1. P30 - 1 Workshop: Using Visualization in Teaching Introductory E&M AAPT National Summer Meeting, Edmonton, Alberta, Canada. Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

2. P30 - 2 MIT Class: Electromagnetic Waves

3. P30 - 3 Maxwell’s Equations 0 0 Solve in free space (no charge/current) to get…

4. P30 - 4 Electromagnetic Radiation

5. P30 - 5 A Question of Time…

6. P30 - 6 Electromagnetic Waves: Plane Waves

7. P30 - 7 Traveling Waves Consider f(x) = x=0 What is g(x,t) = f(x-vt)? x=0 t=0 x=vt 0 t=t 0 x=2vt 0 t=2t 0 f(x-vt) is traveling wave moving to the right!

8. P30 - 8 Traveling Sine Wave: Space Look at t = 0: g(x,0) = y = y 0 sin(kx): x Amplitude (y 0 ) What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y 0 sin(k(x+vt)) = y 0 sin(kx+kvt)

9. P30 - 9 Traveling Sine Wave: Time Amplitude (y 0 ) Look at x=0:

10. P30 - 10 Traveling Sine Wave

11. P30 - 11 Electromagnetic Waves Remember: Hz

12. P30 - 12 Electromagnetic Radiation: Plane Waves Watch 2 Ways: 1) Sine wave traveling to right (+x) 2)Collection of out of phase oscillators (watch one position) Don’t confuse vectors with heights – they are magnitudes of E (gold) and B (blue)

13. P30 - 13 Traveling EM Wave: Space At a fixed time (e.g. t=0): x Amplitude (E 0 )

14. P30 - 14 Traveling EM Wave: Time Amplitude (E 0 ) At x=0, just a function of time:

15. P30 - 15 Traveling E & B Waves

16. P30 - 16 PRS Question: Wave

17. P30 - 17 PRS: Wave The graph shows a plot of the function y = cos(k x). The value of k is (m) :00 1.½ m -1 2.¼ m -1  m -1  /2 m -1 5.I don’t know

18. P30 - 18 PRS Answer: Wave = 4 m  k = 2  / =  /2 m -1 y = cos (  x /2) is 1 at x = –4 m, 0 m, 4 m, etc. Answer: 4. k =  /2 m -1 (m)

19. P30 - 19 Properties of EM Waves Travel (through vacuum) with speed of light At every point in the wave and any instant of time, E and B are in phase with one another, with E and B fields perpendicular to one another, and to the direction of propagation (they are transverse):

20. P30 - 20 Direction of Propagation

21. P30 - 21 PRS Question: Direction of Propagation

22. P30 - 22 PRS: Direction of Propagation The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the 0 1.+x direction 2.–x direction 3.+z direction 4.–z direction 5.I don’t know

23. P30 - 23 PRS Answer: Propagation The propagation direction is given by the direction of E x B (Yellow x Blue) Answer: 4. The wave is moving in the –z direction

24. P30 - 24 PRS Questions: Traveling Wave

25. P30 - 25 PRS: Traveling Wave The B field of a plane EM wave is The electric field of this wave is given by 1. 2. 3. 4. 5.I don’t know :20

26. P30 - 26 PRS Answer: Traveling Wave From the argument of the sin(ky -  t), we know the wave propagates in the +y direction. Answer: 4.

27. P30 - 27 Group Problem: Plane Waves 1)Plot E, B at each of the ten points pictured for t=0 2)Why is this a “plane wave?”

28. P30 - 28 Electromagnetic Radiation Both E & B travel like waves: But there are strict relations between them: Here, E y and B z are “the same,” traveling along x axis

29. P30 - 29 Amplitudes of E & B E y and B z are “the same,” just different amplitudes

30. P30 - 30 Electromagnetic Radiation: Plane Waves

31. P30 - 31 How Do Maxwell’s Equations Lead to EM Waves? Derive Wave Equation Optional

32. P30 - 32 Wave Equation Start with Ampere-Maxwell Eq:

33. P30 - 33 Wave Equation So in the limit that dx is very small: Apply it to red rectangle: Start with Ampere-Maxwell Eq:

34. P30 - 34 Wave Equation Now go to Faraday’s Law

35. P30 - 35 Wave Equation Faraday’s Law: So in the limit that dx is very small: Apply it to red rectangle:

36. P30 - 36 1D Wave Equation for E Take x-derivative of 1st and use the 2nd equation

37. P30 - 37 1D Wave Equation for E This is an equation for a wave. Let:

38. P30 - 38 1D Wave Equation for B Take x-derivative of 1st and use the 2nd equation

39. P30 - 39 Electromagnetic Waves Both E & B travel like waves: But there are strict relations between them: Here, E y and B z are “the same,” traveling along x axis

40. P30 - 40 Energy in EM Waves Energy densities: Consider cylinder: What is rate of energy flow per unit area?

41. P30 - 41 Poynting Vector and Intensity units: Joules per square meter per sec Direction of energy flow = direction of wave propagation Intensity I:

42. P30 - 42 Momentum & Radiation Pressure EM waves transport energy: This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled: They also transport momentum: And exert a pressure:

43. P30 - 43 Standing Waves

44. P30 - 44 Standing Waves What happens if two waves headed in opposite directions are allowed to interfere?

45. P30 - 45 Standing Waves Most commonly seen in resonating systems: Musical Instruments, Microwave Ovens

46. P30 - 46 Group Work: Standing Waves

47. P30 - 47 Generating Plane Electromagnetic Radiation

48. P30 - Shake a Sheet of Charge generating plane wave radiationgenerating plane wave radiation applet

49. P30 - 2) If sheet position is What is B(x,t)? What is E(x,t)? What Direction? Group Problem: B Field Generation Sheet (blue) has uniform charge density  Starting time T ago pulled down at velocity v 1) What is B field? (HINT: Change drawing perspective) sheet

50. P30 - You Made a Plane Wave! generating plane wave

51. P30 - How to Think About E-Field E-Field lines like strings tied to plane This is the field you calculated & that propagates

52. P30 - 52 Group Problem: Energy in Wave You Found: 1) What is total power per unit area radiated away? 2) Where is that energy coming from? 3) Calculate power generated to see efficiency

53. P30 - 53 Generating Electric Dipole Electromagnetic Waves

54. P30 - Quarter-Wavelength Antenna Accelerated charges are the source of EM waves. Most common example: Electric Dipole Radiation. t = 0t = T/4t = T/2t = T

55. P30 - Why are Radio Towers Tall? AM Radio stations have frequencies 535 – 1605 kHz. WLW 700 Cincinnati is at 700 kHz. Tower is 747 ft tall

56. P30 - Quarter-Wavelength Antenna

57. P30 - Quarter-Wavelength Antenna

58. P30 - Spark Gap Transmitter

59. P30 - 59 Spark Gap Generator: An LC Oscillator

60. P30 - 60 Our spark gap antenna 1) Charge gap (RC) 2) Breakdown! (LC) 3) Repeat

61. P30 - Spark Gap Transmitter

62. P30 - PRS Question: Spark Gap Antenna

63. P30 - Spark Gap Antenna

64. P30 - Spark Gap Antenna

65. P30 - Demonstration: Antenna

66. P30 - Polarization

67. P30 - Polarization of TV EM Waves Why oriented as shown? Why different lengths?

68. P30 - Demonstration: Microwave Polarization

69. P30 - Experiment 8: Microwaves

70. P30 - Standing Waves What happens if two waves headed in opposite directions are allowed to interfere?

71. P30 - PRS Questions: Angular Distribution & Polarization of Radiation