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Chapter 32 Inductance and Circuit Oscillations

Main Points of Chapter 32 Inductance and Inductors Energy in Inductors and in the Magnetic Field RL Circuits LC Circuits RLC Circuits, Damped Oscillations, and Energy

32-1 Inductance and Inductors Faraday’s Law: Changing current in a circuit will induce emf in that circuit as well as others nearby Self-Inductance: Circuit induces emf in itself Mutual Inductance: Circuit induces emf in second circuit

32-1 Inductance and Inductors emf induced through self-inductance: (32-2) The inductance L is a proportionality constant that depends on the geometry of the circuit

32-1 Inductance and Inductors emf induced in circuit 2 by changing currents in circuit 1, through mutual inductance: (32-5)

32-1 Inductance and Inductors (32-5) The mutual inductance M depends only on the geometry of the two-circuit system subscripts are omitted, as M 21 = M 12

32-1 Inductance and Inductors Units of inductance: Henry (32-6) Modification of Kirchhoff’s loop rule: In moving across an inductor of inductance L along (or against) the presumed direction of the current I, the potential change is ΔV = –L dI/dt (or +L dI/dt, respectively). Magnetic materials will change self- inductance by changing magnetic flux

32-2 Energy in Inductors Work must be done to create current through inductor This changes the energy stored in the inductor Starting from zero current: (32-13)

32-3 Energy in Magnetic Fields The energy in a solenoid depends on the current, and therefore on the magnetic field created by the current: (32-15) giving the energy density of the magnetic field: (32-16)

32-4 Time Dependence in RL Circuits When the switch closes, the inductor keeps the current from attaining its maximum value immediately. That is when the current is changing most rapidly, and when the potential drop across the conductor is at a maximum.

32-4 Time Dependence in RL Circuits Current as a function of time: (32-19)

32-5 Oscillations in LC Circuits Start with charged capacitor It will discharge through inductor, and then recharge in opposite sense If no resistance, will continue indefinitely

32-5 Oscillations in LC Circuits Charge on capacitor oscillates with frequency ω: Charge as a function of time: Here, Q 0 is the original charge and φ sets the phase at t = 0. (32-23) (32-25)

32-6 Damped Oscillations in RLC Circuits Charge equation: (32-28) Solution: where and (32-30) (32-31) (32-32)

32-6 Damped Oscillations in RLC Circuits

For a certain value of R, ω′ = 0. This is called critical damping. (32-33)

32-7 Energy in LC and RLC Circuits In a pure LC circuit, energy is transferred back and forth between the capacitor’s electric field and the inductor’s magnetic field. Including a resistor causes I 2 R losses, which show up as heat.

Summary of Chapter 32 Definition of inductance: Induced emf: (32-1) (32-2) emf induced in a second loop: (32-5) Energy in an inductor: (32-13) Energy density of a magnetic field: (32-16)

Summary of Chapter 32, cont. LC circuit oscillations: (32-23) RLC circuit oscillations: (32-30)