Skip to Content
OpenStax Logo
Buy book
  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Check Your Understanding

5.1

The force would point outward.

5.2

The net force would point 58°58° below the −x-axis.

5.3

E=14πε0qr2r^E=14πε0qr2r^

5.4

We will no longer be able to take advantage of symmetry. Instead, we will need to calculate each of the two components of the electric field with their own integral.

5.5

The point charge would be Q=σabQ=σab where a and b are the sides of the rectangle but otherwise identical.

5.6

The electric field would be zero in between, and have magnitude σε0σε0 everywhere else.

Conceptual Questions

1.

There are mostly equal numbers of positive and negative charges present, making the object electrically neutral.

3.

a. yes; b. yes

5.

Take an object with a known charge, either positive or negative, and bring it close to the rod. If the known charged object is positive and it is repelled from the rod, the rod is charged positive. If the positively charged object is attracted to the rod, the rod is negatively charged.

7.

No, the dust is attracted to both because the dust particle molecules become polarized in the direction of the silk.

9.

Yes, polarization charge is induced on the conductor so that the positive charge is nearest the charged rod, causing an attractive force.

11.

Charging by conduction is charging by contact where charge is transferred to the object. Charging by induction first involves producing a polarization charge in the object and then connecting a wire to ground to allow some of the charge to leave the object, leaving the object charged.

13.

This is so that any excess charge is transferred to the ground, keeping the gasoline receptacles neutral. If there is excess charge on the gasoline receptacle, a spark could ignite it.

15.

The dryer charges the clothes. If they are damp, the presence of water molecules suppresses the charge.

17.

There are only two types of charge, attractive and repulsive. If you bring a charged object near the quartz, only one of these two effects will happen, proving there is not a third kind of charge.

19.

a. No, since a polarization charge is induced. b. Yes, since the polarization charge would produce only an attractive force.

21.

The force holding the nucleus together must be greater than the electrostatic repulsive force on the protons.

23.

Either sign of the test charge could be used, but the convention is to use a positive test charge.

25.

The charges are of the same sign.

27.

At infinity, we would expect the field to go to zero, but because the sheet is infinite in extent, this is not the case. Everywhere you are, you see an infinite plane in all directions.

29.

The infinite charged plate would have E=σ2ε0E=σ2ε0 everywhere. The field would point toward the plate if it were negatively charged and point away from the plate if it were positively charged. The electric field of the parallel plates would be zero between them if they had the same charge, and E would be E=σε0E=σε0 everywhere else. If the charges were opposite, the situation is reversed, zero outside the plates and E=σε0E=σε0 between them.

31.

yes; no

33.

At the surface of Earth, the gravitational field is always directed in toward Earth’s center. An electric field could move a charged particle in a different direction than toward the center of Earth. This would indicate an electric field is present.

35.

10

Problems

37.

a. 2.00×10−9C(11.602×10−19e/C)=1.248×1010electrons2.00×10−9C(11.602×10−19e/C)=1.248×1010electrons;
b. 0.500×10−6C(11.602×10−19e/C)=3.121×1012electrons0.500×10−6C(11.602×10−19e/C)=3.121×1012electrons

39.

3.750×1021e6.242×1018e/C=–600.8C3.750×1021e6.242×1018e/C=–600.8C

41.

a. 2.0×10−9C(6.242×1018e/C)=1.248×1010e2.0×10−9C(6.242×1018e/C)=1.248×1010e;

b. 9.109×10−31kg(1.248×1010e)=1.137×10−20kg,9.109×10−31kg(1.248×1010e)=1.137×10−20kg, 1.137×10−20kg2.5×10−3kg=4.548×10−18or4.545×10−16%1.137×10−20kg2.5×10−3kg=4.548×10−18or4.545×10−16%

43.

5.00×10−9C(6.242×1018e/C)=3.121×1010e5.00×10−9C(6.242×1018e/C)=3.121×1010e;
3.121×1010e+1.0000×1012e=1.0312×1012e3.121×1010e+1.0000×1012e=1.0312×1012e

45.

atomic mass of copper atom times 1u=1.055×10−25kg1u=1.055×10−25kg;
number of copper atoms =4.739×1023atoms=4.739×1023atoms;
number of electrons equals 29 times number of atoms or 1.374×1025electrons1.374×1025electrons; 2.00×10−6C(6.242×1018e/C)1.374×1025e=9.083×10−13or9.083×10−11%2.00×10−6C(6.242×1018e/C)1.374×1025e=9.083×10−13or9.083×10−11%

47.

244.00u(1.66×10−27kg/u)=4.050×10−25kg244.00u(1.66×10−27kg/u)=4.050×10−25kg;
4.00kg4.050×10−25kg=9.877×1024atoms9.877×1024(94)=9.284×1026protons4.00kg4.050×10−25kg=9.877×1024atoms9.877×1024(94)=9.284×1026protons;
9.284×1026(1.602×10−19C/p)=1.487×108C9.284×1026(1.602×10−19C/p)=1.487×108C

49.

a. charge 1 is 3μC3μC; charge 2 is 12μC12μC, F31=2.16×10−4NF31=2.16×10−4N to the left,
F32=8.63×10−4NF32=8.63×10−4N to the right,
Fnet=6.47×10−4NFnet=6.47×10−4N to the right;
b. F31=2.16×10−4NF31=2.16×10−4N to the right,
F32=9.59×10−5NF32=9.59×10−5N to the right,
Fnet=3.12×10−4NFnet=3.12×10−4N to the right,

Three charges are shown. Charge 1 is a 3 micro Coulomb charge at the bottom left. Charge 2 is a 12 micro Coulomb charge at the bottom right, 1 meter to the right of charge 1. Charge 3 is a minus 2 nano Coulomb charge 0.5 meters above charge 2. The charges define a right triangle, with charge 2 at the right angle. The angle at the vertex with charge one is theta. The forces on charge three are shown. F 3 1 points down and to the left, toward charge 1. Force F 3 2 points vertically down.

;
c. F31x=−2.76×10−5Ni^F31x=−2.76×10−5Ni^,
F31y=−1.38×10−5Nj^F31y=−1.38×10−5Nj^,
F32y=−8.63×10−4Nj^F32y=−8.63×10−4Nj^
Fnet=−3.86×10−5Ni^8.83×10−4Nj^Fnet=−3.86×10−5Ni^8.83×10−4Nj^

51.

F=230.7NF=230.7N

53.

F=53.94NF=53.94N

55.

The tension is T=0.049NT=0.049N. The horizontal component of the tension is 0.0043N0.0043N
d=0.088m,q=6.1×10−8Cd=0.088m,q=6.1×10−8C.
The charges can be positive or negative, but both have to be the same sign.

57.

Let the charge on one of the spheres be nQ, where n is a fraction between 0 and 1. In the numerator of Coulomb’s law, the term involving the charges is nQ(1n)Q.nQ(1n)Q. This is equal to (nn2)Q2(nn2)Q2. Finding the maximum of this term gives 12n=0n=1212n=0n=12

59.

Define right to be the positive direction and hence left is the negative direction, then F=−0.05NF=−0.05N

61.

The particles form triangle of sides 13, 13, and 24 cm. The x-components cancel, whereas there is a contribution to the y-component from both charges 24 cm apart. The y-axis passing through the third charge bisects the 24-cm line, creating two right triangles of sides 5, 12, and 13 cm.
Fy=2.56NFy=2.56N in the negative y-direction since the force is attractive. The net force from both charges is Fnet=−5.12Nj^Fnet=−5.12Nj^.

63.

The diagonal is 2a2a and the components of the force due to the diagonal charge has a factor cosθ=12cosθ=12;
Fnet=[kq2a2+kq22a212]i^[kq2a2+kq22a212]j^Fnet=[kq2a2+kq22a212]i^[kq2a2+kq22a212]j^

65.

a. E=2.0×10−2NCE=2.0×10−2NC up;
b. F=2.0×10−6NF=2.0×10−6N down

67.

a. E=2.88×1011N/CE=2.88×1011N/C;
b. E=1.44×1011N/CE=1.44×1011N/C;
c. F=4.61×10−8NF=4.61×10−8N on alpha particle;
F=4.61×10−8NF=4.61×10−8N on electron

69.

E=(−2.0i^+3.0j^)NE=(−2.0i^+3.0j^)N

71.

F=3.204×10−14NF=3.204×10−14N,
a=3.517×1016m/s2a=3.517×1016m/s2

73.

q=2.78×10−9Cq=2.78×10−9C

75.

a. E=1.15×1012N/CE=1.15×1012N/C;
b. F=1.47×10−6NF=1.47×10−6N

77.

If the q2q2 is to the right of q1,q1, the electric field vector from both charges point to the right. a. E=2.70×106N/CE=2.70×106N/C;
b. F=54.0NF=54.0N

79.

There is 45°45° right triangle geometry. The x-components of the electric field at y=3my=3m cancel. The y-components give E(y=3m)=2.83×103N/CE(y=3m)=2.83×103N/C.
At the origin we have a a negative charge of magnitide
q=−2.83×10−6Cq=−2.83×10−6C.

81.

E(z)=3.6×104N/Ck^E(z)=3.6×104N/Ck^

83.

dE=14πε0λdx(x+a)2,E=λ4πε0[1l+a1a]dE=14πε0λdx(x+a)2,E=λ4πε0[1l+a1a]

85.

σ=0.02C/m2E=2.26×109N/Cσ=0.02C/m2E=2.26×109N/C

87.

At P1P1: E(y)=14πε0λLyy2+L24j^14πε0qa2(a2)2+L24j^=1πε0qaa2+L2j^E(y)=14πε0λLyy2+L24j^14πε0qa2(a2)2+L24j^=1πε0qaa2+L2j^
At P2:P2: Put the origin at the end of L.
dE=14πε0λdx(x+a)2,E=q4πε0l[1l+a1a]i^dE=14πε0λdx(x+a)2,E=q4πε0l[1l+a1a]i^

89.

a. E(r)=14πε02λxbi^+14πε02λyaj^E(r)=14πε02λxbi^+14πε02λyaj^; b. 14πε02(λx+λy)ck^14πε02(λx+λy)ck^

91.

a. F=3.2×10−17Ni^F=3.2×10−17Ni^,
a=1.92×1010m/s2i^a=1.92×1010m/s2i^;
b. F=−3.2×10−17Ni^F=−3.2×10−17Ni^,
a=−3.51×1013m/s2i^a=−3.51×1013m/s2i^

93.

m=6.5×10−11kgm=6.5×10−11kg,
E=1.6×107N/CE=1.6×107N/C

95.

E=1.70×106N/CE=1.70×106N/C,
F=1.53×10−3NTcosθ=mgTsinθ=qEF=1.53×10−3NTcosθ=mgTsinθ=qE,
tanθ=0.62θ=32.0°tanθ=0.62θ=32.0°,
This is independent of the length of the string.

97.

circular arc dEx(i^)=14πε0λdsr2cosθ(i^)dEx(i^)=14πε0λdsr2cosθ(i^),
Ex=λ4πε0r(i^)Ex=λ4πε0r(i^),
dEy(i^)=14πε0λdsr2sinθ(j^)dEy(i^)=14πε0λdsr2sinθ(j^),
Ey=λ4πε0r(j^)Ey=λ4πε0r(j^);
y-axis: Ex=λ4πε0r(i^)Ex=λ4πε0r(i^);
x-axis: Ey=λ4πε0r(j^)Ey=λ4πε0r(j^),
E=λ2πε0r(i^)+λ2πε0r(j^)E=λ2πε0r(i^)+λ2πε0r(j^)

99.

a. W=12m(v2v02)W=12m(v2v02), Qq4πε0(1r1r0)=12m(v2v02)r0r=4πε0Qq12rr0m(v2v02)Qq4πε0(1r1r0)=12m(v2v02)r0r=4πε0Qq12rr0m(v2v02); b. r0rr0r is negative; therefore, v0>vv0>v, r,andv0:Qq4πε0(1r0)=12mv02v0=Qq2πε0mr0r,andv0:Qq4πε0(1r0)=12mv02v0=Qq2πε0mr0

101.


Figure a shows a positive 20 micro Coulomb charge on the left, a negative 20 micro Coulomb charge on the right, and the field lines due to the charges. The field lines come out of the positive charge and converge coming into the negative charge. The outer field lines extend beyond the drawing area and so we see them bending to the right, toward the negative charge, but only see part of the line. The density of lines coming out of the positive is the same as the density going into the negative. Figure b shows a positive 20 micro Coulomb charge on the left, a positive 20 micro Coulomb charge on the right, and the field lines due to the charges. The field lines come out of the positive charges and diverge, bending away from the far charge. The density of lines is the same near each of the charges. Figure c shows a positive 20 micro Coulomb charge on the left, a negative 30 micro Coulomb charge on the right, and the field lines due to the charges. The field lines come out of the positive charge. More lines go into the negative 20 micro Coulomb charge than come out of the positive 20 micro Coulomb charge. All of the lines coming out of the positive charge terminate at the negative, while the outer lines going into the negative start at infinity.
103.


Four charges are shown at the corners of a square. At the top left is positive 10 nano Coulombs. At the top right is negative 10 nano Coulombs. At the bottom left is negative 10 nano Coulombs. At the bottom right is positive 10 nano Coulombs. The field lines are also shown. They come out of the positive charges and curve toward and end at the negative charges. The lowest density is near the center of the square.
105.

Ex=0,Ex=0,
Ey=14πε0[2q(x2+a2)a(x2+a2)]Ey=14πε0[2q(x2+a2)a(x2+a2)]
xa12πε0qax3xa12πε0qax3,
Ey=q4πε0[2ya+2ya(ya)2(y+a)2]Ey=q4πε0[2ya+2ya(ya)2(y+a)2]
ya1πε0qay3ya1πε0qay3

107.

The net dipole moment of the molecule is the vector sum of the individual dipole moments between the two O-H. The separation O-H is 0.9578 angstroms:
p=1.889×10−29Cmi^p=1.889×10−29Cmi^

Additional Problems

109.

Fnet=[−8.99×1093.0×10−6(5.0×10−6)(3.0m)28.99×1099.0×10−6(5.0×10−6)(3.0m)2]i^Fnet=[−8.99×1093.0×10−6(5.0×10−6)(3.0m)28.99×1099.0×10−6(5.0×10−6)(3.0m)2]i^,
−8.99×1096.0×10−6(5.0×10−6)(3.0m)2j^=−0.06Ni^0.03Nj^−8.99×1096.0×10−6(5.0×10−6)(3.0m)2j^=−0.06Ni^0.03Nj^

111.

Charges Q and q form a right triangle of sides 1 m and 3+3m.3+3m. Charges 2Q and q form a right triangle of sides 1 m and 3m.3m.
Fx=0.049N,Fx=0.049N,
Fy=0.093NFy=0.093N,
Fnet=0.036Ni^+0.09Nj^Fnet=0.036Ni^+0.09Nj^

113.

W=0.054JW=0.054J

115.

a. E=14πε0(q(2a)2qa2)i^E=14πε0(q(2a)2qa2)i^; b. E=34πε0qa2(j^)E=34πε0qa2(j^); c. E=2πε0qa212(j^)E=2πε0qa212(j^)

117.

Enet=E1+E2+E3+E4=(4.65i^+1.44j^)×107N/CEnet=E1+E2+E3+E4=(4.65i^+1.44j^)×107N/C

119.

F=qE0(1+x/a)W=12m(v2v02)F=qE0(1+x/a)W=12m(v2v02),
12mv2=qE0(15a2)J12mv2=qE0(15a2)J

121.

Electric field of wire at x: E(x)=14πε02λyxi^E(x)=14πε02λyxi^,
dF=λyλx2πε0(lnblna)dF=λyλx2πε0(lnblna)

123.


A rod of length L is shown, aligned with the x-axis with the left end at the origin. A point P is shown on the z axis, a distance a above the left end of the rod. A small segment of the rod is labeled as d x and is a distance x to the right of the left end of the rod. The line from dx to point P makes an angle of theta with the x axis. The vector d E, drawn with its tail at point P, points away from the segment d x.


dEx=14πε0λdx(x2+a2)xx2+a2dEx=14πε0λdx(x2+a2)xx2+a2,
Ex=λ4πε0[1L2+a21a]i^Ex=λ4πε0[1L2+a21a]i^,
dEz=14πε0λdx(x2+a2)ax2+a2dEz=14πε0λdx(x2+a2)ax2+a2,
Ez=λ4πε0aLL2+a2k^Ez=λ4πε0aLL2+a2k^,
Substituting z for a, we have:
E(z)=λ4πε0[1L2+z21z]i^+λ4πε0zLL2+z2k^E(z)=λ4πε0[1L2+z21z]i^+λ4πε0zLL2+z2k^

125.

There is a net force only in the y-direction. Let θθ be the angle the vector from dx to q makes with the x-axis. The components along the x-axis cancel due to symmetry, leaving the y-component of the force.
dFy=14πε0aqλdx(x2+a2)3/2dFy=14πε0aqλdx(x2+a2)3/2,
Fy=12πε0qλa[l/2((l/2)2+a2)1/2]Fy=12πε0qλa[l/2((l/2)2+a2)1/2]

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-2/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-2/pages/1-introduction
Citation information

© Oct 6, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.