1. The Ups and Downs of Circuits

2. The End is Near! • Quiz – Nov 18th – Material since last quiz. (Induction) • Exam #3 – Nov 23rd – WEDNESDAY • LAST CLASS – December 2nd • FINAL EXAM – 12/5 10:00-12:50 Room MAP 359 • Grades by end of week. Hopefully Maybe.

3. r A circular region in the xy plane is penetrated by a uniform magnetic field in the positive direction of the z axis. The field's magnitude B (in teslas) increases with time t (in seconds) according to B = at, where a is a constant. The magnitude E of the electric field set up by that increase in the magnetic field is given in the Figure as a function of the distance r from the center of the region. Find a.[0.030] T/s VG

4. i For the next problem, recall that

5. R L

6. For the circuit of Figure 30-19, assume that = 11.0 V, R = 6.00W , and L = 5.50 H. The battery is connected at time t = 0. 6W 5.5H (a) How much energy is delivered by the battery during the first 2.00 s?[23.9] J (b) How much of this energy is stored in the magnetic field of the inductor? [7.27] J(c) How much of this energy is dissipated in the resistor?[16.7] J

7. Let’s put an inductor and a capacitor in the SAME circuit.

8. At t=0, the charged capacitor is connected to the inductor. What would you expect to happen??

9. Current would begin to flow…. Energy Density in Capacitor High Low Low High Energy Flows from Capacitor to the Inductor’s Magnetic Field

10. Energy Flow Energy

12. High Low Low High LC Circuit

13. When t=0, i=0 so B=0 When t=0, voltage across the inductor = Q0/C

15. The Math Solution:

16. Energy

17. Inductor

18. The Capacitor

19. Add ‘em Up …

20. Add Resistance

21. Actual RLC:

22. New Feature of Circuits with L and C • These circuits can produce oscillations in the currents and voltages • Without a resistance, the oscillations would continue in an un-driven circuit. • With resistance, the current will eventually die out. • The frequency of the oscillator is shifted slightly from its “natural frequency” • The total energy sloshing around the circuit decreases exponentially • There is ALWAYS resistance in a real circuit!

23. Types of Current • Direct Current • Create New forms of life • Alternating Current • Let there be light

24. Alternating emf DC Sinusoidal

25. Sinusoidal Stuff “Angle” Phase Angle

26. Same Frequency with PHASE SHIFT f

27. Different Frequencies

28. Note – Power is delivered to our homes as an oscillating source (AC) This makes AC Important!

29. Producing AC Generator x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

30. The Real World

31. A

33. The Flux:

34. OUTPUT WHAT IS AVERAGE VALUE OF THE EMF ??

35. Average value of anything: h T Area under the curve = area under in the average box

36. Average Value For AC:

37. So … • Average value of current will be zero. • Power is proportional to i2R and is ONLY dissipated in the resistor, • The average value of i2 is NOT zero because it is always POSITIVE

38. Average Value

39. RMS

40. Usually Written as:

41. R E ~ Example: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit:

42. Power

43. More Power - Details

44. Resistive Circuit • We apply an AC voltage to the circuit. • Ohm’s Law Applies

45. Consider this circuit CURRENT ANDVOLTAGE IN PHASE

46. The next group of slides is stolen from Dr. Braunstein.

47. V(t) Vp wt p 2p fv -Vp An “AC” circuit is one in which the driving voltage and hence the current are sinusoidal in time. Alternating Current Circuits V = VP sin (wt - fv ) I = IP sin (wt - fI ) wis theangular frequency (angular speed) [radians per second]. Sometimes instead ofwwe use thefrequency f [cycles per second] Frequency f [cycles per second, or Hertz (Hz)]w= 2p f

48. V(t) Vp wt p 2p fv -Vp Phase Term V = VP sin (wt - fv )

49. V(t) Vp wt p 2p fv -Vp V = VP sin (wt - fv ) I = IP sin (wt - fI ) Alternating Current Circuits I(t) Ip Irms Vrms t fI/w -Ip Vp and Ip are the peak current and voltage. We also use the “root-mean-square” values: Vrms = Vp / and Irms=Ip / fv andfI are called phase differences (these determine when V and I are zero). Usually we’re free to set fv=0 (but not fI).

50. Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.