Introductory Physics Volume Two

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120 Time Varying Fields 6.4 dφ The quantity ɛ e 0 dt is sometimes called the displacement current because it acts like a current in Ampere’s Law. ⊲ Problem 6.2 You have a parallel plate capacitor with circular plates. The capacitor is being charged so that the electric field between the plates is increasing at a constant rate: E = at, with a = 4.5 × 10 10 V · m −1 s −1 . Use the extended version of Ampere’s Law to find the magnetic field at a distance r = 4.0cm from the center of the capacitor and half way between the plates. Assume that the field is circular about the axis of the capacitor. § 6.4 Inductance Suppose that you have a loop of wire in a region where there is no source of magnetic field. If you run a current through the wire there will now be a magnetic field due to the current. This magnetic field will create a magnetic flux through the loop. Because the field produced is proportional to the current, the flux will also be proportional to the current. φ m = LI The proportionality constant L is called the self inductance of the loop. The induced EMF on the loop is the product of the inductance and the rate of change of the current. E = L dI dt Circuit elements are manufactured with specific values of inductance, and are called inductors. The SI unit of inductance is called the Henry, abbreviated as H. Inductors are essentially a coil of wire, sometimes wrapped around a core of material designed to increase the inductance. The sign of E has been chosen in such a way as to facilitate the application if Kirchhoff’s loop rule to inductors. It is the same convention that was used for the resistor. + ε – + ΔV – I For an inductor Lenz’s law can be interpreted as “the induced EMF across an inductor is in such a direction as to oppose the change in the current through the inductor”. For example, suppose that a current is flowing through an inductor, if you now try and reduce the current, an EMF will be generated in the inductor that will tend to keep the current going. Similarly if you try to increase the current an EMF will be generated that will tend to stop the current from increasing. I
6.4 Inductance 121 Example Suppose that you have the circuit shown. By some means you have established a current in the circuit (perhaps it is an induced current caused by some external magnetic field) such + ε – that at t = 0 the current is I 0 . At t = 0 the external cause of the current disappears. What will happen to the current? – ΔV + I Since the inductor opposes the change in the current the current will not stop abruptly. Instead it will decrease gradually, the inductor will drive the circuit for a while. Note that this implies that an inductor can store energy, much like a capacitor can. Let us see how we would do the circuit analysis of this system. Kirchhoff’s Loop rule give us that E + ∆V = 0 −→ L dI dt + IR = 0 −→ dI dt = −R L I −→ I(t) = I 0 e − R L t Thus we see that the current will decay exponentially. I ⊲ Problem 6.3 The circuit below is assembled with the switch open. At the time t = 0 the switch is closed. + ε – V S + – I – ΔV + (a) What is the current as a function of time? (b) What is the EMF as a function of time? (c) What is the voltage on the resistor as a function of time? (d) Check to see if Kirchhoff’s loop rule is satisfied at all times. I
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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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Page 91 and 92: 4.6 More Examples 91 The force on t
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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 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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