Copyright © 2009 Pearson Education, Inc. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

30-2 Self-Inductance Example 30-5: Coaxial cable inductance. Determine the inductance per unit length of a coaxial cable whose inner conductor has a radius r 1 and the outer conductor has a radius r 2. Assume the conductors are thin hollow tubes so there is no magnetic field within the inner conductor, and the magnetic field inside both thin conductors can be ignored. The conductors carry equal currents I in opposite directions.

30-3 Energy Stored in a Magnetic Field

A circuit consisting of an inductor and a resistor will begin with most of the voltage drop across the inductor, as the current is changing rapidly. With time, the current will increase less and less, until all the voltage is across the resistor LR Circuits

The sum of potential differences around the loop gives Integrating gives the current as a function of time: The time constant of an LR circuit is..

If the circuit is then shorted across the battery, the current will gradually decay away: 30-4 LR Circuits.

Example 30-6: An LR circuit. At t = 0, a 12.0-V battery is connected in series with a 220-mH inductor and a total of 30-Ω resistance, as shown. (a) What is the current at t = 0? (b) What is the time constant? (c) What is the maximum current? (d) How long will it take the current to reach half its maximum possible value? (e) At this instant, at what rate is energy being delivered by the battery, and (f) at what rate is energy being stored in the inductor’s magnetic field?

30-5 LC Circuits and Electromagnetic Oscillations An LC circuit is a charged capacitor shorted through an inductor.

30-5 LC Circuits and Electromagnetic Oscillations Summing the potential drops around the circuit gives a differential equation for Q: This is the equation for simple harmonic motion, and has solutions..

30-5 LC Circuits and Electromagnetic Oscillations Substituting shows that the equation can only be true for all times if the frequency is given by The current is sinusoidal as well:

30-5 LC Circuits and Electromagnetic Oscillations The charge and current are both sinusoidal, but with different phases.

30-5 LC Circuits and Electromagnetic Oscillations The total energy in the circuit is constant; it oscillates between the capacitor and the inductor:

34.(II) A 425-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor. Problem 34

30-6 LC Oscillations with Resistance ( LRC Circuit) Any real (nonsuperconducting) circuit will have resistance.

30-6 LC Oscillations with Resistance (LRC Circuit) Now the voltage drops around the circuit give

30-6 LC Oscillations with Resistance ( LRC Circuit) This figure shows the three cases of underdamping A ( R 2 4L/C), and critical damping B ( R 2 = 4L/C).

30-6 LC Oscillations with Resistance ( LRC Circuit) The angular frequency for underdamped oscillations is given by The charge in the circuit as a function of time is..