The Language of Physics

Since the ball is in equilibrium the first condition of equilibrium is appliedas

2 F, = 0,

T, - w = 0,T sin 0 = w,

Fe T cos i

2FX = 0Fe - r, = oFc = T cos 8

w T sin 8 tan i

Fc = tan 8

kq2 _ mgtan 8

Solving for the charge q on each ball we get

Q =r2mg

k tan 8

The angle 6, found from the geometry of the figure 18.18(a), is

,r/2ft = nrlc-l_£_

COS

I

,5.00 cm25.0 cm

78.5C

Therefore, the charge on each ball is

-, / r2mgV k tan 8

/(0.100 m)2(5.00 X 10-3 kg)(9.80 m/s2)V (9.00 X 109 N m2/C2)tan 78.5°

= 1.05 X 10-7C

The tension in the string is

Tsin 8 = w

_ w _ m gsin 8 sin 8

_ (5.00 X 10-3kg)(9.80m/s2)sin 78.5°

= 5.00 X 10-2N

ElectrostaticsThe study of electric charges at rest underthe action of electric forces (p. 517).The fundamental principle of electrostaticsLike electric charges repel each other,whereas unlike electric charges attract eachother (p. 518).

QuarksElementary particles of matter. There aresix quarks. They are up, down, strange,charm, bottom, and top. The proton andneutron are made of quarks, but theelectron is not (p. 520).ConductorsMaterials that permit the free flow ofelectric charge through them (p. 521).

Insulators or dielectricsMaterials that do not permit the free flowof electric charge through them (p. 521).Coulomb's lawThe force between point charges qt and q2is directly proportional to the product oftheir charges and inversely proportional tothe square of the distance separating them.The direction of the force lies along the lineseparating the charges (p. 523).

530 Electricity and Magnetism

Summary of Important Equations

Coulomb's law(18.1)

Force caused by multiple chargesF = F, + F2 +

F3 + F4 + • • ■ (18.3)

Questions for Chapter 18

1. Describe the process of charging byinduction.

t2. What did Ben Franklin have to dowith the classification of electricalcharge?

3. The difference between one chemicalelement and another is the number ofprotons that each contains. Is itpossible for one chemical element tohave the same number of neutrons asanother chemical element?

4. Discuss the planetary model of theatom and state its good points and itslimitations.

t5. Is it possible for a quark to exist if ithas not been seen? If it cannot beisolated does it make any sense todescribe particles as though theywere made up of quarks?

t6. Describe the process of lightning inthe atmosphere. How do the earthand the clouds pick up electriccharge? If air is an insulator, howcan a lightning bolt ever reach theground?

7. Can you think of a way that youcould use electrostatics to measurethe humidity of the atmosphere?

8. How could you paint a metallicobject with a minimum of paintusing the principles of electrostatics?

t9. Describe the phenomenon known as"St. Elmo's Fire" in terms ofelectrostatics.

10. Why do clothes taken from a clothesdryer sometimes cling to the body?

Problems for Chapter 18

18.4 Coulomb's Law1. How many electrons are contained in

a charge of 3 juC?2. A point charge of 4.00 liC is placed

25.0 cm from another point charge of— 5.00 fiC Calculate the magnitudeand direction of the force on eachcharge.

3. A point charge qt of 2.53 liC isplaced 1.00 m in front of a secondpoint charge q2 of 8.64 /xC. Find themagnitude and direction of the forceon qt.

4. If the force of repulsion between twoprotons is equal to the weight of theproton, how far apart are theprotons?

5. The force between two point chargesis 3.5 X 10"2 N. What is the force ifthe distance separating the charges isdoubled?

6. What equal positive charges wouldhave to be placed on the earth andthe moon to neutralize thegravitational force between them?

7. What is the velocity of an electron inthe hydrogen atom if the centripetalforce is supplied by the coulombforce between the electron andproton? The radius of the electronorbit is 5.29 X 10"n m.

8. Find the electrical force on an alphaparticle when it is 6.00 X 10~" mfrom an aluminum nucleus. (Hint:

find the number of protons in analpha particle and the number ofprotons in an aluminum nucleus.)

9. Two identical metal spheres areplaced 15.0 cm apart. A charge of6.00 juC is placed on one sphere,whereas a charge of —2.00 pC isplaced on the other. What is theforce on each sphere? If the twospheres are brought together andtouched and then separated to theiroriginal separation, what will be theforce on each sphere?

10. Two identical metal spheres attracteach other with a force of 5.00 X10"6 N when they are 5.00 cm apart.The spheres are then touchedtogether and then removed to theoriginal separation where now a forceof repulsion of 1.00 X 10"6 N isobserved. What is the charge on eachsphere after touching and beforetouching?

11. Two point charges repel each otherwith a force of 3.00 X 10"5 N whenthey are 20.0 cm apart. Find theforce if the distance is reduced to5.00 cm.

18.5 Multiple Charges12. How far from a charge qt = 3 pC

should you place a charge of q2 = 9nC, such that the charge q2 experiences the force 3.98 X 10"2 N?

13. Three charges qt = 2.00 fiC, q2 =5.00 liC, and q3 = 8.00 liC areplaced along the x-axis at 0.00 cm,45.0 cm, and 72.4 cm, respectively.Find the force on each charge.

14. Three identical charges qx — q2 =q3 = 1.00 fiC are placed along thex-axis at x = 0\ 0.750, and 2.00 m.What is the magnitude and directionof the resultant force on eachcharge?

15. Three charges of 2.00 /iC, -4.00IxC, and 6.00 fiC are placed on thesame line, each 15.0 cm apart. Findthe resultant force on each charge.

16. Two charges are separated as shown.Where should a third charge beplaced on the line between them suchthat the resultant force on it will bezero? Does it matter if the thirdcharge is positive or negative?

D = 50 cm

q, =4 x 10-6C g2 = 8X 10"6C

17. Repeat problem 16 but now letcharge qt be negative, and find anyposition on the line, either to the leftof q-i, between qt and q2, or to theright of q2, where a third charge canbe placed that experiences a zeroresultant force.

Chapter 18 Electrostatics 531

tl8.

tl9.

Three charges of 2.00 liC, -4.00liC, and 6.00 itC are placed at thevertices of an equilateral triangle oflength 10.0 cm on a side. Find theresultant force on each charge.If <7i = 5.00 fiC = q2 = q3 = q4 arelocated on the corners of a square oflength 20.0 cm, find the resultantforce on q3.

Addi t ional Problemst23. Find the force on charge q5 = 5.00

nC, located at the center of a square25.0 cm on a side if q, = q2 = 3.00iiC and <?3 = q4 = 6.00 /xC.

t20. Charges of 2.54 llC, -7.86 /tC, 5.34/uC, and —3.78 fiC are placed on thecorners of a square of side 23.5 cm.Find the resultant force on the firstcharge.

t21. Find the force on charge q3 in thediagram. The distance separatingcharges ' qt and q2 is 5.00 cm.

(~ \ q ,=5x 10"6C

0/= 1 m

q2=-5x 10^6C

■ e(?3 = 1 X 10"6C

24. Four electrons are located at thecorners of a rectangle, 4.00 cm by3.00 cm, one electron at each corner.Find the magnitude and direction ofthe net force on each electron due tothe other three.

t25. Two small, equally charged spheresof mass 0.500 g are suspended fromthe same point by a silk fiber 50.0 cmlong. The repulsion between themkeeps them 15.0 cm apart. What isthe charge on each sphere?Two pith balls of 10.0-g mass arehung from ends of a string 25.0 cmlong, as shown. When the balls arecharged with equal amounts ofcharge, the threads separate to anangle of 30.0°. What is the chargeon each ball?

t26.

8 cm

i 1 0 c m i

28. Suppose that equal charges are to beplaced on the earth and on a 54.5-kgwoman so as to render the womaneffectively weightless. How muchwould the charge on each body haveto be?

t29. Find the force on q3 in the diagram if<?, = 8.00 liC, q2 = -8.00 fiC, andq3 = 8.00/xC.

1.00m

t22. Find the resultant force on charge q3in the diagram if qt = 2.00 <xC, q2= -7 ixC, and q3 = 5.00 iiC.

0.600 m

0.750 m

20.0 cm 1.11 m

t30. A charge of 15.0 ixC is on a metallicsphere 10.0-cm radius. It is thentouched to a sphere of 5.00-cmradius, until the surface chargedensity is the same on both spheres.What is the charge on each sphereafter they are separated?

t31. Two small spheres carrying chargesqx = 7.00 liC and q2 = 5.00 /uC areseparated by 20.0 cm. If q2 were freeto move, what would its initialacceleration be? Sphere 2 has themass m2 = 15.0 g.

t27. Two 10.0-g pith balls are hung fromthe ends of two 25.0-cm long stringsas shown. When an equal andopposite charge is placed on eachball, their separation is reduced from10.0 cm to 8.00 cm. Find the tensionin each string and the charge on eachball.

532 Electricity and Magnetism

t32. Where should a fourth charge, q4 =3.00 /zC/be placed to give a net forceof zero on charge q3l Charges q, =2.00 liC, q2 = 4.00 liC, and q3 =2.00 liC.

t33. Charge q, = 3.00 fiC is located atthe coordinates (0,2) and charge q2= 6.00 iiC is located at thecoordinates (1,0) of a Cartesiancoordinate system. Find thecoordinates of a third charge thatwill experience a zero net force.

t34. The configuration of a positivecharge q separated by a distance 2afrom a negative charge — q, is calledan electric dipole. Show that theforce exerted by an electric dipole ona point charge q0, located as shownin the diagram varies as 1/r3 whilethe force between a point charge qand the point charge q0 varies as1/r2. Which force is the weaker?

36. A charge of 3.00 /xC is located at theorigin and a charge of —3.00 fiC islocated on the x-axis at the pointx = 10.0 cm, forming an electricdipole. Find the force exerted by thisdipole on a proton located at thepoint x = 25.0 cm and y = 35.0 cm.

t37. A plastic rod 50.0 cm long has acharge +qt = 2.00 fiC at each end.The rod is then hung from a stringand placed so that each charge isonly 5.00 cm from negative chargesq2 = — 10.0 /iC, as shown in thediagram. Find the torque acting onthe string.

<71

Dipole and point chargea ■ ©

Two point charges

35. A charge of 3.00 fiC is located at theorigin and a charge of —3.00 llC islocated on the x-axis at the pointx = 10.0 cm, forming an electricdipole. Find the force exerted by thisdipole on a proton located on thex-axis at the point x = 40.0 cm.

<7o

q2

38. A 10.0-g object with a 10.0-liCcharge is suspended from a verticalspring of spring constant 50 N/m. Acharge of —5.00 liC is put on theground directly below the verticalspring. As a result of the presence ofthe negative charge, the springstretches by 18.0 cm relative to itsequilibrium position before thenegatively charged mass was placedbelow it. Find the distance betweenthe two charged masses in this newequilibrium position. Assume thatthey are point objects.

t39. A charge of 5.00 nC is uniformlydistributed over a copper ring 2.00cm in radius. What force will thisring exert on a point charge of 8.00nC that is placed 3.00 m away fromthe ring? Indicate what assumptionsyou make to solve this problem.

t40. A charge of 2.50 j*C is placed at thecenter of a hollow sphere of charge of8.00 llC. What is the resultant forceon the charge placed at the center ofthe sphere? Indicate whatassumptions you make to solve thisproblem.

Interact ive Tutor ia lsy 41. Coulomb's law. Two charges q, =

2.00 X 10-" C and q2 = 3.00 X10"6 C are separated by a distancer = 1.00 m. Calculate theelectrostatic force F of repulsionacting on charge 1 as the distance ofseparation is increased from r = 1 tor = 10 m. Show how the force Fvaries with the distance r.

H 42. Two electrons of charge q\ = q2 = e= 1.60 X 10"'9 C are positioned atthe coordinates (0,1) and (0,-1) (inmeters) of a Cartesian coordinatesystem. Calculate the net force F3 onan electron 173 as it is moved from x= 0 to x = 10.0 m along the x-axis.

H 43. Coulomb's law and multiple charges.Two charges <?, = 8.32 X 10" C andq2 = —2.55 X 106C lie on thex-axis and are separated by thedistance rl2 = 0.823 m. A thirdcharge q3 = 3.87 X 106 C is locateda distance r23 = 0.475 m fromcharge q2, and the line betweencharge 2 and 3 makes an angle 4> =60.0° with respect to the x-axis. Findthe resultant force on (a) charge 3,(b) charge 2, and (c) charge 1.

5)44. Coulomb's law and a continuouscharge distribution. A rod of chargeof length L = 0.100 m lies on thex-axis. One end of the rod lies at theorigin and the other end is on thepositive x-axis. A charge q' = 7.36X 10~6 C is uniformly distributedover the rod. Find the force exertedon a point charge q = 2.95 X10~6 C that lies on the x-axis atdistance x0 = 0.175 m from theorigin of the coordinate system.

>

Chapter 18 Electrostat ics 5 3 3