Physics of scientist ha

If a negative charge is initially at rest in an electric field, charge Q is unchanged, would this affect the potential at will it mOVe toward a region of higher;,potential or lower point

To determine the electric field surrounding a collection of two or more pointcharges requires adding up the electric fields due to each charge Since the electricfield is a vector, this can often be a chore To find the electric potential due to a

Potentials add as scalars collection of point charges is far easier, since the electric potential is a scalar, and

(Fields add as vectors) lience you only need to add numbers together without concern for direction This

is a major advantage in using electric potential We do have to include the signs ofcharges, however

600 CHAPTER 23 Electric Potential

604 CHAPTER 23

An important device th~t makes use of voltage, and that allows us to "visualize"

voltages in the sense of displaying graphically how a voltage changes in time, is

the cathode ray tube (CRT) A CRT used in this way is an oscilloscope-but an

even more common use of a CRT is as the picture tube of television sets and

com-puter monitors

The operation of a CRT depends first of all on the phenomenon of

thermionic emission, discovered by Thomas Edison (1847-1931) in the course of

experiments on developing the electric light bulb To understand how thermionic

emission occurs, consider two small plates telectrodes) inside an evacuated "bulb"

or "tube" as shown in Fig 23-18, to which is applied a potential difference (by a

battery, say) The negative electrode is called the cathode, the positive one the

anode If the negative cathode is heated (usually by an electric current, as in a

lightbulb) so that it becomes hot and glowing, it is found that negative charge

leaves the cathode and flows to the positive anode These negative charges are ~

now called electrons, but originally they: were called cathode rays since they

seemed to come from the cathode

We can understand how electrons might be "boiled off" a hot metal plate if

we treat electrons like molecules in <\ gas This makes sense if electrons are

rela-tively free to move about inside a metal, which is consistent with metals being good

conductors However, electrons don't readily escape from the metal If an electron

were to escape outside the metal surface, a net positive charge would remain

behind, and this would attract the electron back To escape, an electron needs a

cer-tain minimum kinetic energy, just as molecules in a liquid must have a minimum

kinetic energy to "evaporate" into the gaseous state We saw in Chapter 18 that the

average kinetic energy (K) of molecules in a gas is proportional to the absolute

temperature T We can apply this idea, but only very roughly, to free electrons in a

metal as if they made up an "electron gas." Of course, some electrons have more

kinetic energy than average and others less At room temperature, very few

elec-trons would have sufficient energy to escape At high temperature, K is larger and

many electrons escape-just as molecules evaporate from liquids, which occurs

more readily at high temperatures Thus, significant thermionic emission occurs only

at elevated temperatures

*SECTION 23-9 Cathode Ray Tube: TV and Computer Monitors, Oscilloscope 605

The cathode-ray tube (CRT) derives its name from the fact that inside an

evacuated glass tube, a beam of cathode rays (electrons) is directed to variousparts of a screen to produce a "picture." A simple CRT is diagrammed inFig 23-19 Electrons emitted by the heated cathode are accelerated by a high volt-ag'e (5,000-50,000 V) applied to the anode The electrons pass out of this "elec-tron gun" through a small hole in the anode The inside of the tube face is coatedwith a fluorescent material that glows when struck by electrons A tiny bright spot

is thus visible where the electron beam strikes the screen Two horizontal and twovertical plates deflect the beam of electrons when a voltage is applied to them Theelectrons are deflected toward whichever plate is positive By varying the voltage onthe deflection plates, the bright spot can be placed at any point on the screen Todaymany CRTs use magnetic deflection coils (Chapter 27) instead of electric plates

In the picture tube or monitor for a computer or television set, the electron

beam is made to sweep over the screen in the manner shown in Fig 23-20 Thebeam is swept horizontally by the horizontal deflection plates or coils When thehorizontal deflecting field is maximum in one direction, the beam is at one edge ofthe screen As the field decreases to zero, the beam moves to the center; and as thefield increases to a maximum in the opposite direction, the beam approaches theop.posite edge When the beam reaches this edge, the voltage or current abruptlycha~ges to return the beam to the opposite side of the screen Simultaneously, thebeam is deflected downward slightly by the vertical deflection plates (or coils), andthefl another horizontal sweep is made For standard television in the UnitedStates, 525 lines constitutes a complete sweep over the entire screen (High-defini-tion TV provides more than double this number of lines, giving greater picturesharpness.) The complete picture of 525 lines is swept out infa-s Actually, a singleverticaf sweep takes -los and involves every other line The lines in between arethen swept out over the next -los (called interlacing) We see a picture because theimage is retained by the fluorescent screen and by our eyes for about fas The pic-ture we see consists of the varied brightness of the spots on the screen The bright-ness at any point is controlled by the grid (a "porous" electrode, such as a wiregrid, that allows passage of electrons) which can limit the flow of electrons bymeans of the voltage applied to it: the more negative this voltage, the more elec-trons are repelled and the fewer pass through The voltage on the grid is deter-mined by the video signal (a voltage) sent out by the TV station and received bythe TV set Accompanying this signal are signals that synchronize the grid voltage

to the horizontal and vertical sweeps

An oscilloscope is a device for amplifying, measuring, and visually observing

an electrical signal (a "signal" is usually a time-varying voltage), especially rapidlychanging signals The signal is displayed on the screen of a CRT.Innormal opera-tion, the electron beam is swept horizontally at a uniform rate in time by using atime varying potential difference applied to the horizontal deflection plates Thesignal to be displayed is applied, after amplification, to the vertical deflection plates.The visible "trace" on the screen, which could be an ECG (Fig 23-21), a voltage in

an electronic device being repaired, or a signal from an experiment, is thus a plot ofthe signal voltage (vertically) versus time (horizontally)

1 If two points are at the same potential, does this mean that 12 Suppose the charged ring of Example 23-8 was not

uni-no work is done in moving a test charge from one point to formly charged, so that the density of charge was twice as the other? Does this imply that no force must be exerted? great near the top as near the bottom Assuming the total

2 If a negative charge is initially at rest in an electric field, charge Q is unchanged, would this affect the potential at

will it mOVe toward a region of higher;,potential or lower point P on the axis (Fig 23-li)? Would it affect the value ofpotential? What about a posltive charge? How does the E at that point? Is there a discrepancy here? Explain.potential energy of the charge change in each of these two 13 Consider a metal conductor in the shape of a football If it

carries a total charge Q, where would you expect the charge

3 State clearly the difference (a) between electric potential density uto be greatest, at the ends or along the flatter sides?

and electric field, (b) between electric potential and electric Explain (Hint: Near the surface of a conductor, E = U/EO')

identi-4 An electron is accelerated by a potential difference of, say, cal conducting sphere is neutral The two are initially

insu-0.10 V How much greater would its final speed be if it were lated, but then they are placed in contact (a) What can youaccelerated with four times as much voltage? say about the potential of each when they are in contact?

5 Can a particle ever move from a region of low ,electric (b) Will charge flow from one to the other? If so, how much?

potential to one of high potential and yet have its electric (c) If the spheres do not have the same radius, how are your

potential energy decrease? Explain. answers to parts (a) and (b) altered?

15 At a particular point, the electric field points due north In

6 If V = 0 at a point in space, must E = O? If E = 0 at

what direction(s) will the rate of change of potential be some point, must V = 0 at that point? Explain Give

(a) greatest, (b) least, and (c) zero?

examples for each.

16 If you know V at a point in space, can you calculate E at

7 When dealing with practical devices, we often take the that point? If you know E at a point can you calculate Vground (the Earth) to be 0 V. (a) If instead we said the at that point? If not, what else must be known in each case?

ground was -10 V,how would this affect V and E at other

17 Equipotentiallines are spaced 1.00 V apart Does the distance

points? (b) Does the fact that the Earth carries a net charge

between the lines in different regions of space tell you anything

affect the choice of V at its surface?

about the relative strengths of E in those regions? If so, what?

8 Can two equipotentiallines cross? Explain.

18 If the electric field E is uniform in a region, what can you

9 Draw in a few equipotentiallines in Fig 21-33b and c. infer about the electric potential V? If V is uniform in a

10 What can you say about the electric field in a region of region of space, what can you infer about E?

space that has the same potential throughout? 19 Is the electric potential energy of two unlike charges

posi-11 A satellite orbits the Earth along a gravitational equipoten- tive or negative? What about two like charges? What is the tialline What shape must the orbit be? significance of the sign of the potential energy in each case?

Questions 607

o Problems

1 (I) How much work is needed to move a-7.0-ILC charge from

of magnitude 150 V1m near its surface (a) What is the

potential of the Earth's surface relative to V = 0 atground to a point whose potential is +6.00 V higher? r = oo? (b) If the potential of the Earth is chosen to be

2 (I) How much work is needed to move a proton from a zero, what is the potential at infinity? (Ignore the fact thatpoint with a potential of +100 V to a point where it is -50 V? positive charge in the ionosphere approximately cancels the

3 (I) How much kinetic energy will an electron gain (in joules) Earth's net charge; how would this affect your answer?)

if it falls through a potential difference of 21,000 V 14 (II) A 32-cm-diameter conducting sphere is charged to

4 (I) An electron acquires 16.4 x'lO-)6 J of kinetic energy charge density u? (b) At what distance will the potentialwhen it is accelerated by an electric field from plate A to due to the sphere be only 10 V?

plate B What is the potential difference between the plates, 15 (II) An insulated spherical conductor of radius r) carries a

and which plate is at the higher potentia!.? charge Q.A second conducting sphere of radius rz and

ini-5 (II) The work done by an external force to move a tially uncharged is then connected to the first by a long -8.1O-ILC charge from point a to point b is 8.00 X 10- 4 1 If ducting wire (a) After the connection, what can you saythe charge was started from rest and had 2.10 X 10- 4 J' of about the electric potential of each sphere? (b) How muchkinetic energy when it reached point b, what must be the charge is transferred to the second sphere? Assume the con-potential difference between a and b? nected spheres are far apart compared to their radii (Why

con-make this assumption?)

6 (I) The electric field between two parallel plates connected

points that are distances Ra and Rb from a very long

to a 45-V battery is 1500 V 1m How far apart are the plates? (» Ra or Rb) straight wire carrying a uniform charge per

7 (I) An electric field -of 640 V 1m is desired between two

unit length A.

parallel plates 11.0 mm apart How large a voltage should 17 (II) A nonconducting sphere of radius ra carries a total

be applied?

charge Q distributed uniformly throughout its volume.

8 (I) How strong is the electric field between two parallel

Determine the electric potential as a function of the distance plates 5.0 mm apart if the potential difference between them

r from the center of the sphere for (a) r >ra and (b) r <ra·

9 (I) What is the maximum amount of charge that a spherical

18 (III) Repeat Problem 17 assuming the charge density PE

conductor of radius 5.0 cm can hold in air?.

increases as the square of the distance from the center of

10 (I) What minimum radius must a large conducting sphere

the sphere, and PE = 0 at the center.

of an electrostatic generating machine have if it is to carry 19 (III) A very long conducting cylinder (length L) of radius Ra30,000 V without discharge into the air? How much charge

(Ra« L) carries a uniform surface charge density u( C/m Z).

will it carry?

The cylinder is at an electric potential Va· What is the

poten-11 (II) A uniform electric field E = -300 N/Ci points in the

tial, at points far from the end, at a distance r from the center

of the cylinder? IJ"etermine for (a) r >Ra and (b) r <Ra·

negative x direction as shown in Fig 23,)2 The x and y (c) Is V = 0' at r = 00 (assume L = oo)? Explain.

coordinates of points A, B, and C are given on the diagram

(in meters) Determine the differences in potential (a) V BA,

20 (III) A hollow spherical conductor, carrying a net charge

(b) V CB, and (c) V CA'

+Q, has mner radius r) and outer radius rz = 2r)

(Fig 23-23) At the center of the sphere is a point charge

+QI2 (a) Write the electric field strength E in all three

regions as a function of r Then determine the potential as a function of r, the distance from the center, for (b) r> rz, (c) r) <r <rz, and (d) 0<r <r 1• (e) Plot both V and E

as a function of r from r = 0 to r = 2rz·

40 (II) Calculate the electric potential due to a dipole whose

dipole moment is 4.8 X 10-30C· m at a point 1.1 X 10-9 m

away if this point is far from the dipole, and: (a) along the

axis of the dipole nearer the positive charge; (b) 45° above

the axis but nearer the positive charge; (c) 45° above the

axis but nearer the negative charge.

41 (II) (a) In Example 23-10 part b, calculate the electric

potential without using the dipole approximation, Eq 23-7;

that is, don't assume r » t (b) What is the percent error in

this case when the dipole approximation is used?

42 (III) Show that if an electric dipole is placed in a uniform

electric field, then a torque is exerted on it equal to

pE sin cf>, where cf>is the angl~ between the dipole moment

vector and the direction of the electric field as shown in

Fig 23-29 What is the net force on the dipole? How are

your answers affected if the field is nonuniform? Note that

the dipole moment vector p is defined so that its magnitude

is Qt and its direction is pointing from the negative end to

the positive end as shown.

Section 23-7

44 (I) What is the potential gradient just outside the surface of

a uranium nucleus (Q = +92e) whose diameter is about

48 (III) Use the results of Problems 32 and 33 to determine the electric field due to the uniformly charged rod of Fig 23-28

for points (a) along the y axis and (b) along the x axis.

Section 23-8

49 (I) Determine the mutual electrostatic potential energy (in electron volts) of two protons in a uranium (235U) nucleus

(a) if they are at the surface, on opposite sides of the nucleus,

and (b) if one is at the center and the other is at the surface.

The diameter of a 235Unucleus is about 15 X 10-15m Ignore the other protons for this calculation.

50 (I) How much work must be done to bring three electrons from a great distance apart to within 1.0 X 10-10m from one another?

51 (I) What potential difference is needed to give a helium nucleus (Q= 3.2 X 10-19C) 48 keY of kinetic energy?

52 (I) What is the speed of (a) a 3.5 keY (kinetic energy) tron and (b) a 3.5 keY proton?

elec-53 (II) Write the total electrostatic potential energy, U, for

(a) four point charges and' (b) five point charges Draw a

diagram defining all quantities.

54 (II) An alpha particle (which is a helium nucleus, Q = +2e,

m = 6.64 X 10-27kg) is emitted in a radioactive decay with kinetic energy 5.53 MeV What is its speed?

55 (II) An electron starting from rest acquires 2.0 ke V of

kinetic energy in moving from point A to point B (a) How

much kinetic energy would a proton acquire, starting from

rest at B anli moving to point A? (b) Determine the ratio of

their speeds at the end of their respective trajectories.

56 (II) Four equal point charges, Q, are fixed at the corners of

a square of side b (a) What is their total electrostatic tial energy? (b) How much potential energy will a fifth charge, Q, have at the center of the square (relative to

poten-V = 0 at r = oo)? (c) If constrained to remain in that plane, is the fifth charge in stable or unstable equilibrium?

If unstable, what maximum kinetic energy could it acquire?

(d) Repeat part (c) for a negative (-Q) charge.

57 (II) Repeat Problem 56, parts aand b,assuming that two of the charges, on diagonally opposite corners, are replaced by -Q charges.

58 (II) Determine the total electrostatic potential energy of a

conducting sphere of radius ro that carries a total charge Qdistributed uniformly on its surface.

59 (III) Determine the total electrostatic potential energy -of a nonconducting sphere of radius rocarrying a total charge Qdistributed uniformly throughout its volume.

Molecular description Let us now examine, from the molecular point of view, why the capacitance of a

of dielectrics capacitor should increase when a dielectric is inserted between the plates Consider

a capacitor whose plates are separated by an air gap This capacitor has a charge+Q on one plate and -Q on the other (Fig 24-14a) The capacitor is isolated (notconnected to a battery) so charge cannot flow to or from the plates The potential

difference between the plates, V o, is given by Eq 24-1: Q = Co V o; the subscripts

(0) refer to the situation when only air is between the plates Now we insert adielectric between the plates (Fig 24-14b) The molecules of the dielectric may be

polar That is, although the molecules are neutral, they may have a permanent

dipole moment (as water does) Because of the electric field between the plates,the molecules will tend to become oriented as shown in Fig 24-14b; they won't beperfectly aligned because of thermal motion (Chapter 18), but they will usually be

at least partially aligned (the stronger the electric field the more alignment).Even if the, molecules are not polar, the electric field between the plates willinduce some separation of charge in the molecules (induced dipole moment).Although the electrons do not leave the molecules, they will move slightly withinthe molecules toward the positive plate So the situation is still as illustrated inFig 24-"14b

The net effect in either case is as if there were a net negative charge on theouter egge of the dielectric facing the positive plate, and a net positive charge onthe opposite side, as shown in Fig 24-14c

We can visualize that some of the electric field lines do not pass through thedielectric but instead end on charges induced on the surface of the dielectric asshown Hence the electric field within the dielectric is less than in air Now imagine apositive test charge within the dielectric Because the electric field is less, the force

a test charge feels is reduced'by some factor K (equal, as we shall see, to the dielectric

co~\tant) Because the force on our test charge is reduced by a factor K, the workneeded to move it from one plate to the other is reduced by a factor K.(We assume

624 CHAPTER 24 Capacitance, Dielectrics, Electric Energy Storage

1 Suppose two nearby conductors -(,arry the same negative

charge Can there be a potential difference between them? If

so, can the definition of capacitance, C =Q/V, be used here?

2. Suppose the separation of plates d in a parallel-plate

capac-itor is not very small compared to the dimensions of the

plates Would you expect Eq 24-2 to give an overestimate

or underestimate of the true capacitance? Explain.

3. Suppose one of the plates of a parallel-plate capacitor was

moved so that the area of overlap was reduced by hal( but

they are still parallel How would this affect the capacitance?

4. Explain how the relation for the capacitance 01 a cylindrical

capacitor, Example 24-2, makes sense intuitively Use

argu-ments such as those just after Eq 24-2.

5. Describe a simple method of measuring EO using a capacitor.

6 When a battery is connected to a capacitor, why do the two

plates acquire charges of the same magnitude? Will this be

true if the two conductors are different sizes or shapes?

7. A large copper sheet of thickness I is placed between the

parallel plates of a capacitor, but does not touch the plates.

How will this affect the capacitance?

8 Suppose three identical capacitors are connected to a

bat-tery Will they store more energy if connected in series or

in parallel?

9 The parallel plates of an isolated capacitor carry opposite

charges, Q If the separation of the plates is increased, is a

force required? Is the potential difference changed? What

happens to the work done in the pulling process?

10. How does the energy iri a capacitor change if (a) the tial difference is doubled, (b) the charge on each plate is doubled, and (c) the separation of the plates is doubled, as

poten-the capacitor remains connected to a battery?

11. For dielectrics consisting of polar molecules, how would you expect the dielectric constant to change with temperature?

12. An isolated charged capacitor has horizontal plates If a thin dielectric is inserted a short way between the plates, Fig 24-16, how wiJl it move when it is then released?

13. Suppose a battery remains connected to the capacitor in Question 12 What then will happen when the dielectric is released?

14. A dielectric is pulled from between the plates of a capacitor which remains connected to a battery What changes occur

to the capacitance, charge on the plates, potential difference, energy stored, and electric field?

15. How does the energy stored in a capacitor change when a

dielectric is inserted if (a) the capacitor is isolated so Q

doesn't change; (b) the capacitor remains connected to a

battery soVdoesn't change?

Questions 627

16 We have seen that the capacitance C depends on the size,

shape, and position of the two conductors, as well as on the

dielectric constant K What then did we mean when we said

that C is a constant in Eq 24-1 ?

17 What value might we assign to the dielectric constant for a

good conductor? Explain.

18 Dissolving Power 0tWater The very high dielectric constant

of water, K = 80 (Table 24-1), has a profound effect on

materials in that it allows many of them to be dissolved in

water For example, ordinary table salt, NaCI (sodium

chlo-ride), whose crystal structure (Fig 24-17a) is held together

by the attractive forces between the ions Na+ and Cl-, is

eas-ily dissolved when placed in water Explain why we would

expect that the electric field produced by each ion would be

reduced by a factor equal to the dielectric constant; that is,

discuss the extension of Eq 24-10 to the field of a point

charge in a dielectric, and thus explain (using this simple

mod~l) how salt is dissolved (see Fig 24-17b) .

1 (I) The two plates of a capacitor hold +2500JLC and 9 (I) A OAO-JLF capacitor is desired What area must the

-2500 JLC of charge, respectively, when the potential differ- plates have if they are to be separated by a 4.0-mm air gap?ence is 950 V What is the capacitance?

HJ (I) What is the capacitance per unit length (F1m)of a

coax-2 (I) A 12,OOO-pFcapacitor holds 28.0 ~ 10- 8 C of charge ial cable whose inner conductor has a 1.O-mm diameter and What is the voltage across the ,capacitor? the outer cylindrical sheath has a 5.0-mm diameter? Assume

3 (I) The potential difference between tWQparallel wires in the space between is filled with air.

11 (I) Determine the capacitance of the Earth, assuming it to air is 12.0 V They carry equal and opposite charge of mag-

be a spherical conductor.

nitude 75 pc What is the capacitance of the two wires?

12 (II) Use Gauss's law to show that E = 0 inside the inner

4 (I) How much charge flows from a 12-V battery when it is conductor of a cylindrical capacitor (see Fig 24-5 andconnected to a15.6-JLF capacitor?

Example 24-2) as well as outside the outer cylinder.

5 (I) The charge on a capacitor increases by 16JLC when 13 (II) Dry air will break down if the electric field exceeds the voltage across it increases from 28 V tQ 48 V What is the about 3.0 X 10 6 V1m.What amount of charge can be placed capacitance of the capacitor? on a capacitor if the area of each plate is 8.5 cmz?

6 (II) A capacitor C) carries a charge Qo.It is then connected 14 (II) An electric field of 2.80x 10 5 V1mis desired between

two parallel plates each of area 21.0 cmz and separated by directly to a second, initially uncharged, capacitor C z What

0.250 cm of air What charge must be on each plate?

charge will each carry now? What will be the potential

dif-15 (II) In the limit of a very small separation of the two ference across each?

cylin-ders of a cylindrical capacitor (Ra - Rb « Ra in Fig 24-5)

7 (II) It takes 25 J of energy to move a 0.20-mC charge from

show that the relation derived in Example 24-2 reduces to one plate of a 16-JLF capacitor to the other How much that of a parallel-plate capacitor (Eq 24-2).

charge is on each plate?

16 (II) Suppose a capacitor carries a charge of ±4.2JLC, and

8 (II) A 2AO-JLF capacitor is charged to 880 V and a4.00-JLF an electric field of 2.0 kVImm is desired between the plates

capacitor is charged to 560 V (a) These capacitors are then which are separated by 4.0 mm of air What must each disconnected from their batteries, and the positive plates are plate's area be?

now connected to each other and the negative plates are con- 17 (II) How strong is the electric field between the plates of anected to each other What will be the potential difference 0.80-JLF air-gap capacitor if they are 2.0 mm apart and each

across each capacitor and the charge on each? (b) What is has a charge of 72JLC?

the voltage and charge for each capacitor if plates of oppo- 18 (II) Show that for a spherical capacitor (Example 24-3), ifsite sign are connected?

the spacing between shells is very small (ra - rb « ra),

the formula reduces to that for a parallel-plate capacitor.

628 CHAPTER 24 Capacitance, Dielectrics, Electric Energy Storage

19 (II) A large metal sheet of thickness I is placed between, 27 (II) Consider three capacitors, of capacitance 3000 pF,and parallel to, the plates of the parallel-plate capacitor of 5000 pF, and 0.010 p,F.What are the maximum and minimumfig 24-3 It does not touch the plates, and extends beyond capacitances that you can form from these? How do you

their edges (a) What is now the net ci!pacitance in terms of make the connection in each case?

A doand I? (b) If I = ~d, by what factor does the capaci- 28 (II) A0.20-p,F and a0.30-p,F capacitor are connected in seriestana: change when the sheet is inserted? to a 9.0-V battery Calculate (a) the potential difference across

Section 24- 3

each capacitor and (b) the charge on each (c) Repeat parts

(a) and (b) assuming the two capacitors are in parallel.

• (I) Six1.8-p,F capacitors are connected in parallel What is 29 (II) In Fig. 24-21, suppose C)=C2 =C3=C4=C.the equivalent capacitance? What is their equivalent capaci- (a) Determine the equivalent capacitance between points a

lance if connected in series? and b.(b) Determine the charge on each capacitor and the

ZL (I) The capacitance of a port,ion of a circuit is to be reduced potential difference across each if V ba = V.

from 3600 pF to 1600 pF What" capacitance can be added to 30 (II) Suppose in Fig 24-21 that C1=C2 =C3=16.0 p,Fthe circuit to produce this effect without removing existing and C4=36.0 p,F If the charge on C2 is Q2 = 12.4p,C,circuit elements? Must any existing connections be broken determine the charge on each of the other capacitors, the

Z! (D) Suppose three parallel-plate capacitors, whose plates the entire combination.

have areas A), A2' and A3 and separations d), d2, and d3,

are connected in parallel Show, using only Eq 24-2, that

Eq~ 24-3 is valid.

a (D) (a) Determine the equivalent capacitance of the circuit

shown in Fig 21-18. (b) If C) = C 2 = 2C 3 = 14.0p,F, how

much charge is stored on each capacitor when V = 25.0 V?

41 (I) How much energy is stored by the electric field between two square plates, 8.0 cm on a side, separated by a 1.5 mm air gap? The charges on the plates are equal and opposite and of magnitude 420 J-tc.

42 (II) A parallel-plate capacitor has fixed charges +Q and

-Q. The separation of the plates is then doubled (a) By

what factor does the energy stored in the electric field

change? (b) How much work must be done if the separation

of the plates is doubled from d to 2d? The area of each plate is A.

43 (II) In Fig 24-18, let V = 10.0 V and C j = Cz = C 3 =

2200 pF How much energy is stored in the capacitor network?

44 (II) What is the total energy stored in the network of capacitors in Fig 24-21, Problem 29?

45 (II) How much energy must a 12-V battery expend to fully

charge a 0.15-J-tF and a 0.20-J-tF capacitor when they are placed (a) in parallel, (b) in series? (c) How much charge

flowed from the battery in each case?

46 (II) (a) Suppose the outer radius Ra of a cylindrical

.capacitor was doubled, but the charge was kept constant By what factor would the stored energy change? Where would

the energy come from? (b) Repeat, assuming the voltage

remains constant.

47 (II) A 3.0-J-tF capacitor is charged by a 12-V battery It is

disconnected from the battery and then connected to an uncharged 5.0-J-tF capacitor. Determine the total stored

energy (a) before the two capacitors are connected and

(b) after they are connected (c) What is the change in

so the charge remains constant?

49 (II) (a) Show that each plate of a parallel-plate capacitor exerts a force

CHAPTER 24 Capacitance, Dielectrics, Electric Energy Storage

68 A homemade capacitor is assembled by placing two 9-inch pie 76 A 3.5-/.LF capacitor is charged by a 12.4-V battery and then pans 10 cm apart and connecting them to the opposite ter- is disconnected from the battery When this capacitor (C1) is minals of a 9-V battery Estimate (a) the capacitance, then connected to a second (initially uncharged) capacitor,

(b) the charge on each plate, (c) the electric field halfway C 2, the voltage on the first drops to 5.2 V What is the value

between the plates, (d) the work done by the battery to of C 2 ?

charge the plates (e) Which of the above values change if a 77 The power supply for a pulsed nitrogen laser has a0.060-/.LF

69 How does the energy stored in a capacitor change if (a) the (a) Estimate how much energy could be stored in this

potential difference is doubled, (b) the charge on each plate capacitor (b) If 10 percent of this stored electrical energy is

is doubled, and (c) the separation of the plates is doubled, converted to light energy in a pulse that is 10 microseconds

as the capacitor remains connected to a battery? long, what is the power of the laser pulse?

70 A huge 7.0-F capacitor has enough stored energy to heat 78 The variable capacitance of an old radio tuner consists of 2.5 kg of water from 20°C to 95°C What is the potential dif- four plates connected together placed alternately between ference acrQss the plates? four other plates, also connected together (Fig 24-32) Each

71 An uncharged capacitor is connected to a 24.0-V battery plate is separated from its neighbor by 1.0 mm of air Oneuntil it is fully charged, after which it is ,disconnected from set of plates can move so that the area of overlap of eachthe battery A slab of paraffin is then inserted between the plate varies from 2.0 cm2 to 9.0 cm 2. (a) Are these seven

plates What will now be the voltage between the plates? capacitors connected in series or in parallel? (b) Determine

72 It takes TS.5 J of energy to move a B.O-mC charge from one the range of capacitance values.

plate of a12.0-/.LF capacitor to the other How much charge

is on each plate?

73 A coaxial cable, Fig. 24-31, consists of an inner cylindrical

conducting wire of radius Rb surrounded by a dielectric

insu-lator Surrounding the dielectric insulator is ap outer

con-ducting sheath of radius Ra, which is usua:Jly "grounded."

(a) Determine an exp}ession for the capacitance per unit

length of a cable whose insulator has dielectric constant K.

(b) For a given cable, Rb = 3.5 mm and Ra = 9.0 mm The

dielectric constant of the dielectric insulator is K = 2.6.

Suppose that there is a potential of 1.0 kV between the inner

conducting wire and the outer conducting sheath Find the

capacitance per meter of the cable

79 A high-voltage supply can be constructed from a variable capacitor with interleaving plates which can be rotated as in Fig 24-32 A version of this type of capacitor with more plates has a capacitance which can be varied from 10 pF to

1 pF (a) Initially, this capacitor is charged by a 10,000-volt

power supply when "the capacitance is 10 pF It is then connected from the power supply and the capacitance reduced to 1.0 pF by rotating the plates What is the voltage

dis-across the capacitor now? (b) What is a major disadvantage

74 The electric field between the plates of a paper-separated of this as a high-voltage power supply?

(K = 3.75) capacitor is 9.21 X 10 4 Vim The plates are 80 A 150-pF capacitor is connected in series with an unknown1.95 mm apart and the charge on each plate is 0.475/.LC. capacitor, and as a series combination they are connected to Determine the capacitance of this capacitor and the area of a 25.0-V battery If the 150-pF capacitor stores 125 pC of

75 A parallel-plate capacitor is isolated with a charge ± Q on 81 A circuit contains a single 330-pF capacitor hooked across aeach plate If the separation of the plates is halved and a battery It is desired to store three times as much energy bydielectric (constant K) is inserted in place of air, by what adding a single capacitor to this one How would you hookfactor does the energy storage change? To what do you it up and what would its value be?

attribute the change in stored potential energy? How does 82 What are the values of effective capacitance which can be the new value of the electric field between the plates com- obtained by connecting four identical capacitors, each hav-

632 CHAPTER 24 Capacitance, Dielectrics, Electric Energy Storage

The glow of the thin wire filament of

a light bulb is caused by the electric

current passing through it Electric

energy is transformed to thermal

energy (via collisionsbetween

movingelectrons and atoms of the

wire), which causes the wire's

temperature to become so high that

it glows.Electric current and electric

power in electric circuits are of basic

importance in everyday life.We

examine both dc and ac in this

chapter, and_includethe microscopic

analysisof electric current, as well as

a look at electric hazards

Inthe previous four chapters we have been studying static electricity: electriccharges at rest In this chapter we begin our study of charges in motion, and

we call a flow of charge an electric current

In everyday life we are familiar with electric currents flowing in wires and otherconductors Indeed, most practical electrical devices depend on electric current: cur-rent flowing through a light bulb, current inlhe heating element of a stove or electriche~ter, and of course currents in electronic devices Electric currents can exist in con-ductors such as wires and also in other devices, such as the CRT of a television orcomputer monitor whose charged electrons flow through space (Section 23-9)

In electrostatic situations, we saw (Sections 21-9 and 22-3) that the electricfield must be zero inside a conductor (if it weren't, the charges would move) Butwhen charges are moving in a conductor, there can be an electric field in the con-ductor Indeed, an electric field is needed to get charges into motion, and to keepthem in motion in any normal conductor

We first look at electric current from a macroscopic point of view: that is, rent as measured in a laboratory Later in the chapter we look at currents from amicroscopic (theoretical) point of view as a flow of electrons in a wire

cur-We can control the flow of charge using electric fields and electric potential(voltage), concepts we have just been discussing In order to have a current in awire, a potential difference is needed, which can be provided by a battery

Until the year 1800,the technical development of electricity consisted mainly

of producing a static charge by friction It all changed in 1800 when Aless~ndroVolta (1745-1827; Fig 25-1) invented the electric battery, and with it produced thefirst steady flow of electric charge-that is, a steady electric current

125-11 The Electric Battery

The events that led to the discovery ofthe battery are interesting For not only was

this an important discovery, but it also gave rise to a famous scientific debate

In the 1780s, Luigi Galvani (1737-1798), professor at the University of

Bologna, carried out a series of experiments on the contraction of a frog's leg

mus-cle through electricity produced by static electricity Galvani found that

contrac-tion of the muscle could also be produced when dissimilar metals were inserted

into the frog Galvani believed that the source of the electric charge was in the frog

muscle or nerve itself, and t~at the metal merely transmitted the charge to the

proper points When he published his work in 1791, he termed this charge "animal

electricity." Many wondered, including Galvani himself, if he had discovered the

long-sought "life-force."

Volta, at the University of Pavia 200 km away, was skeptical of Galvani's results,

and came to believe that the source of the electricity was not in the animal itself,

but rather in the contact between the dissimilar metals Volta realized that a moist

conductor, such as a frog muscle or moisture at the contact point of two dissimilar

metals, was necessary in the circuit if it was to be effective He also saw that the

contracting frog muscle was a sensitive instrument for detecting electric "tension"

or "electromotive force" (his words for what we now call potential), in fact more

sensitive than the best available electroscopes that he and others had developed t

Volta's research found that certain -combinations of metals produced a

greater effect than others, and, using his measurements, he listed them in order of

effectiveness (This "electrochemical series" is still used by chemists today.) He

also found that carbon could be used in place of one of the metals

Volta then conceived his greatest contribution to science Between a disc of

zinc and one of silver, he placed a piece of cloth or paper soalced in salt solution

or dilute acid and piled a "battery" of such couplings, one on top of another, as

shown in Fig 25-2 This "pile" or "batt>ery" produced a much increased potential

difference Indeed, when strips of metal connected to the two ends of the pile

were brought close, a spark was produced Volta had designed and built the first

electric battery

A battery produces electricity by transforming chemical energy into

electri-cal energy Today a great variety of electric cells and batteries are available, from

flashlight batteries to the storage battery ofa car The simplest batteries contain

two plates or rods made of dissimilar metals (one can be carbon) called electrodes

The electrodes are immersed in a solution, such as a dilute acid, called the

electrolyte Such a device is properly calied an electric cell, and several cells

con-nected together is a battery, although today even a single cell is called a battery

The chemical reactions involved in most electric cells are quite complicated Here

we describe how one very simple ce1l works, emphasizing the physical aspects

The cell shown in Fig 25-3 uses dilute sulfuric acid as the electrolyte One of

the electrodes is made of carbon, the other of zinc That part of each electrode

out-side the solution is called the terminal, and connections t<;>wires and circuits are

made here The acid reacts with the zinc electrode and tends to dissolve it Each

zinc atom leaves two electrons behind and enters the solution as a positive ion

The zinc electrode thus acquires a negative charge As the electrolyte becomes

positively charged, electrons are pulled off the carbon electrode Thus the carbon

electrode becomes positively charged Because there is an opposite charge on the

two electrodes, there is a potential difference between the two terminals In a cell

"Yolta's most sensitive electroscope measured about 40 Y per degree (angle of leaf separation).

~onetheless, he was able to estimate the potential differences produced by dissimilar metals in

contact: for a silver-zinc contact he got about 0.7 Y, remarkably close to today's value of 0.78 V.

SECTION 25-1 The Electric Battery 635