11Chapter 11IInductance and Magnetic EnergyMutual InductanceExample 11.1 Mutual Inductance of Two Concentric Coplanar LoSelf-InductanceExample 11.2 Self-Inductance of a SolenoidExample 11.3 Self-Inductance of a ToroidExample 11.4 Mutual Inductance of a Coil Wrapped Around a SoEnergy Stored in Magnetic FieldsExample 11.5 Energy Stored in a SolenoidAnimation 11.1: Creating and Destroying Magnetic EnergyAnimation 11.2: Magnets and Conducting RingsRL CircuitsSelf-Inductance and the Modified Kirchhoff's Loop RuleRising CurrentDecaying CurrentLC OscillationsThe RLC Series CircuitSummaryAppendix 1: General Solutions for the RLC Series CircuitQuality FactorAppendix 2: Stresses Transmitted by Magnetic FieldsAnimation 11.3: A Charged Particle in a Time-Varying MagnetiProblem-Solving StrategiesCalculating Self-InductanceCircuits containing inductorsSolved ProblemsEnergy stored in a toroidMagnetic Energy DensitySolution:Mutual InductanceRL CircuitRL CircuitLC CircuitConceptual QuestionsAdditional ProblemsSolenoidSelf-InductanceCoupled InductorsRL CircuitRL CircuitInductance of a Solenoid With and Without Iron CoreRLC CircuitSpinning CylinderSpinning LoopChapter 11 Inductance and Magnetic Energy 11.1 Mutual Inductance ............................................................................................ 11-3 Example 11.1 Mutual Inductance of Two Concentric Coplanar Loops ............... 11-5 11.2 Self-Inductance ................................................................................................. 11-5 Example 11.2 Self-Inductance of a Solenoid........................................................ 11-6 Example 11.3 Self-Inductance of a Toroid........................................................... 11-7 Example 11.4 Mutual Inductance of a Coil Wrapped Around a Solenoid ........... 11-8 11.3 Energy Stored in Magnetic Fields .................................................................. 11-10 Example 11.5 Energy Stored in a Solenoid ........................................................ 11-11 Animation 11.1: Creating and Destroying Magnetic Energy............................ 11-12 Animation 11.2: Magnets and Conducting Rings ............................................. 11-13 11.4 RL Circuits ...................................................................................................... 11-15 11.4.1 Self-Inductance and the Modified Kirchhoff's Loop Rule....................... 11-15 11.4.2 Rising Current.......................................................................................... 11-18 11.4.3 Decaying Current..................................................................................... 11-20 11.5 LC Oscillations ............................................................................................... 11-21 11.6 The RLC Series Circuit ................................................................................... 11-26 11.7 Summary......................................................................................................... 11-28 11.8 Appendix 1: General Solutions for the RLC Series Circuit ............................ 11-30 11.8.1 Quality Factor .......................................................................................... 11-32 11.9 Appendix 2: Stresses Transmitted by Magnetic Fields .................................. 11-33 Animation 11.3: A Charged Particle in a Time-Varying Magnetic Field......... 11-37 11.10 Problem-Solving Strategies .......................................................................... 11-38 11.10.1 Calculating Self-Inductance................................................................... 11-38 11.10.2 Circuits containing inductors ................................................................. 11-39 11.11 Solved Problems ........................................................................................... 11-39 11.11.1 Energy stored in a toroid........................................................................ 11-39 11.11.2 Magnetic Energy Density ...................................................................... 11-40 11.11.3 Mutual Inductance ................................................................................. 11-41 11.11.4 RL Circuit............................................................................................... 11-42 11.11.5 RL Circuit............................................................................................... 11-44 11.11.6 LC Circuit............................................................................................... 11-45 11.12 Conceptual Questions ................................................................................... 11-47 11-111.13 Additional Problems ..................................................................................... 11-48 11.13.1 Solenoid ................................................................................................. 11-48 11.13.2 Self-Inductance ...................................................................................... 11-48 11.13.3 Coupled Inductors.................................................................................. 11-48 11.13.4 RL Circuit............................................................................................... 11-49 11.13.5 RL Circuit............................................................................................... 11-50 11.13.6 Inductance of a Solenoid With and Without Iron Core ......................... 11-50 11.13.7 RLC Circuit ............................................................................................ 11-51 11.13.8 Spinning Cylinder .................................................................................. 11-52 11.13.9 Spinning Loop........................................................................................ 11-52 11-2Inductance and Magnetic Energy 11.1 Mutual Inductance Suppose two coils are placed near each other, as shown in Figure 11.1.1 Figure 11.1.1 Changing current in coil 1 produces changing magnetic flux in coil 2. The first coil has N1 turns and carries a current I1 which gives rise to a magnetic field 1BG. Since the two coils are close to each other, some of the magnetic field lines through coil 1 will also pass through coil 2. Let 21Φdenote the magnetic flux through one turn of coil 2 due to I1. Now, by varying I1 with time, there will be an induced emf associated with the
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