Introductory Physics Volume Two

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42 Electric Potential 2.2 shown. The following graph is a section through the equipotentials for the same three charges (there are a few more equipotentials drawn in this section graph). +2 +1 -3 Below is a graph of the same equipotentials with a few electric field lines graphed also. +2 +1 -3 Notice that while the field lines go from positive charges to negative charges, the equipotentials encircle charges. Also notice that wherever a field line crosses an equipoential, the field line is perpendicular to the equipotential. This must happen, as we will now show. Imagine that you move a charge a small distance ⃗ dr along the surface of an equipotential. The change in electric potential must be zero since the beginning and ending points are both on the same equipotential. But we also know that the change in electric potential is given by, dV = − ⃗ E · ⃗dr So since dV = 0, it must be the case that E ⃗ · ⃗dr = 0, and ⃗E · ⃗dr = 0 −→ E ⃗ is perpendicular to dr ⃗ since ⃗ E ≠ 0 and ⃗ dr ≠ 0.
2.3 Conductors in Equilibrium 43 ⊲ Problem 2.5 (a) Sketch the equipotentials for two like charges. (b) Sketch the equipotentials for two opposite charges. § 2.3 Conductors in Equilibrium A conductor allows it’s free charges to move through it with some amount of ease. So, if there is a force acting on a free charge then that charge will move. Thus if there is an electric field inside a conductor then the free charges in the conductor will move. In some situations, for example if the conductor is electrically insulated, the charges will eventually reach an equilibrium configuration in which they no longer move around. This makes sense, since if you apply a field to an isolated conductor the charges will move around to adjust to the new force but in the end they must settle down into the equilibrium configuration, which is the state with the lowest energy. But this tells us something about the electric field inside a conductor once it has reached equilibrium. Since the charges are not moving, there can be no electric field inside the conductor. Theorem: Conductor in Equilibrium: Field The electric field is zero inside a conductor at equilibrium. This in turn tells us that there can be no net charge inside the volume of a conductor if the conductor is in equilibrium. We can argue this as follows. Suppose that there was a net charge in some volume inside a conductor. By Gauss’s Law we know that there must be a net electric flux through the surface of this volume, but the flux is the integral of the electric field over the surface, so the electric field cannot be zero if there is a region with a net charge. But by the previous theorem we know that if there is an electric field inside a conductor then the conductor is not in equilibrium. Thus there can be a net charge inside a conductor only if the conductor is not at equilibrium. Theorem: Conductor in Equilibrium: Charge There is no net charge in any volume inside a conductor in equilibrium. If a charge is placed on a conductor it will reside on the surface of the conductor. We can arrive at one more very important result by using the fact that there is no field in a conductor in equilibrium. Since the field is zero, this means that ∆V = − ∫ ⃗ E · ⃗ dr = 0 for any path inside the conductor. This tells us that a conductor in equilibrium is all at the same electric potential.
Page 1 and 2: 1 Introductory Physics Volume Two S
Page 3 and 4: 1.1 Electric Charge 3 Demo 1 Electr
Page 5 and 6: 1.1 Electric Charge 5 Then charge a
Page 7 and 8: - - - - - 1.2 Coulomb’s Law 7 How
Page 9 and 10: 1.2 Coulomb’s Law 9 Example Consi
Page 11 and 12: 1.3 Electric Field 11 + - Compute t
Page 13 and 14: 1.3 Electric Field 13 points a well
Page 15 and 16: 1.3 Electric Field 15 Look back now
Page 17 and 18: 1.5 Continuous Charge Distributions
Page 19 and 20: 1.6 Gauss’s Law 19 § 1.6 Gauss
Page 21 and 22: 1.6 Gauss’s Law 21 Suppose that y
Page 23 and 24: 1.7 More Examples 23 ⊲ Problem 1.
Page 25 and 26: 1.7 More Examples 25 The net force
Page 27 and 28: 1.7 More Examples 27 out the net fo
Page 29 and 30: 1.7 More Examples 29 -4q +2q -q E -
Page 31 and 32: 1.7 More Examples 31 The area vecto
Page 33 and 34: 1.8 Homework 33 ⊲ Problem 1.14 Ri
Page 35 and 36: 1.9 Summary 35 § 1.9 Summary Defin
Page 37 and 38: 2.1 Electric Potential 37 2 Electri
Page 39 and 40: 2.1 Electric Potential 39 Example S
Page 41: 2.2 Equipotentials 41 ⊲ Problem 2
Page 45 and 46: 2.4 Capacitors 45 § 2.4 Capacitors
Page 47 and 48: 2.5 Energy in an Electric Field 47
Page 49 and 50: 2.7 More Examples 49 ⊲ Problem 2.
Page 51 and 52: 2.7 More Examples 51 The resulting
Page 53 and 54: 2.8 Homework 53 add up potential du
Page 55 and 56: 2.8 Homework 55 ⊲ Problem 2.27 Ho
Page 57 and 58: 3.2 Electric Current 57 3 Circuits
Page 59 and 60: 3.3 Ohm’s Law 59 Some materials,
Page 61 and 62: 3.5 Kirchhoff’s Rules 61 ⊲ Prob
Page 63 and 64: 3.6 Resistors in Combination 63 par
Page 65 and 66: 3.8 Capacitor Circuits 65 Theorem:
Page 67 and 68: 3.9 More Examples 67 Note that the
Page 69 and 70: 3.9 More Examples 69 This mean that
Page 71 and 72: 3.9 More Examples 71 15Ω 5Ω 10Ω
Page 73 and 74: 3.9 More Examples 73 First combine
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Page 77 and 78: 3.10 Homework 77 4 µF + - 2 µF +
Page 79 and 80: 3.11 Summary 79 § 3.11 Summary Def
Page 81 and 82: 4.2 Magnetic Force on a Current 81
Page 83 and 84: 4.3 Trajectories Under Magnetic For
Page 85 and 86: 4.4 Lorentz Force 85 with a charge
Page 87 and 88: 4.5 Hall Effect 87 But this charge
Page 89 and 90: 4.6 More Examples 89 -e F E v F B =
Page 91 and 92: 4.6 More Examples 91 The force on t
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4.7 Homework 93 Imagine the closed
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4.7 Homework 95 circular orbits. Wr
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5.1 Sources of Magnetic Field 97 5
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5.1 Sources of Magnetic Field 99
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5.2 Ampere’s Law 101 and ⃗ dr s
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5.2 Ampere’s Law 103 This satisfi
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5.4 More Examples 105 distance r fr
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5.4 More Examples 107 up the integr
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5.4 More Examples 109 Use the Biot-
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5.5 Homework 111 The magnetic force
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5.5 Homework 113 3.00 A 1.00 A 1 mm
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6.2 Faraday’s Law 115 6 Time Vary
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6.2 Faraday’s Law 117 Definition:
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6.3 Maxwell’s Extension of Ampere
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6.4 Inductance 121 Example Suppose
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6.7 AC Circuit Elements 123 + V S-
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6.8 Phasor Diagrams 125 Theorem: Im
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6.8 Phasor Diagrams 127 What we wan
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6.9 Homework 129 ⊲ Problem 6.6 Sh
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6.9 Homework 131 (b) A small circul
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6.9 Homework 133 maximum current in
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6.10 Summary 135 § 6.10 Summary De
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7.1 Maxwell Equations 137 7 Wave Op
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7.2 Describing Oscillations 139 tio
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7.3 Describing Waves 141 For a harm
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7.3 Describing Waves 143 Definition
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7.4 Electromagnetic Waves 145 § 7.
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7.5 Interference of Waves 147 I I A
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7.7 Interference of Two Sources 149
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7.8 Far Field Approximation 151 §
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7.9 Thin Film Interference 153 pass
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7.11 Homework 155 Second Min First
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7.11 Homework 157 θ 2 d θ 1 θ 2
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7.12 Summary 159 § 7.12 Summary De
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8.2 Reflection 161 8 Geometric Opti
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8.4 Snell’s Law 163 We see then t
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8.5 Virtual Image Caused by Refract
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8.6 Thin Lens Equation 167 back in
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8.7 Virtual Image in a Converging L
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8.8 Diverging Lenses 171 (b) virtua
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8.10 Multi-Lens Optical Systems 173
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8.11 Homework 175 We can also compu
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@ Summary 177 § 8.12 Summary Facts
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A Hints 179 1.14 Since this is only
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A Hints 181 2.27 Imagine that you b
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A Hints 183 4.12 Consider the vecto
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A Hints 185 7.26 The distance from
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B Index 187 † Ohm’s Law: Resist
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B Index 189
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C Power Series Expansions 191 The F
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D Unit Prefixes 193 § D.3 Fundamen
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Introductory Physics Volume Two

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