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Chapter 15

524 Chapter 15 Electric Forces and Electric Fields

CONCEPTUAL QUESTIONS1. A glass object is charged to ! 3 nC by rubbing it with a silk

cloth. In the rubbing process, have protons been added tothe object or have electrons been removed from it?

2. Why must hospital personnel wear special conductingshoes while working around oxygen in an operatingroom. What might happen if the personnel wore shoeswith rubber soles?

3. Two insulated rods are oppositely charged on their ends.They are mounted at the centers so that they are free torotate, and then held in the position shown in FigureQ15.3 in a view from above. The rods rotate in the planeof the paper. Will the rods stay in those positions whenreleased? If not, into what position(s) will they move? Willthe final configuration(s) be stable?

4. Explain from an atomic viewpoint why charge is usuallytransferred by electrons.

5. Explain how a positively charged object can be used toleave another metallic object with a net negative charge.Discuss the motion of charges during the process.

6. If a suspended object A is attracted to a charged object B,can we conclude that A is charged? Explain.

7. If a metal object receives a positive charge, does its massincrease, decrease, or stay the same? What happens to itsmass if the object receives a negative charge?

8. When defining the electric field, why is it necessary tospecify that the magnitude of the test charge be very small?

9. In fair weather, there is an electric field at the surface ofthe Earth, pointing down into the ground. What is theelectric charge on the ground in this situation?

10. A student stands on a thick piece of insulating material,places her hand on top of a Van de Graaff generator, andthen turns on the generator. Does she receive a shock?

11. An uncharged, metallic-coated Styrofoam ball is sus-pended in the region between two vertical metal plates. Ifthe two plates are charged, one positively and one nega-tively, describe the motion of the ball after it is broughtinto contact with one of the plates.

12. Is it possible for an electric field to exist in empty space?Explain.

13. There are great similarities between electric and gravita-tional fields. A room can be electrically shielded so thatthere are no electric fields in the room by surrounding itwith a conductor. Can a room be gravitationally shielded?

Why or why not? [Hint: There are two kinds of charge innature, but only one kind of mass.]

14. Would life be different if the electron were positivelycharged and the proton were negatively charged? Doesthe choice of signs have any bearing on physical andchemical interactions? Explain.

15. Explain why Gauss’s law cannot be used to calculatethe electric field of (a) a polar molecule consisting of apositive and a negative charge separated by a very smalldistance, (b) a charged disk, and (c) three point chargesat the corner of a triangle.

16. Why should a ground wire be connected to the metal sup-port rod for a television antenna?

17. A balloon negatively charged by rubbing clings to a wall.Does this mean that the wall is positively charged? Whydoes the balloon eventually fall?

18. A spherical surface surrounds a point charge q. Describewhat happens to the total flux through the surface if(a) the charge is tripled, (b) the volume of the sphere isdoubled, (c) the surface is changed to a cube, (d) thecharge is moved to another location inside the surface,and (e) the charge is moved outside the surface.

19. A charged comb often attracts small bits of dry paper thatthen fly away when they touch the comb. Explain.

20. Answer the given questions with one of the statementsthat follow and defend your answer. The answer to eachpart is either !Q , 0, or "Q. A spherical conductingobject A with a charge of !Q is lowered through a holeinto a metal container B that is initially uncharged.(a) When A is at the center of B, but not touching it, the

charge on the inner surface of B is _____.(b) The charge on the outer surface of B is _____.(c) Object A is now allowed to touch the inner surface of

B. The charge on A is now _____.(d) The charge on the inner surface of B is now _____.(e) The charge on the outer surface of B is now _____.

21. A positively charged ball hanging from a nonconductingstring is brought near a nonconducting object. Based onthe behavior of the ball–string combination, the ball is seento be attracted to the object. From this experiment, it is notpossible to determine whether the object is negativelycharged or neutral. Why not? What additional experimentwould help you decide between these two possibilities?

22. An electron moving horizontally passes between two hori-zontal plates, the upper charged negatively, the lowerpositively. A uniform, upward-directed electric field existsin the region between the plates, and this field exertsan electric force downward on the electron. Describe themovement of the electron in this region.

+ – – +

Figure Q15.3

PROBLEMS1, 2, 3 = straightforward, intermediate, challenging ! = full solution available in Student Solutions Manual/Study Guide

= coached solution with hints available at www.cp7e.com = biomedical application

Section 15.3 Coulomb’s Law1. A charge of 4.5 # 10"9 C is located 3.2 m from a charge

of "2.8 # 10"9 C. Find the electrostatic force exerted byone charge on the other.

2. The Moon and Earth are bound together by gravity. If,instead, the force of attraction were the result of eachhaving a charge of the same magnitude but opposite in

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Problems 525

sign, find the quantity of charge that would have to beplaced on each to produce the required force.

3. An alpha particle (charge ! " 2.0e) is sent at high speedtoward a gold nucleus (charge ! " 79e). What is theelectrical force acting on the alpha particle when it is 2.0 # 10$14 m from the gold nucleus?

4. Four point charges are situated at the corners of a squarewith sides of length a, as in Figure P15.4. Find the expres-sion for the resultant force on the positive charge q.

The nucleus of 8Be, which consists of 4 protons and 4 neutrons, is very unstable and sponta-neously breaks into two alpha particles (helium nuclei,each consisting of 2 protons and 2 neutrons). (a) What isthe force between the two alpha particles when they are5.00 # 10$15 m apart, and (b) what will be the magnitudeof the acceleration of the alpha particles due to thisforce? Note that the mass of an alpha particle is 4.0026 u.

6. A molecule of DNA (deoxyribonucleic acid) is 2.17 %mlong. The ends of the molecule become singly ionized—negative on one end, positive on the other. The helicalmolecule acts like a spring and compresses 1.00% uponbecoming charged. Determine the effective spring con-stant of the molecule.

7. Suppose that 1.00 g of hydrogen is separated into electronsand protons. Suppose also that the protons are placed at theEarth’s North Pole and the electrons are placed at the SouthPole. What is the resulting compressional force on the Earth?

8. An electron is released a short distance above the surfaceof the Earth. A second electron directly below it exerts anelectrostatic force on the first electron just great enoughto cancel the gravitational force on it. How far below thefirst electron is the second?

Two identical conducting spheres are placed with their cen-ters 0.30 m apart. One is given a charge of 12 # 10$9 C, theother a charge of $18 # 10$9 C. (a) Find the electrostaticforce exerted on one sphere by the other. (b) The spheresare connected by a conducting wire. Find the electrostaticforce between the two after equilibrium is reached.

10. Calculate the magnitude and direction of the Coulombforce on each of the three charges shown in Figure P15.10.

9.

5.

11. Three charges are arranged as shown in Figure P15.11.Find the magnitude and direction of the electrostaticforce on the charge at the origin.

12. Three charges are arranged as shown in Figure P15.12.Find the magnitude and direction of the electrostaticforce on the 6.00-nC charge.

13. Three point charges are located at the corners of anequilateral triangle as in Figure P15.13. Calculate the netelectric force on the 7.00-%C charge.

14. Two small beads having positive charges 3q and q arefixed at the opposite ends of a horizontal insulating rod,extending from the origin to the point x ! d. As shown inFigure P15.14, a third small charged bead is free to slideon the rod. At what position is the third bead in equilib-rium? Can it be in stable equilibrium?

–q

–q+q

a

a

–q

x

y

Figure P15.4

3.00 cm 2.00 cm

6.00 C 1.50 C –2.00 Cm m m

Figure P15.10 (Problems 10 and 18)

0.100 m

x

–3.00 nC

5.00 nC 0.300 m 6.00 nC

y

Figure P15.11

0.500 m

0.500 m

0.500 m

2.00 nC

3.00 nC

6.00 nC

Figure P15.12

0.500 m

7.00 C

2.00 C –4.00 C

60.0°x

y m

mm

+

+

Figure P15.13

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Problems 525










9.

5.





–q

–q+q

a

a

–q

x

y

Figure P15.4

3.00 cm 2.00 cm

6.00 C 1.50 C –2.00 Cm m m


0.100 m

x

–3.00 nC

5.00 nC 0.300 m 6.00 nC

y

Figure P15.11

0.500 m

0.500 m

0.500 m

2.00 nC

3.00 nC

6.00 nC

Figure P15.12

0.500 m

7.00 C

2.00 C –4.00 C

60.0°x

y m

mm

+

+

Figure P15.13

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Problems 525










9.

5.





–q

–q+q

a

a

–q

x

y

Figure P15.4

3.00 cm 2.00 cm

6.00 C 1.50 C –2.00 Cm m m


0.100 m

x

–3.00 nC

5.00 nC 0.300 m 6.00 nC

y

Figure P15.11

0.500 m

0.500 m

0.500 m

2.00 nC

3.00 nC

6.00 nC

Figure P15.12

0.500 m

7.00 C

2.00 C –4.00 C

60.0°x

y m

mm

+

+

Figure P15.13

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Problems 525










9.

5.





–q

–q+q

a

a

–q

x

y

Figure P15.4

3.00 cm 2.00 cm

6.00 C 1.50 C –2.00 Cm m m


0.100 m

x

–3.00 nC

5.00 nC 0.300 m 6.00 nC

y

Figure P15.11

0.500 m

0.500 m

0.500 m

2.00 nC

3.00 nC

6.00 nC

Figure P15.12

0.500 m

7.00 C

2.00 C –4.00 C

60.0°x

y m

mm

+

+

Figure P15.13

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Problems 525










9.

5.





–q

–q+q

a

a

–q

x

y

Figure P15.4

3.00 cm 2.00 cm

6.00 C 1.50 C –2.00 Cm m m


0.100 m

x

–3.00 nC

5.00 nC 0.300 m 6.00 nC

y

Figure P15.11

0.500 m

0.500 m

0.500 m

2.00 nC

3.00 nC

6.00 nC

Figure P15.12

0.500 m

7.00 C

2.00 C –4.00 C

60.0°x

y m

mm

+

+

Figure P15.13

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Two small metallic spheres, each of mass 0.20 g, are sus-pended as pendulums by light strings from a commonpoint as shown in Figure P15.15. The spheres are giventhe same electric charge, and it is found that they come toequilibrium when each string is at an angle of 5.0° withthe vertical. If each string is 30.0 cm long, what is the mag-nitude of the charge on each sphere?

15.

16. A charge of 6.00 ! 10"9 C and a charge of "3.00 ! 10"9 Care separated by a distance of 60.0 cm. Find the position atwhich a third charge, of 12.0 ! 10"9 C, can be placed sothat the net electrostatic force on it is zero.

Section 15.4 The Electric Field17. An object with a net charge of 24 #C is placed in a uni-

form electric field of 610 N/C, directed vertically. What isthe mass of the object if it “floats” in the electric field?

18. (a) Determine the electric field strength at a point 1.00 cmto the left of the middle charge shown in Figure P15.10.(b) If a charge of "2.00 #C is placed at this point, whatare the magnitude and direction of the force on it?

An airplane is flying through a thunder-cloud at a height of 2 000 m. (This is a very dangerousthing to do because of updrafts, turbulence, and thepossibility of electric discharge.) If there are chargeconcentrations of $40.0 C at a height of 3 000 m withinthe cloud and "40.0 C at a height of 1 000 m, what is theelectric field at the aircraft?

20. An electron is accelerated by a constant electric field ofmagnitude 300 N/C. (a) Find the acceleration of the elec-tron. (b) Use the equations of motion with constant accel-eration to find the electron’s speed after 1.00 ! 10"8 s,assuming it starts from rest.

21. A Styrofoam® ball covered with a conducting paint has amass of 5.0 ! 10"3 kg and has a charge of 4.0 #C. Whatelectric field directed upward will produce an electricforce on the ball that will balance its weight?

E:

19.

22. Each of the protons in a particle beam has a kineticenergy of 3.25 ! 10"15 J. What are the magnitude anddirection of the electric field that will stop these protonsin a distance of 1.25 m?A proton accelerates from rest in a uniform electric field of640 N/C. At some later time, its speed is 1.20 ! 106 m/s.(a) Find the magnitude of the acceleration of the proton.(b) How long does it take the proton to reach this speed?(c) How far has it moved in that interval? (d) What is itskinetic energy at the later time?

24. Three charges are at the corners of an equilateral trian-gle, as shown in Figure P15.24. Calculate the electric fieldat a point midway between the two charges on the x-axis.

23.

25. Three identical charges (q % " 5.0 #C) lie along a circleof radius 2.0 m at angles of 30°, 150°, and 270°, as shownin Figure P15.25. What is the resultant electric field at thecenter of the circle?

26. Two point charges lie along the y -axis. A charge of q1 %" 9.0 #C is at y % 6.0 m, and a charge of q 2 % " 8.0 #C isat y % " 4.0 m. Locate the point (other than infinity) atwhich the total electric field is zero.

27. In Figure P15.27, determine the point (other than infin-ity) at which the total electric field is zero.

d

+3q

+q

y

x

Figure P15.14

0.20 g 0.20 g

30.0 cmu

Figure P15.15

y

– 5.00 nC

60.0°

8.00 nC

3.00 nC

0.500 m

x

Figure P15.24

1.0 m

–2.5 mC 6.0 mC

Figure P15.27

30°

150°

q

qq

x

y

270°

Figure P15.25

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15.








E:

19.



23.




d

+3q

+q

y

x

Figure P15.14

0.20 g 0.20 g

30.0 cmu

Figure P15.15

y

– 5.00 nC

60.0°

8.00 nC

3.00 nC

0.500 m

x

Figure P15.24

1.0 m

–2.5 mC 6.0 mC

Figure P15.27

30°

150°

q

qq

x

y

270°

Figure P15.25

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15.








E:

19.



23.




d

+3q

+q

y

x

Figure P15.14

0.20 g 0.20 g

30.0 cmu

Figure P15.15

y

– 5.00 nC

60.0°

8.00 nC

3.00 nC

0.500 m

x

Figure P15.24

1.0 m

–2.5 mC 6.0 mC

Figure P15.27

30°

150°

q

qq

x

y

270°

Figure P15.25

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15.








E:

19.



23.




d

+3q

+q

y

x

Figure P15.14

0.20 g 0.20 g

30.0 cmu

Figure P15.15

y

– 5.00 nC

60.0°

8.00 nC

3.00 nC

0.500 m

x

Figure P15.24

1.0 m

–2.5 mC 6.0 mC

Figure P15.27

30°

150°

q

qq

x

y

270°

Figure P15.25

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19.



23.




d

+3q

+q

y

x

Figure P15.14

0.20 g 0.20 g

30.0 cmu

Figure P15.15

y

– 5.00 nC

60.0°

8.00 nC

3.00 nC

0.500 m

x

Figure P15.24

1.0 m

–2.5 mC 6.0 mC

Figure P15.27

30°

150°

q

qq

x

y

270°

Figure P15.25

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Chapter 16

562 Chapter 16 Electrical Energy and Capacitance

CONCEPTUAL QUESTIONS1. (a) Describe the motion of a proton after it is released

from rest in a uniform electric field. (b) Describe thechanges (if any) in its kinetic energy and the electric po-tential energy associated with the proton.

2. Describe how you can increase the maximum operatingvoltage of a parallel-plate capacitor for a fixed plate sepa-ration.

3. A parallel-plate capacitor is charged by a battery, and thebattery is then disconnected from the capacitor. Becausethe charges on the capacitor plates are opposite in sign,they attract each other. Hence, it takes positive work toincrease the plate separation. Show that the externalwork done when the plate separation is increased leadsto an increase in the energy stored in the capacitor.

4. Distinguish between electric potential and electricalpotential energy.

5. Suppose you are sitting in a car and a 20-kV power linedrops across the car. Should you stay in the car or getout? The power line potential is 20 kV compared to thepotential of the ground.

6. Why is it important to avoid sharp edges or points onconductors used in high-voltage equipment?

7. Explain why, under static conditions, all points in a con-ductor must be at the same electric potential.

8. If you are given three different capacitors C1, C2, and C3,how many different combinations of capacitance can youproduce, using all capacitors in your circuits.

9. Why is it dangerous to touch the terminals of a high-volt-age capacitor even after the voltage source that chargedthe battery is disconnected from the capacitor? What canbe done to make the capacitor safe to handle after thevoltage source has been removed?

10. The plates of a capacitor are connected to a battery.What happens to the charge on the plates if the connect-ing wires are removed from the battery? What happens tothe charge if the wires are removed from the battery andconnected to each other?

11. Can electric field lines ever cross? Why or why not? Canequipotentials ever cross? Why or why not?

12. Is it always possible to reduce a combination of capacitorsto one equivalent capacitor with the rules developed inthis chapter? Explain.

13. If you were asked to design a capacitor for which a smallsize and a large capacitance were required, what factorswould be important in your design?

14. Explain why a dielectric increases the maximum operat-ing voltage of a capacitor even though the physical size ofthe capacitor doesn’t change.

15. (a) Capacitors connected in parallel all have the same(i) charge on them, (ii) potential difference across them,or (iii) neither of the above. (b) Capacitors connected inseries all have the same (i) charge on them, (ii) potentialdifference across them, or (iii) neither of the above.

16. (a) The equivalent capacitance for a group of capaci-tors connected in parallel is (i) greater than the capaci-tance of any of the capacitors in the group, (ii) lessthan the capacitance of any of the capacitors in thegroup, or (iii) neither of the above. (b) The equivalentcapacitance for a group of capacitors connected inseries is (i) greater than the capacitance of any of thecapacitors in the group, (ii) less than the capacitanceof any of the capacitors in the group, or (iii) neither ofthe above.

17. Suppose scientists had chosen to measure small energiesin proton volts rather than electron volts. What differ-ence would this make?

18. (a) Under what conditions can the equation VB ! VA "! Ex#x be used? (b) Can the equation be used to findthe difference in potential between two points in an elec-tric field set up by a point charge? (c) Can the equationbe used to find the difference in potential between twopoints in the electric field between the plates of acharged parallel-plate capacitor?


= coached problem with hints available at www.cp7e.com = biomedical application

Section 16.1 Potential Difference and Electric Potential1. A proton moves 2.00 cm parallel to a uniform electric

field of E " 200 N/C. (a) How much work is done by thefield on the proton? (b) What change occurs in the po-tential energy of the proton? (c) What potential differ-ence did the proton move through?

2. A uniform electric field of magnitude 250 V/m is directedin the positive x-direction. A 12-$C charge moves fromthe origin to the point (x, y) " (20 cm, 50 cm). (a) Whatwas the change in the potential energy of this charge?(b) Through what potential difference did the chargemove?A potential difference of 90 mV exists between the innerand outer surfaces of a cell membrane. The inner surface

3.

is negative relative to the outer surface. How much workis required to eject a positive sodium ion (Na%) from theinterior of the cell?

4. An ion accelerated through a potential difference of 60.0 V has its potential energy decreased by 1.92 & 10!17 J.Calculate the charge on the ion.

5. The potential difference between the accelerating platesof a TV set is about 25 kV. If the distance between theplates is 1.5 cm, find the magnitude of the uniform elec-tric field in the region between the plates.

6. To recharge a 12-V battery, a battery charger must move3.6 & 105 C of charge from the negative terminal to thepositive terminal. How much work is done by the charger?Express your answer in joules.

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CONCEPTUAL QUESTIONS1. (a) Describe the motion of a proton after it is released

from rest in a uniform electric field. (b) Describe thechanges (if any) in its kinetic energy and the electric po-tential energy associated with the proton.

2. Describe how you can increase the maximum operatingvoltage of a parallel-plate capacitor for a fixed plate sepa-ration.

3. A parallel-plate capacitor is charged by a battery, and thebattery is then disconnected from the capacitor. Becausethe charges on the capacitor plates are opposite in sign,they attract each other. Hence, it takes positive work toincrease the plate separation. Show that the externalwork done when the plate separation is increased leadsto an increase in the energy stored in the capacitor.

4. Distinguish between electric potential and electricalpotential energy.

5. Suppose you are sitting in a car and a 20-kV power linedrops across the car. Should you stay in the car or getout? The power line potential is 20 kV compared to thepotential of the ground.

6. Why is it important to avoid sharp edges or points onconductors used in high-voltage equipment?

7. Explain why, under static conditions, all points in a con-ductor must be at the same electric potential.

8. If you are given three different capacitors C1, C2, and C3,how many different combinations of capacitance can youproduce, using all capacitors in your circuits.

9. Why is it dangerous to touch the terminals of a high-volt-age capacitor even after the voltage source that chargedthe battery is disconnected from the capacitor? What canbe done to make the capacitor safe to handle after thevoltage source has been removed?

10. The plates of a capacitor are connected to a battery.What happens to the charge on the plates if the connect-ing wires are removed from the battery? What happens tothe charge if the wires are removed from the battery andconnected to each other?

11. Can electric field lines ever cross? Why or why not? Canequipotentials ever cross? Why or why not?

12. Is it always possible to reduce a combination of capacitorsto one equivalent capacitor with the rules developed inthis chapter? Explain.

13. If you were asked to design a capacitor for which a smallsize and a large capacitance were required, what factorswould be important in your design?

14. Explain why a dielectric increases the maximum operat-ing voltage of a capacitor even though the physical size ofthe capacitor doesn’t change.

15. (a) Capacitors connected in parallel all have the same(i) charge on them, (ii) potential difference across them,or (iii) neither of the above. (b) Capacitors connected inseries all have the same (i) charge on them, (ii) potentialdifference across them, or (iii) neither of the above.

16. (a) The equivalent capacitance for a group of capaci-tors connected in parallel is (i) greater than the capaci-tance of any of the capacitors in the group, (ii) lessthan the capacitance of any of the capacitors in thegroup, or (iii) neither of the above. (b) The equivalentcapacitance for a group of capacitors connected inseries is (i) greater than the capacitance of any of thecapacitors in the group, (ii) less than the capacitanceof any of the capacitors in the group, or (iii) neither ofthe above.

17. Suppose scientists had chosen to measure small energiesin proton volts rather than electron volts. What differ-ence would this make?

18. (a) Under what conditions can the equation VB ! VA "! Ex#x be used? (b) Can the equation be used to findthe difference in potential between two points in an elec-tric field set up by a point charge? (c) Can the equationbe used to find the difference in potential between twopoints in the electric field between the plates of acharged parallel-plate capacitor?


= coached problem with hints available at www.cp7e.com = biomedical application

Section 16.1 Potential Difference and Electric Potential1. A proton moves 2.00 cm parallel to a uniform electric

field of E " 200 N/C. (a) How much work is done by thefield on the proton? (b) What change occurs in the po-tential energy of the proton? (c) What potential differ-ence did the proton move through?

2. A uniform electric field of magnitude 250 V/m is directedin the positive x-direction. A 12-$C charge moves fromthe origin to the point (x, y) " (20 cm, 50 cm). (a) Whatwas the change in the potential energy of this charge?(b) Through what potential difference did the chargemove?A potential difference of 90 mV exists between the innerand outer surfaces of a cell membrane. The inner surface

3.

is negative relative to the outer surface. How much workis required to eject a positive sodium ion (Na%) from theinterior of the cell?

4. An ion accelerated through a potential difference of 60.0 V has its potential energy decreased by 1.92 & 10!17 J.Calculate the charge on the ion.

5. The potential difference between the accelerating platesof a TV set is about 25 kV. If the distance between theplates is 1.5 cm, find the magnitude of the uniform elec-tric field in the region between the plates.

6. To recharge a 12-V battery, a battery charger must move3.6 & 105 C of charge from the negative terminal to thepositive terminal. How much work is done by the charger?Express your answer in joules.

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Problems 563

Oppositely charged parallel plates areseparated by 5.33 mm. A potential difference of 600 Vexists between the plates. (a) What is the magnitude of theelectric field between the plates? (b) What is the magni-tude of the force on an electron between the plates?(c) How much work must be done on the electron tomove it to the negative plate if it is initially positioned2.90 mm from the positive plate?

8. Calculate the speed of a proton that is accelerated fromrest through a potential difference of 120 V. (b) Calcu-late the speed of an electron that is accelerated throughthe same potential difference.

9. A 4.00-kg block carrying a charge Q ! 50.0 "C is con-nected to a spring for which k ! 100 N/m. The block lieson a frictionless horizontal track, and the system isimmersed in a uniform electric field of magnitude E !5.00 # 105 V/m directed as in Figure P16.9. (a) If theblock is released at rest when the spring is unstretched(at x ! 0), by what maximum amount does the spring ex-pand? (b) What is the equilibrium position of the block?

7. 14. Three charges are situated at corners of a rectangle as inFigure P16.13. How much energy would be expended inmoving the 8.00-"C charge to infinity?

15. Two point charges Q1 ! $ 5.00 nC and Q 2 ! % 3.00 nCare separated by 35.0 cm. (a) What is the electric poten-tial at a point midway between the charges? (b) What isthe potential energy of the pair of charges? What is thesignificance of the algebraic sign of your answer?

16. A point charge of 9.00 # 10%9 C is located at the origin.How much work is required to bring a positive charge of3.00 # 10%9 C from infinity to the location x ! 30.0 cm?

17. The three charges in Figure P16.17 are at the vertices ofan isosceles triangle. Let q ! 7.00 nC, and calculate theelectric potential at the midpoint of the base.

18. An electron starts from rest 3.00 cm from the center of auniformly charged sphere of radius 2.00 cm. If thesphere carries a total charge of 1.00 # 10%9 C, how fastwill the electron be moving when it reaches the surfaceof the sphere?

In Rutherford’s famous scattering ex-periments that led to the planetary model of the atom,alpha particles (having charges of $ 2e and masses of6.64 # 10%27 kg) were fired toward a gold nucleus withcharge $ 79e. An alpha particle, initially very far from thegold nucleus, is fired at 2.00 # 107 m/s directly towardthe nucleus, as in Figure P16.19. How close does the al-pha particle get to the gold nucleus before turningaround? Assume the gold nucleus remains stationary.

19.

km, Q

E

x = 0

Figure P16.9

4.00 cm

q

–q –q

2.00 cm

Figure P16.17

6.00 cm

3.00 cm

8.00 mC

2.00 mC 4.00 mC


2e++

+

+ ++ +

+ +

d

v = 079e

++

Figure P16.19

10. On planet Tehar, the free-fall acceleration is the same asthat on Earth, but there is also a strong downward electricfield that is uniform close to the planet’s surface. A 2.00-kgball having a charge of 5.00 "C is thrown upward at aspeed of 20.1 m/s. It hits the ground after an interval of4.10 s. What is the potential difference between the start-ing point and the top point of the trajectory?

Section 16.2 Electric Potential and Potential Energy Due to Point ChargesSection 16.3 Potentials and Charged ConductorsSection 16.4 Equipotential Surfaces11. (a) Find the electric potential 1.00 cm from a proton.

(b) What is the electric potential difference between twopoints that are 1.00 cm and 2.00 cm from a proton?

12. Two point charges are on the y -axis, one of magnitude3.0 # 10%9 C at the origin and a second of magnitude6.0 # 10%9 C at the point y ! 30 cm. Calculate the poten-tial at y ! 60 cm.(a) Find the electric potential, taking zero at infinity, atthe upper right corner (the corner without a charge) ofthe rectangle in Figure P16.13. (b) Repeat if the 2.00-"Ccharge is replaced with a charge of % 2.00 "C.

13.

20. Starting with the definition of work, prove that the sur-face must be perpendicular to the local electric field atevery point on an equipotential surface.

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Problems 563

Oppositely charged parallel plates areseparated by 5.33 mm. A potential difference of 600 Vexists between the plates. (a) What is the magnitude of theelectric field between the plates? (b) What is the magni-tude of the force on an electron between the plates?(c) How much work must be done on the electron tomove it to the negative plate if it is initially positioned2.90 mm from the positive plate?

8. Calculate the speed of a proton that is accelerated fromrest through a potential difference of 120 V. (b) Calcu-late the speed of an electron that is accelerated throughthe same potential difference.

9. A 4.00-kg block carrying a charge Q ! 50.0 "C is con-nected to a spring for which k ! 100 N/m. The block lieson a frictionless horizontal track, and the system isimmersed in a uniform electric field of magnitude E !5.00 # 105 V/m directed as in Figure P16.9. (a) If theblock is released at rest when the spring is unstretched(at x ! 0), by what maximum amount does the spring ex-pand? (b) What is the equilibrium position of the block?

7. 14. Three charges are situated at corners of a rectangle as inFigure P16.13. How much energy would be expended inmoving the 8.00-"C charge to infinity?

15. Two point charges Q1 ! $ 5.00 nC and Q 2 ! % 3.00 nCare separated by 35.0 cm. (a) What is the electric poten-tial at a point midway between the charges? (b) What isthe potential energy of the pair of charges? What is thesignificance of the algebraic sign of your answer?

16. A point charge of 9.00 # 10%9 C is located at the origin.How much work is required to bring a positive charge of3.00 # 10%9 C from infinity to the location x ! 30.0 cm?

17. The three charges in Figure P16.17 are at the vertices ofan isosceles triangle. Let q ! 7.00 nC, and calculate theelectric potential at the midpoint of the base.

18. An electron starts from rest 3.00 cm from the center of auniformly charged sphere of radius 2.00 cm. If thesphere carries a total charge of 1.00 # 10%9 C, how fastwill the electron be moving when it reaches the surfaceof the sphere?

In Rutherford’s famous scattering ex-periments that led to the planetary model of the atom,alpha particles (having charges of $ 2e and masses of6.64 # 10%27 kg) were fired toward a gold nucleus withcharge $ 79e. An alpha particle, initially very far from thegold nucleus, is fired at 2.00 # 107 m/s directly towardthe nucleus, as in Figure P16.19. How close does the al-pha particle get to the gold nucleus before turningaround? Assume the gold nucleus remains stationary.

19.

km, Q

E

x = 0

Figure P16.9

4.00 cm

q

–q –q

2.00 cm

Figure P16.17

6.00 cm

3.00 cm

8.00 mC

2.00 mC 4.00 mC


2e++

+

+ ++ +

+ +

d

v = 079e

++

Figure P16.19

10. On planet Tehar, the free-fall acceleration is the same asthat on Earth, but there is also a strong downward electricfield that is uniform close to the planet’s surface. A 2.00-kgball having a charge of 5.00 "C is thrown upward at aspeed of 20.1 m/s. It hits the ground after an interval of4.10 s. What is the potential difference between the start-ing point and the top point of the trajectory?

Section 16.2 Electric Potential and Potential Energy Due to Point ChargesSection 16.3 Potentials and Charged ConductorsSection 16.4 Equipotential Surfaces11. (a) Find the electric potential 1.00 cm from a proton.

(b) What is the electric potential difference between twopoints that are 1.00 cm and 2.00 cm from a proton?

12. Two point charges are on the y -axis, one of magnitude3.0 # 10%9 C at the origin and a second of magnitude6.0 # 10%9 C at the point y ! 30 cm. Calculate the poten-tial at y ! 60 cm.(a) Find the electric potential, taking zero at infinity, atthe upper right corner (the corner without a charge) ofthe rectangle in Figure P16.13. (b) Repeat if the 2.00-"Ccharge is replaced with a charge of % 2.00 "C.

13.

20. Starting with the definition of work, prove that the sur-face must be perpendicular to the local electric field atevery point on an equipotential surface.

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21. A small spherical object carries a charge of 8.00 nC. Atwhat distance from the center of the object is the poten-tial equal to 100 V? 50.0 V? 25.0 V? Is the spacing of theequipotentials proportional to the change in voltage?

Section 16.6 CapacitanceSection 16.7 The Parallel-Plate Capacitor22. (a) How much charge is on each plate of a 4.00-!F ca-

pacitor when it is connected to a 12.0-V battery? (b) Ifthis same capacitor is connected to a 1.50-V battery, whatcharge is stored?

23. Consider the Earth and a cloud layer 800 m above theplanet to be the plates of a parallel-plate capacitor. (a) Ifthe cloud layer has an area of 1.0 km2 " 1.0 # 106 m2,what is the capacitance? (b) If an electric field strengthgreater than 3.0 # 106 N/C causes the air to break downand conduct charge (lightning), what is the maximumcharge the cloud can hold?

24. The potential difference between a pair of oppositelycharged parallel plates is 400 V. (a) If the spacing be-tween the plates is doubled without altering the chargeon the plates, what is the new potential difference be-tween the plates? (b) If the plate spacing is doubledwhile the potential difference between the plates is keptconstant, what is the ratio of the final charge on one ofthe plates to the original charge?An air-filled capacitor consists of two parallel plates, eachwith an area of 7.60 cm2 and separated by a distance of1.80 mm. If a 20.0-V potential difference is applied to theseplates, calculate (a) the electric field between the plates,(b) the capacitance, and (c) the charge on each plate.

26. A 1-megabit computer memory chip contains many60.0 # 10$15-F capacitors. Each capacitor has a platearea of 21.0 # 10$12 m2. Determine the plate separationof such a capacitor. (Assume a parallel-plate configura-tion). The diameter of an atom is on the order of 10$10 m " 1 Å. Express the plate separation in angstroms.

27. A parallel-plate capacitor has an area of 5.00 cm2, andthe plates are separated by 1.00 mm with air betweenthem. The capacitor stores a charge of 400 pC. (a) Whatis the potential difference across the plates of the capaci-tor? (b) What is the magnitude of the uniform electricfield in the region between the plates?

28. A small object with a mass of 350 mg carries a charge of30.0 nC and is suspended by a thread between the verticalplates of a parallel-plate capacitor. The plates are separatedby 4.00 cm. If the thread makes an angle of 15.0° with thevertical, what is the potential difference between the plates?

Section 16.8 Combinations of Capacitors29. A series circuit consists of a 0.050-!F capacitor, a 0.100-!F

capacitor, and a 400-V battery. Find the charge (a) oneach of the capacitors and (b) on each of the capacitors ifthey are reconnected in parallel across the battery.

30. Three capacitors, C1 " 5.00 !F, C2 " 4.00 !F, and C3 " 9.00 !F, are connected together. Find the effectivecapacitance of the group (a) if they are all in parallel,and (b) if they are all in series.

31. (a) Find the equivalent capacitance of the capacitors inFigure P16.31. (b) Find the charge on each capacitorand the potential difference across it.

25.

Two capacitors give an equivalent capacitance of 9.00 pFwhen connected in parallel and an equivalent capaci-tance of 2.00 pF when connected in series. What is thecapacitance of each capacitor?

33. Four capacitors are connected as shown in Figure P16.33.(a) Find the equivalent capacitance between points aand b. (b) Calculate the charge on each capacitor if a15.0-V battery is connected across points a and b.

32.

34. Consider the combination of capacitors in Figure P16.34.(a) What is the equivalent capacitance of the group?(b) Determine the charge on each capacitor.

Find the charge on each of the capaci-tors in Figure P16.35.

35.

36. To repair a power supply for a stereo amplifier, an elec-tronics technician needs a 100-!F capacitor capable ofwithstanding a potential difference of 90 V between itsplates. The only available supply is a box of five 100-!Fcapacitors, each having a maximum voltage capability of50 V. Can the technician substitute a combination ofthese capacitors that has the proper electrical character-istics, and if so, what will be the maximum voltage acrossany of the capacitors used? [Hint: The technician maynot have to use all the capacitors in the box.]

12.0 V

4.00 mF

3.00 mF

2.00 mF

Figure P16.31

6.00 mF

20.0 mF

3.00 mF15.0 mF

a b

Figure P16.33

36.0 V

24.0 mF

8.00 mF

2.00 mF4.00 mF

Figure P16.34

5.00 mF24.0 V

+

–1.00 mF

8.00 mF 4.00 mF

Figure P16.35

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25.



32.



35.


12.0 V

4.00 mF

3.00 mF

2.00 mF

Figure P16.31

6.00 mF

20.0 mF

3.00 mF15.0 mF

a b

Figure P16.33

36.0 V

24.0 mF

8.00 mF

2.00 mF4.00 mF

Figure P16.34

5.00 mF24.0 V

+

–1.00 mF

8.00 mF 4.00 mF

Figure P16.35

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25.



32.



35.


12.0 V

4.00 mF

3.00 mF

2.00 mF

Figure P16.31

6.00 mF

20.0 mF

3.00 mF15.0 mF

a b

Figure P16.33

36.0 V

24.0 mF

8.00 mF

2.00 mF4.00 mF

Figure P16.34

5.00 mF24.0 V

+

–1.00 mF

8.00 mF 4.00 mF

Figure P16.35

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25.



32.



35.


12.0 V

4.00 mF

3.00 mF

2.00 mF

Figure P16.31

6.00 mF

20.0 mF

3.00 mF15.0 mF

a b

Figure P16.33

36.0 V

24.0 mF

8.00 mF

2.00 mF4.00 mF

Figure P16.34

5.00 mF24.0 V

+

–1.00 mF

8.00 mF 4.00 mF

Figure P16.35

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problems chapter 15 - physicsatthebay.com & 16 h_w.pdf · 11.three charges are arranged as...

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