Dr. Jie ZouPHY Chapter 32 Inductance

Dr. Jie ZouPHY 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 ZouPHY Self inductance 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 = N  B /I; SI unit: henry (H). 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 ZouPHY Mutual induction 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.

Dr. Jie ZouPHY RL circuits An inductor: A circuit element that has a large self-inductance is called an inductor. An inductor in a circuit opposes changes in the current in that circuit. A RL circuit: Kirchhoff’s rule: 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.

Dr. Jie ZouPHY Energy in a magnetic field Energy stored in an inductor: U = (1/2)LI 2. This expression represents the energy stored in the magnetic field of the inductor when the current is I. Magnetic energy density: u B = B 2 /2  0 The energy density is proportional to the square of the field magnitude.

Dr. Jie ZouPHY Oscillations in an LC circuit Total energy of the circuit: U = U C + U L = Q 2 /2C + (1/2)LI 2. If the LC circuit is resistanceless and non- radiating, the total energy of the circuit must remain constant in time: dU/dt = 0. We obtain Solving for Q: Q = Q max cos(  t +  ) Solving for I: I = dQ/dt = -  Q max sin(  t +  ) Natural frequency of oscillation of the LC circuit:

Dr. Jie ZouPHY Oscillations in an LC circuit- from an energy point of view

Dr. Jie ZouPHY The RLC circuit The rate of energy transformation to internal energy within a resistor: dU/dt = - I 2 R Equation for Q: Compare this with the equation of motion for a damped block- spring system: Solving for Q: Q = Q max e -Rt/2L cos(  d t)