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Chapter 32 Inductance Dr. Jie Zou PHY 1361 1 Outline Self-inductance (32.1) Mutual induction (32.4) RL circuits (32.2) Energy in a magnetic field (32.3) Oscillations in an LC circuit (32.5) The RLC circuit (32.6, 33.5) Dr. Jie Zou PHY 1361 2 Self inductance Due to self-induction, the current in the circuit does not jump from zero to its maximum value instantaneously when the switch is thrown closed. Dr. Jie Zou Self induction: the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf L set up in this case is called a self-induced emf. L = -L(dI/dt) L = - L/(dI/dt): Inductance is a measure of the opposition to a change in current. Inductance of an N-turn coil: L = NB/I; SI unit: henry (H). PHY 1361 3 Mutual induction Dr. Jie Zou Mutual induction: Very often, the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits. This condition induces an emf through a process known as mutual induction. An application: An electric toothbrush uses the mutual induction of solenoids as part of its battery-charging system. PHY 1361 4 RL circuits An inductor: A circuit element that has a large self-inductance is called an inductor. A RL circuit: Dr. Jie Zou An inductor in a circuit opposes changes in the current in that circuit. Kirchhoff’s rule: IR L dI 0 dt Solving for I: I = (/R)(1 – e-t/) = L/R: time constant of the RL circuit. If L 0, i.e. removing the inductance from the circuit, I reaches maximum value (final equilibrium value) /R instantaneously. PHY 1361 5 Energy in a magnetic field Energy stored in an inductor: U = (1/2)LI2. dI IR L 0 dt dI 2 I I R LI dt dU dI LI dt dt I 0 0 U dU LIdI L IdI Dr. Jie Zou Magnetic energy density: uB = B2/20 I This expression represents the energy stored in the magnetic field of the inductor when the current is I. 1 2 LI 2 The energy density is proportional to the square of the field magnitude. PHY 1361 6 Oscillations in an LC circuit Total energy of the circuit: U = UC + UL = Q2/2C + (1/2)LI2. If the LC circuit is resistanceless and nonradiating, the total energy of the circuit must remain constant in time: dU/dt = 0. We obtain 2 d Q 1 Q 2 dt LC Solving for Q: Q = Qmaxcos(t + ) Solving for I: I = dQ/dt = - Qmaxsin(t + ) Natural frequency of oscillation of the LC circuit: 1 Dr. Jie Zou PHY 1361 LC 7 Oscillations in an LC circuitfrom an energy point of view Dr. Jie Zou PHY 1361 8 The RLC circuit The rate of energy transformation to internal energy within a resistor: dU/dt = - I 2R d 2Q dQ Q 0 Equation for Q: L 2 R dt Dr. Jie Zou C Compare this with the equation of motion for a damped blockspring system: d 2 x dx m dt dt 2 b dt kx 0 Solving for Q: Q = Qmaxe-Rt/2Lcos(dt) PHY 1361 9