24LEARNING GOALSBy studying this chapter, you will learn:
The nature of capacitors, and howto calculate a quantity that meas-ures their ability to store charge.
How to analyze capacitors con-nected in a network.
How to calculate the amount ofenergy stored in a capacitor.
What dielectrics are, and how theymake capacitors more effective.
CAPACITANCE AND DIELECTRICS
?The energy used in acameras flash unit isstored in a capacitor,which consists of twoclosely spaced conduc-tors that carry oppositecharges. If the amountof charge on the con-ductors is doubled, bywhat factor does thestored energy increase?
When you set an old-fashioned spring mousetrap or pull back the stringof an archers bow, you are storing mechanical energy as elastic poten-tial energy. A capacitor is a device that stores electric potential energyand electric charge. To make a capacitor, just insulate two conductors from eachother. To store energy in this device, transfer charge from one conductor to theother so that one has a negative charge and the other has an equal amount of posi-tive charge. Work must be done to move the charges through the resulting poten-tial difference between the conductors, and the work done is stored as electricpotential energy.
Capacitors have a tremendous number of practical applications in devices suchas electronic flash units for photography, pulsed lasers, air bag sensors for cars,and radio and television receivers. Well encounter many of these applications inlater chapters (particularly Chapter 31, in which well see the crucial role playedby capacitors in the alternating-current circuits that pervade our technologicalsociety). In this chapter, however, our emphasis is on the fundamental propertiesof capacitors. For a particular capacitor, the ratio of the charge on each conductorto the potential difference between the conductors is a constant, called the capaci-tance. The capacitance depends on the sizes and shapes of the conductors and onthe insulating material (if any) between them. Compared to the case in whichthere is only vacuum between the conductors, the capacitance increases when aninsulating material (a dielectric) is present. This happens because a redistributionof charge, called polarization, takes place within the insulating material. Study-ing polarization will give us added insight into the electrical properties of matter.
Capacitors also give us a new way to think about electric potential energy. Theenergy stored in a charged capacitor is related to the electric field in the spacebetween the conductors. We will see that electric potential energy can beregarded as being stored in the field itself. The idea that the electric field is itself astorehouse of energy is at the heart of the theory of electromagnetic waves andour modern understanding of the nature of light, to be discussed in Chapter 32.
24 .1 Capacitors and Capacitance 817816 C HAPTE R 24 Capacitance and Dielectrics
24.1 Capacitors and CapacitanceAny two conductors separated by an insulator (or a vacuum) form a capacitor(Fig. 24.1). In most practical applications, each conductor initially has zero netcharge and electrons are transferred from one conductor to the other; this is calledcharging the capacitor. Then the two conductors have charges with equal magni-tude and opposite sign, and the net charge on the capacitor as a whole remainszero. We will assume throughout this chapter that this is the case. When we saythat a capacitor has charge or that a charge is stored on the capacitor, wemean that the conductor at higher potential has charge and the conductor atlower potential has charge (assuming that is positive). Keep this in mind inthe following discussion and examples.
In circuit diagrams a capacitor is represented by either of these symbols:
24.1 Any two conductors and insu-lated from each another form a capacitor.
When the separation of the platesis small compared to their size,the fringing of the field is slight.
Plate a, area A
Plate b, area A
(a) Arrangement of the capacitor plates
QPotentialdifference 5 Vab
(b) Side view of the electric field ES
24.2 A charged parallel-plate capacitor.
Calculating Capacitance: Capacitors in VacuumWe can calculate the capacitance of a given capacitor by finding the potentialdifference between the conductors for a given magnitude of charge andthen using Eq. (24.1). For now well consider only capacitors in vacuum; that is,well assume that the conductors that make up the capacitor are separated byempty space.
The simplest form of capacitor consists of two parallel conducting plates, eachwith area separated by a distance that is small in comparison with theirdimensions (Fig. 24.2a). When the plates are charged, the electric field is almostcompletely localized in the region between the plates (Fig. 24.2b). As we dis-cussed in Example 22.8 (Section 22.4), the field between such plates is essen-tially uniform, and the charges on the plates are uniformly distributed over theiropposing surfaces. We call this arrangement a parallel-plate capacitor.
We worked out the electric-field magnitude for this arrangement in Exam-ple 21.13 (Section 21.5) using the principle of superposition of electric fields andagain in Example 22.8 (Section 22.4) using Gausss law. It would be a good ideato review those examples. We found that where is the magnitude(absolute value) of the surface charge density on each plate. This is equal to themagnitude of the total charge on each plate divided by the area of the plate,or so the field magnitude can be expressed as
The field is uniform and the distance between the plates is so the potential dif-ference (voltage) between the two plates is
From this we see that the capacitance of a parallel-plate capacitor in vacuum is
(capacitance of a parallel-plate capacitor in vacuum)
The capacitance depends only on the geometry of the capacitor; it is directlyproportional to the area of each plate and inversely proportional to their sepa-ration The quantities and are constants for a given capacitor, and is auniversal constant. Thus in vacuum the capacitance is a constant independentof the charge on the capacitor or the potential difference between the plates. Ifone of the capacitor plates is flexible, the capacitance C changes as the plateseparation d changes. This is the operating principle of a condenser microphone(Fig. 24.3).
When matter is present between the plates, its properties affect the capaci-tance. We will return to this topic in Section 24.4. Meanwhile, we remark that ifthe space contains air at atmospheric pressure instead of vacuum, the capacitancediffers from the prediction of Eq. (24.2) by less than 0.06%.
In Eq. (24.2), if is in square meters and in meters, is in farads. The unitsof are so we see that
Because (energy per unit charge), this is consistent with our defini-tion Finally, the units of can be expressed as
P0 5 8.85 3 10212 F/m1 F/m,
1 C2/N # m2 5P01 F 5 1 C/V.1 V 5 1 J/C
1 F 5 1 C2/N # m 5 1 C2/JC2/N # m2,P0
Vab 5 Ed 51P0
Es 5 Q/A,AQ
sE 5 s/P0 ,
In either symbol the vertical lines (straight or curved) represent the conductorsand the horizontal lines represent wires connected to either conductor. One com-mon way to charge a capacitor is to connect these two wires to opposite terminalsof a battery. Once the charges and are established on the conductors, thebattery is disconnected. This gives a fixed potential difference between theconductors (that is, the potential of the positively charged conductor withrespect to the negatively charged conductor ) that is just equal to the voltage ofthe battery.
The electric field at any point in the region between the conductors is propor-tional to the magnitude of charge on each conductor. It follows that the poten-tial difference between the conductors is also proportional to If we doublethe magnitude of charge on each conductor, the charge density at each point dou-bles, the electric field at each point doubles, and the potential difference betweenconductors doubles; however, the ratio of charge to potential difference does notchange. This ratio is called the capacitance of the capacitor:
(definition of capacitance) (24.1)
The SI unit of capacitance is called one farad (1 F), in honor of the 19th-centuryEnglish physicist Michael Faraday. From Eq. (24.1), one farad is equal to onecoulomb per volt
CAUTION Capacitance vs. coulombs Dont confuse the symbol for capacitance(which is always in italics) with the abbreviation C for coulombs (which is neveritalicized).
The greater the capacitance of a capacitor, the greater the magnitude ofcharge on either conductor for a given potential difference and hence thegreater the amount of stored energy. (Remember that potential is potential energyper unit charge.) Thus capacitance is a measure of the ability of a capacitor tostore energy. We will see that the value of the capacitance depends only on theshapes and sizes of the conductors and on the nature of the insulating materialbetween them. (The above remarks about capacitance being independent of and do not apply to certain special types of insulating materials. We wontdiscuss these materials in this book, however.)
1 F 5 1 farad 5 1 C/V 5 1 coulomb/volt11 C/V 2 :
24.3 Inside a condenser micro