Physics 1502: Lecture 22 Today’s Agenda

1. Announcements: RL - RV - RLC circuits Homework 06: due next Wednesday … Induction / AC current Physics 1502: Lecture 22Today’s Agenda

2. X X X X X X X X X Induction Self-Inductance, RL Circuits long solenoid Energy and energy density

3. L/R L/R 2L/R 2L/R e/R e/R I I 0 0 t t 0 e VL VL -e 0 t t e on e off

4. RC 2RC RC 2RC Ce Ce q q 0 0 t t 0 e/R I I - e/R 0 t t Charging Discharging

8. Suppose you have two coils with multiple turns close to each other, as shown in this cross-section Coil 1 Coil 2 B N1 N2 Mutual Inductance • We can define mutual inductance M12 of coil 2 with respect to coil 1 as: It can be shown that :

9. What is the combined (equivalent) inductance of two inductors in series, as shown ? a a Leq L1 L2 b b And: Inductors in Series Note: the induced EMF of two inductors now adds: Since:

10. What is the combined (equivalent) inductance of two inductors in parallel, as shown ? a a L1 L2 Leq b b And finally: Inductors in parallel Note: the induced EMF between points a and be is the same ! Also, it must be: We can define:

11. Consider from point of view of energy! In the RC circuit, any current developed will cause energy to be dissipated in the resistor. In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor! C C R L • Suppose that the circuits are formed at t=0 with the capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why?? LC Circuits • Consider the LC and RC series circuits shown:

12. i i Q +++ Q C L +++ - - - C R - - - LC: current oscillates RC: current decays exponentially 0 i -i 0 t 0 t 1 RC/LC Circuits

13. + + C C L Þ L - - Ý ß - - C C L Ü L + + LC Oscillations(qualitative)

14. Energy transfer in a resistanceless, nonradiating LC circuit. The capacitor has a charge Qmax at t = 0, the instant at which the switch is closed. The mechanical analog of this circuit is a block–spring system.

15. What do we need to do to turn our qualitative knowledge intoquantitative knowledge? • What is the frequency w of the oscillations (when R=0)? (it gets more complicated when R finite…and R is always finite) + + C L - - LC Oscillations(quantitative)

16. where: • w0 determined from equation • f, Q0 determined from initial conditions i + + Q C L - - remember: LC Oscillations(quantitative) • Begin with the looprule: • Guess solution: (just harmonic oscillator!) • Procedure: differentiate above form for Q and substitute into loop equation to find w0.

17. where: • w0 determined from equation • f, Q0 determined from initial conditions i + + Q C L - - which we could have determined from the mass on a spring result: Review: LC Oscillations • Guess solution: (just harmonic oscillator!)

18. The energy in LC circuit conserved ! When the capacitor is fully charged: When the current is at maximum (Io): The maximum energy stored in the capacitor and in the inductor are the same: At any time:

19. At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0. What is the relation betweenw0andw2, the frequency of oscillations when the initial charge =2Q0? 1A (c) w2 = 2 w0 (b) w2 = w0 (a) w2 = 1/2 w0 Lecture 22, ACT 1

20. At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0. • What is the relation betweenI0andI2, the maximum current in the circuit when the initial charge =2Q0? 1B (c) I2 = 4 I0 (b) I2 = 2 I0 (a) I2 = I0 Lecture 22, ACT 1

21. J. C. Maxwell (~1860) summarized all of the work on electric and magnetic fields into four equations, all of which you now know. However, he realized that the equations of electricity & magnetism as then known (and now known by you) have an inconsistency related to the conservation of charge! Gauss’ Law Faraday’s Law Gauss’ Law For Magnetism Ampere’s Law Summary of E&M I don’t expect you to see that these equations are inconsistent with conservation of charge, but you should see a lack of symmetry here!

22. Gauss’ Law: ! Ampere’s Law is the Culprit! • Symmetry: both E and B obey the same kind of equation (the difference is that magnetic charge does not exist!) • Ampere’s Law and Faraday’s Law: • If Ampere’s Law were correct, the right hand side of Faraday’s Law should be equal to zero -- since no magnetic current. • Therefore(?), maybe there is a problem with Ampere’s Law. • In fact, Maxwell proposes a modification of Ampere’s Law by adding another term (the “displacement” current) to the right hand side of the equation! ie

23. Remember: Iin Iout changing electric flux Displacement current FE

26. Can we understand why this “displacement current” has the form it does? circuit Maxwell’s Displacement Current • Consider applying Ampere’s Law to the current shown in the diagram. • If the surface is chosen as 1, 2 or 4, the enclosed current = I • If the surface is chosen as 3, the enclosed current = 0! (ie there is no current between the plates of the capacitor) Big Idea: The Electric field between the plates changes in time. “displacement current” ID = e0 (dfE/dt)= the real current I in the wire.

27. Maxwell’s Equations • These equations describe all of Electricity and Magnetism. • They are consistent with modern ideas such as relativity. • They even describe light