Introductory Physics Volume Two

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124 Time Varying Fields 6.7 element is proportional to the amplitude of the current through the element, V = ZI, just like a resistor. The proportionality constant Z is called the impedance. Let us see how this works for an inductor. Suppose that the current through an inductor is I(t) = I 0 cos ωt. Then we know that the EMF (V L ) on the inductor is V L = L dI dt = −ωLI 0 sin ωt So that the amplitude of the voltage oscillation is ωLI 0 . V L0 = ωLI 0 = Z L I 0 with Z L ≡ ωL So we see that the amplitudes of the voltage and current are proportional. The constant Z L = ωL is the equivalent of the resistance for an inductor. Theorem: Impedance: Inductor In an AC circuit the amplitude of the voltage on an inductor is proportional to the amplitude of the current flowing through the inductor. V L0 = Z L I 0 with Z L ≡ ωL The impedance of an inductor is Z L = ωL. The actual voltage and current are not proportional, since one is a sine function and the other is the cosine function. V L The voltage on an inductor reaches the peak value one quarter of a cycle before the current does. For this reason the voltage on an inductor in an AC circuit is said to lead the current by a phase of 90 ◦ . A capacitor is similar. V C = 1 C Q = 1 ∫ Idt = 1 C ωC I 0 sin ωt 1 So that the amplitude of the voltage oscillation is ωC I 0. V C0 = 1 ωC I 0 = Z C I 0 with Z C ≡ 1 ωC So we see that the amplitudes of the voltage and current are proportional. The constant Z C = 1 ωC is the equivalent of the resistance for an capacitor.
6.8 Phasor Diagrams 125 Theorem: Impedance: Capacitor In an AC circuit the amplitude of the voltage on a capacitor is proportional to the amplitude of the current flowing through the capacitor. V C0 = Z C I 0 with Z C ≡ 1 ωC The impedance of an capacitor is Z C = 1 ωC . The voltage on a capacitor reaches the peak value one quarter of a cycle after the current does. For this reason the voltage on a capacitor in an AC circuit is said to follow the current by a phase of 90 ◦ . § 6.8 Phasor Diagrams This relationship between the leading and following phases is often represented in a phasor diagraph. As you may recall an oscillation can be thought of as the horizontal component of a circular motion. So we can represent the current or voltage in and AC circuit as the horizontal component of a circular motion. The current is represented as a vector with constant length I 0 that is rotating in the counter clockwise direction. The real physical current is the projection of this vector onto the horizontal (Re) axis. One thing that is very nice about the phasor representation is that it allows us to clearly represent in a diagram the phase relationship between different quantities. Let us take the leading phase of the inductor
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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
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3.2 Electric Current 57 3 Circuits
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3.3 Ohm’s Law 59 Some materials,
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3.5 Kirchhoff’s Rules 61 ⊲ Prob
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3.6 Resistors in Combination 63 par
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3.8 Capacitor Circuits 65 Theorem:
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3.9 More Examples 67 Note that the
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3.9 More Examples 69 This mean that
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3.9 More Examples 71 15Ω 5Ω 10Ω
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Page 91 and 92: 4.6 More Examples 91 The force on t
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Page 111 and 112: 5.5 Homework 111 The magnetic force
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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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