19.5

Capacitors and Dielectrics

| The Capacitance of a Capacitor

In Section 18.6 we saw that a parallel plate capacitor consists of two parallel metal plates placed near one another but not touching. This type of capacitor is only one among many. In general, a capacitor consists of two conductors of any shape placed near one another without touching. For a reason that will become clear later on, it is common practice to fill the region between the conductors or plates with an electrically insulating material called a dielectric, as Figure 19-17 illustrates.

Figure 19-17 A parallel plate capacitor consists of two metal plates, one carrying a charge +q and the other a charge −q. The potential of the positive plate exceeds that of the negative plate by an amount V. The region between the plates is filled with a dielectric.

A capacitor stores electric charge. Each capacitor plate carries a charge of the same magnitude, one positive and the other negative. Because of the charges, the electric potential of the positive plate exceeds that of the negative plate by an amount V, as Figure 19-17 indicates. Experiment shows that when the magnitude q of the charge on each plate is doubled, the magnitude V of the electric potential difference is also doubled, so q is proportional to V: . Equation 19.8 expresses this proportionality with the aid of a proportionality constant C, which is the capacitance of the capacitor.

| The Relation Between Charge and Potential Difference for a Capacitor
| The magnitude q of the charge on each plate of a capacitor is directly proportional to the magnitude V of the potential difference between the plates:

| (19.8)

where C is the capacitance.
SI Unit of Capacitance: coulomb/volt = farad (F)

Equation 19.8 shows that the SI unit of capacitance is the coulomb per volt (C/V). This unit is called the farad (F), named after the English scientist Michael Faraday (1791–1867). One farad is an enormous capacitance. Usually smaller amounts, such as a microfarad or a picofarad , are used in electric circuits. The capacitance reflects the ability of the capacitor to store charge, in the sense that a larger capacitance C allows more charge q to be put onto the plates for a given value of the potential difference V.
The physics of random-access memory (RAM) chips. The ability of a capacitor to store charge lies at the heart of the random-access memory (RAM) chips used in computers, where information is stored in the form of the “ones” and “zeros” that comprise binary numbers. Figure 19-18 illustrates the role of a capacitor in a RAM chip. The capacitor is connected to a transistor switch, to which two lines are connected, an address line and a data line. A single RAM chip often contains millions of such transistor–capacitor units. The address line is used by the computer to locate a particular transistor–capacitor combination, and the data line carries the data to be stored. A pulse on the address line turns on the transistor switch. With the switch turned on, a pulse coming in on the data line can cause the capacitor to charge. A charged capacitor means that a “one” has been stored, whereas an uncharged capacitor means that a “zero” has been stored.

Figure 19-18 A transistor–capacitor combination is part of a RAM chip used in computer memories.

| The Dielectric Constant

If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly because of the way in which the dielectric alters the electric field between the plates. Figure 19-19 shows how this effect comes about. In part a, the region between the charged plates is empty. The field lines point from the positive toward the negative plate. In part b, a dielectric has been inserted between the plates. Since the capacitor is not connected to anything, the charge on the plates remains constant as the dielectric is inserted. In many materials (e.g., water) the molecules possess permanent dipole moments, even though the molecules are electrically neutral. The dipole moment exists because one end of a molecule has a slight excess of negative charge while the other end has a slight excess of positive charge. When such molecules are placed between the charged plates of the capacitor, the negative ends are attracted to the positive plate and the positive ends are attracted to the negative plate. As a result, the dipolar molecules tend to orient themselves end to end, as in part b. Whether or not a molecule has a permanent dipole moment, the electric field can cause the electrons to shift position within a molecule, making one end slightly negative and the opposite end slightly positive. Because of the end-to-end orientation, the left surface of the dielectric becomes positively charged, and the right surface becomes negatively charged. The surface charges are shown in red in the picture.

Figure 19-19 (a) The electric field lines inside an empty capacitor. (b) The electric field produced by the charges on the plates aligns the molecular dipoles within the dielectric end to end. (c) The surface charges on the dielectric reduce the electric field inside the dielectric. The space between the dielectric and the plates is added for clarity. In reality, the dielectric fills the region between the plates.

Because of the surface charges on the dielectric, not all the electric field lines generated by the charges on the plates pass through the dielectric. As Figure 19-19c shows, some of the field lines end on the negative surface charges and begin again on the positive surface charges. Thus, the electric field inside the dielectric is less strong than the electric field inside the empty capacitor, assuming the charge on the plates remains constant. This reduction in the electric field is described by the dielectric constant , which is the ratio of the field magnitude E0 without the dielectric to the field magnitude E inside the dielectric:

| (19.9)

Being a ratio of two field strengths, the dielectric constant is a number without units. Moreover, since the field without the dielectric is greater than the field inside the dielectric, the dielectric constant is greater than unity. The value of depends on the nature of the dielectric material, as Table 19-1 indicates.

Table 19-1 Dielectric Constants of Some Common Substances*

Substance | Dielectric Constant, κ
Vacuum | 1
Air | 1.000 54
Teflon | 2.1
Benzene | 2.28
Paper (royal gray) | 3.3
Ruby mica | 5.4
Neoprene rubber | 6.7
Methyl alcohol | 33.6
Water | 80.4

a) |  | Near room temperature.

| The Capacitance of a Parallel Plate Capacitor

The capacitance of a capacitor is affected by the geometry of the plates and the dielectric constant of the material between them. For example, Figure 19-17 shows a parallel plate capacitor in which the area of each plate is A and the separation between the plates is d. The magnitude of the electric field inside the dielectric is given by Equation 19.7b as , where V is the magnitude of the potential difference between the plates. If the charge on each plate is kept fixed, the electric field inside the dielectric is related to the electric field in the absence of the dielectric via Equation 19.9. Therefore,

Since the electric field within an empty capacitor is (see Equation 18.4), it follows that , which can be solved for q to give

A comparison of this expression with (Equation 19.8) reveals that the capacitance C is

| (19.10)

Notice that only the geometry of the plates (A and d) and the dielectric constant affect the capacitance. With C0 representing the capacitance of the empty capacitor , Equation 19.10 shows that . In other words, the capacitance with the dielectric present is increased by a factor of over the capacitance without the dielectric. It can be shown that the relation applies to any capacitor, not just to a parallel plate capacitor. One reason, then, that capacitors are filled with dielectric materials is to increase the capacitance. Example 10 illustrates the effect that increasing the capacitance has on the charge stored by a capacitor.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 10 | | | Storing Electric Charge

The capacitance of an empty capacitor is . The capacitor is connected to a 12-V battery and charged up. With the capacitor connected to the battery, a slab of dielectric material is inserted between the plates. As a result, of additional charge flows from one plate, through the battery, and onto the other plate. What is the dielectric constant of the material?
Reasoning The charge stored by a capacitor is , according to Equation 19.8. The battery maintains a constant potential difference of between the plates of the capacitor while the dielectric is inserted. Inserting the dielectric causes the capacitance C to increase, so that with V held constant, the charge q must increase. Thus, additional charge flows onto the plates. To find the dielectric constant, we will apply Equation 19.8 to the capacitor filled with the dielectric material and then to the empty capacitor.
Knowns and Unknowns The following table summarizes the given data:

Description | Symbol | Value
Capacitance of empty capacitor | C0 |
Potential difference between capacitor plates | V | 12 V
Additional charge that flows when dielectric is inserted |  |
Unknown Variable |  |
Dielectric constant |  | ?

Modeling the Problem

| Step 1 Dielectric Constant
| According to Equation 19.8, the charge stored by the dielectric-filled capacitor is , where is the dielectric constant, is the capacitance of the capacitor with the dielectric inserted, C0 is the capacitance of the empty capacitor, and V is the voltage provided by the battery. Solving for the dielectric constant gives Equation 1 at the right. In this expression, C0 and V are known, and we will deal with the unknown quantity q in Step 2.

| (1)

| Step 2 The Charge Stored by the Dielectric-Filled Capacitor
| The charge q stored by the dielectric-filled capacitor is the charge q0 stored by the empty capacitor plus the additional charge that arises when the dielectric is inserted. Therefore, we have

This result can be substituted into Equation 1, as indicated at the right. is given, and a value for q0 will be obtained in Step 3.

| Step3 The Charge Stored by the Empty Capacitor
| According to Equation 19.8, the charge stored by the empty capacitor is

The substitution of this expression into Equation 2 is shown at the right.

Solution Combining the results of each step algebraically, we find that

The dielectric constant can now be determined:

Related Homework: Problem 45

In Example 10 the capacitor remains connected to the battery while the dielectric is inserted between the plates. The next example discusses what happens if the capacitor is disconnected from the battery before the dielectric is inserted.

| Conceptual Example 11 | | | The Effect of a Dielectric when a Capacitor has a Constant Charge

An empty capacitor is connected to a battery and charged up. The capacitor is then disconnected from the battery, and a slab of dielectric material is inserted between the plates. Does the voltage across the plates increase, remain the same, or decrease?
Reasoning and Solution Our reasoning is guided by the following fact: once the capacitor is disconnected from the battery, the charge on its plates remains constant, for there is no longer any way for charge to be added or removed. According to Equation 19.8, the charge q stored by the capacitor is . Inserting the dielectric causes the capacitance C to increase. Therefore, the voltage V across the plates must decrease in order for q to remain unchanged. The amount by which the voltage decreases from the value initially established by the battery depends on the dielectric constant of the slab.
Related Homework: Problem 46

Capacitors are used often in electronic devices, and Example 12 deals with one familiar application.

| Example 12 | | | A Computer Keyboard

The physics of a computer keyboard. One common kind of computer keyboard is based on the idea of capacitance. Each key is mounted on one end of a plunger, the other end being attached to a movable metal plate (see Figure 19-20). The movable plate is separated from a fixed plate, the two plates forming a capacitor. When the key is pressed, the movable plate is pushed closer to the fixed plate, and the capacitance increases. Electronic circuitry enables the computer to detect the change in capacitance, thereby recognizing which key has been pressed. The separation of the plates is normally but decreases to when a key is pressed. The plate area is , and the capacitor is filled with a material whose dielectric constant is 3.50. Determine the change in capacitance that is detected by the computer.

Figure 19-20 In one kind of computer keyboard, each key, when pressed, changes the separation between the plates of a capacitor.

Reasoning We can use Equation 19.10 directly to find the capacitance of the key, since the dielectric constant , the plate area A, and the plate separation d are known. We will use this relation twice, once to find the capacitance when the key is pressed and once when it is not pressed. The change in capacitance will be the difference between these two values.
Solution When the key is pressed, the capacitance is

| (19.10)

A calculation similar to the one above reveals that when the key is not pressed, the capacitance has a value of . The change in capacitance is an increase of . The change in the capacitance is greater with the dielectric present, which makes it easier for the circuitry within the computer to detect it.

| Energy Storage in a Capacitor

When a capacitor stores charge, it also stores energy. In charging up a capacitor, for example, a battery does work in transferring an increment of charge from one plate of the capacitor to the other plate. The work done is equal to the product of the charge increment and the potential difference between the plates. However, as each increment of charge is moved, the potential difference increases slightly, and a larger amount of work is needed to move the next increment. The total work W done in completely charging the capacitor is the product of the total charge q transferred and the average potential difference . Since the average potential difference is one-half the final potential V, or , the total work done by the battery is . This work does not disappear but is stored as electric potential energy in the capacitor, so that . Equation 19.8 indicates that or, equivalently, that . We can see, then, that our expression for the energy can be cast into two additional equivalent forms by substituting for q or for V. Equations 19.11a, 19.11b and 19.11c summarize these results:

| (19.11a)

| (19.11b)

| (19.11c)

Need more practice?

Interactive LearningWare19.2
The bulb in a flashlight uses a power of 1.2 W. A capacitor stores the energy that this bulb uses in ten minutes of operation. (a) What is the voltage across the plates of this capacitor and (b) what is the charge on one of those plates? Related Homework: Problem 43

It is also possible to regard the energy as being stored in the electric field between the plates. The relation between energy and field strength can be obtained for a parallel plate capacitor by substituting (Equation 19.7b) and (Equation 19.10) into Equation 19.11b:

Since the area A times the separation d is the volume between the plates, the energy per unit volume or energy density is

| (19.12)

It can be shown that this expression is valid for any electric field strength, not just that between the plates of a capacitor.
The physics of an electronic flash attachment for a camera. The energy-storing capability of a capacitor is often put to good use in electronic circuits. For example, in an electronic flash attachment for a camera, energy from the battery pack is stored in a capacitor. The capacitor is then discharged between the electrodes of the flash tube, which converts the energy into light. Flash duration times range from 1/200 to 1/1 000 000 second or less, with the shortest flashes being used in high-speed photography (see Figure 19-21). Some flash attachments automatically control the flash duration by monitoring the light reflected from the photographic subject and quickly stopping or quenching the capacitor discharge when the reflected light reaches a predetermined level.

Figure 19-21 This time-lapse photo of a gymnast was obtained using a camera with an electronic flash attachment. The energy for each flash of light comes from the electrical energy stored in a capacitor. (Benelux Press/Index Stock)

The physics of a defibrillator. During a heart attack, the heart produces a rapid, unregulated pattern of beats, a condition known as cardiac fibrillation. Cardiac fibrillation can often be stopped by sending a very fast discharge of electrical energy through the heart. Emergency medical personnel use defibrillators, such as the one shown in Figure 19-22. A paddle is connected to each plate of a large capacitor, and the paddles are placed on the chest near the heart. The capacitor is charged to a potential difference of about a thousand volts. The capacitor is then discharged in a few thousandths of a second; the discharge current passes through a paddle, the heart, and the other paddle. Within a few seconds, the heart often returns to its normal beating pattern.

Figure 19-22 A portable defibrillator is being used in an attempt to revive this heart attack victim. A defibrillator uses the electrical energy stored in a capacitor to deliver a controlled electric current that can restore normal heart rhythm. (Adam Hart-Davis/SPL/Photo Researchers, Inc.)

Check Your Understanding 4

An empty parallel plate capacitor is connected to a battery that maintains a constant potential difference between the plates. With the battery connected, a dielectric is then inserted between the plates. Do the following quantities decrease, remain the same, or increase when the dielectric is inserted? (a) The electric field between the plates; (b) the capacitance; (c) the charge on the plates; (d) the energy stored in the capacitor. (The answers are given at the end of the book.)
Background: This question emphasizes the importance of the dielectric material in capacitors.
For similar questions (including calculational counterparts), consult Self-Assessment Test 19.2, which is described next.

Test your understanding of the material in Sections 19.4 and 19.5:

· | Equipotential Surfaces and Their Relation to the Electric Field
· | Capacitors and Dielectrics

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