as its size.
- The capacitance C is the amount of charge stored per volt, or
C =
Q
V
- The capacitance of a parallel plate capacitor is C = 0 Ad , when the plates are separated by air or free space. 0 is called the permittivity of free space.
- A parallel plate capacitor with a dielectric between its plates has a capacitance given by
C = κ 0
A
d
where κ is the dielectric constant of the material.
- The maximum electric eld strength above which an insulating material begins to break down and conduct is called dielectric strength.
4 Conceptual Questions
Exercise 1 Does the capacitance of a device depend on the applied voltage? What about the charge stored in it? Exercise 2 Use the characteristics of the Coulomb force to explain why capacitance should be proportional to the plate area of a capacitor. Similarly, explain why capacitance should be inversely proportional to the separation between plates. Exercise 3 Give the reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. What is the independent reason that a dielectric material also allows a greater voltage to be applied to a capacitor? (The dielectric thus increases C and permits a greater V .) Exercise 4 How does the polar character of water molecules help to explain water's relatively large dielectric constant? (Figure 7) Exercise 5 Sparks will occur between the plates of an air- lled capacitor at lower voltage when the air is humid than when dry. Explain why, considering the polar character of water molecules. Exercise 6 Water has a large dielectric constant, but it is rarely used in capacitors. Explain why. Exercise 7 Membranes in living cells, including those in humans, are characterized by a separation of charge across the membrane. Eectively, the membranes are thus charged capacitors with important functions related to the potential dierence across the membrane. Is energy required to separate these charges in living membranes and, if so, is its source the metabolization of food energy or some other source?
Figure 9: The semipermeable membrane of a cell has dierent concentrations of ions inside and out. Diusion moves the K+^ (potassium) and Cl^ (chloride) ions in the directions shown, until the Coulomb force halts further transfer. This results in a layer of positive charge on the outside, a layer of neg- ative charge on the inside, and thus a voltage across the cell membrane. The membrane is normally impermeable to Na+^ (sodium ions).
5 Problems & Exercises
Exercise 8 (Solution on p. 16.) What charge is stored in a 180 μF capacitor when 120 V is applied to it? Exercise 9 Find the charge stored when 5.50 V is applied to an 8.00 pF capacitor. Exercise 10 (Solution on p. 16.) What charge is stored in the capacitor in Example 1 (Capacitance and Charge Stored in a Parallel Plate Capacitor)? Exercise 11 Calculate the voltage applied to a 2 .00 μF capacitor when it holds 3 .10 μC of charge. Exercise 12 (Solution on p. 16.) What voltage must be applied to an 8.00 nF capacitor to store 0.160 mC of charge? Exercise 13 What capacitance is needed to store 3 .00 μC of charge at a voltage of 120 V? Exercise 14 (Solution on p. 16.) What is the capacitance of a large Van de Graa generator's terminal, given that it stores 8. mC of charge at a voltage of 12.0 MV?
Solutions to Exercises in this Module
Solution to Exercise (p. 14)
- 6 mC Solution to Exercise (p. 14)
- 0 mC Solution to Exercise (p. 14) 20.0 kV Solution to Exercise (p. 14) 667 pF Solution to Exercise (p. 15) (a) 4 .4 μF (b) 4. 0 × 10 −−^5 C Solution to Exercise (p. 15) (a) 14.2 kV (b) The voltage is unreasonably large, more than 100 times the breakdown voltage of nylon. (c) The assumed charge is unreasonably large and cannot be stored in a capacitor of these dimensions.
Glossary
De nition 1: capacitor a device that stores electric charge De nition 2: capacitance amount of charge stored per unit volt De nition 3: dielectric an insulating material De nition 4: dielectric strength the maximum electric eld above which an insulating material begins to break down and conduct De nition 5: parallel plate capacitor two identical conducting plates separated by a distance De nition 6: polar molecule a molecule with inherent separation of charge
