Introductory Physics Volume Two

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108 Sources of Magnetic Field 5.4 distance r from an infinite wire was µ0I 2πy . Let us take the limit of a → ∞ of the results we have for the finite wire, and see if we get the same results. ⎛ ⎞ µ 0 I x + B = lim ⎝ a 2 x − a 2 √ a→∞ 4πy (x ) − √ ⎠ + a 2 2 + y 2 (x − a 2 )2 + y 2 ( = µ a 0I 2 √ 4πy ( a − − a 2 √ 2 )2 (− a 2 )2 ) = µ 0I 2πy OK Example A circular wire with radius a carryies a current I. Locate the circle in the xy-plane and compute the magnetic field it causes along a line through its center. Here is the geometry of the loop: z r f r s The path along the circle can be parameterized using t = [0, 2π]: We also have dl ⃗r s = a cos t î + a sin t ĵ dr ⃗ s = −a sin t î + a cos t ĵ dt ⃗r f = z ˆk ⃗r f − ⃗r s = −a cos t î − a sin t ĵ + z ˆk √ | ⃗r f − ⃗r s | = a 2 cos 2 t + a 2 sin 2 t + z 2 = √ a 2 + z 2 Compute the cross product needed for the Biot-Savart law: ⃗ dr s dt × (⃗r f − ⃗r s ) = (−a sin tî + a cos tĵ) × (−a cos tî − a sin tĵ + zˆk) = az cos tî + az sin tĵ + (a 2 sin 2 t + a 2 cos 2 t)ˆk = az cos tî + az sin tĵ + a 2ˆk
5.4 More Examples 109 Use the Biot-Savart law: ⃗B = µ ∫ 0I 4π = µ 0I 4π ∫2π 0 ⃗ drs dt × [⃗r f − ⃗r s ] | ⃗r f − ⃗r s | 3 dt az cos tî + az sin tĵ + a 2ˆk dt (a 2 + z 2 ) 3 2 Since the integrals of the cosine and sine vanish for the interval [0, 2π], the resulting magnetic field is: µ 0 Ia ⃗B 2 ∫2π = ˆk dt 4π (a 2 + z 2 ) 3 2 −→ ⃗ B = µ 0I 2 0 a 2 (a 2 + z 2 ) 3 2 ˆk Example A semi-circular shaped wire with radius a carryies a current I. What is the magnetic field caused at the center of the semi-circle’s diameter? B = ? a Divide the wire into small sections and compute the magentic field contribution due to one small section: I dθ dB dl The direction for all sections is up, by the right hand rule. Use the Biot-Savart law for the magnitude: d ⃗ B = µ 0 4π = Iµ 0 4π Id ⃗ l × ⃗r ′ ˆk r ′ 3 (dl)(a) ˆk, a 3
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1 Introductory Physics Volume Two S
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1.1 Electric Charge 3 Demo 1 Electr
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1.1 Electric Charge 5 Then charge a
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- - - - - 1.2 Coulomb’s Law 7 How
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1.2 Coulomb’s Law 9 Example Consi
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1.3 Electric Field 11 + - Compute t
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1.3 Electric Field 13 points a well
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1.3 Electric Field 15 Look back now
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1.5 Continuous Charge Distributions
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1.6 Gauss’s Law 19 § 1.6 Gauss
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1.6 Gauss’s Law 21 Suppose that y
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1.7 More Examples 23 ⊲ Problem 1.
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1.7 More Examples 25 The net force
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1.7 More Examples 27 out the net fo
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1.7 More Examples 29 -4q +2q -q E -
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1.7 More Examples 31 The area vecto
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1.8 Homework 33 ⊲ Problem 1.14 Ri
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1.9 Summary 35 § 1.9 Summary Defin
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2.1 Electric Potential 37 2 Electri
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2.1 Electric Potential 39 Example S
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2.2 Equipotentials 41 ⊲ Problem 2
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2.3 Conductors in Equilibrium 43
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2.4 Capacitors 45 § 2.4 Capacitors
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2.5 Energy in an Electric Field 47
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2.7 More Examples 49 ⊲ Problem 2.
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2.7 More Examples 51 The resulting
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2.8 Homework 53 add up potential du
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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
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Page 73 and 74: 3.9 More Examples 73 First combine
Page 75 and 76: 3.10 Homework 75 ⊲ Problem 3.16 B
Page 77 and 78: 3.10 Homework 77 4 µF + - 2 µF +
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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
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Page 91 and 92: 4.6 More Examples 91 The force on t
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Page 95 and 96: 4.7 Homework 95 circular orbits. Wr
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Page 101 and 102: 5.2 Ampere’s Law 101 and ⃗ dr s
Page 103 and 104: 5.2 Ampere’s Law 103 This satisfi
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Page 111 and 112: 5.5 Homework 111 The magnetic force
Page 113 and 114: 5.5 Homework 113 3.00 A 1.00 A 1 mm
Page 115 and 116: 6.2 Faraday’s Law 115 6 Time Vary
Page 117 and 118: 6.2 Faraday’s Law 117 Definition:
Page 119 and 120: 6.3 Maxwell’s Extension of Ampere
Page 121 and 122: 6.4 Inductance 121 Example Suppose
Page 123 and 124: 6.7 AC Circuit Elements 123 + V S-
Page 125 and 126: 6.8 Phasor Diagrams 125 Theorem: Im
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Page 137 and 138: 7.1 Maxwell Equations 137 7 Wave Op
Page 139 and 140: 7.2 Describing Oscillations 139 tio
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Page 153 and 154: 7.9 Thin Film Interference 153 pass
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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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