W13D2: Maxwell ’ s Equations and Electromagnetic Waves

1. W13D2:Maxwell’s Equations and Electromagnetic Waves Today’s Reading Course Notes: Sections 13.5-13.7

2. No Math Review next week PS 10 due Week 14 Tuesday May 7 at 9 pm in boxes outside 32-082 or 26-152 Next Reading Assignment W13D3 Course Notes: Sections 13.9, 13.11, 13.12 Announcements

3. Outline Maxwell’s Equations and the Wave Equation Understanding Traveling Waves Electromagnetic Waves Plane Waves Energy Flow and the Poynting Vector

4. 0 0 Maxwell’s Equations in Vacua No charges or currents

5. Wave Equations: Summary Electric & magnetic fields travel like waves satisfying: with speed But there are strict relations between them:

6. Understanding Traveling Wave Solutions to Wave Equation

7. Example: Traveling Wave Consider The variables x and t appear together as x - vt At t = 0: At vt = 2 m: At vt = 4 m: is traveling in the positive x-direction

8. Direction of Traveling Waves Consider The variables x and t appear together as x + vt At t = 0: At vt = 2 m: At vt = 4 m: is traveling in the negative x-direction

9. General Sol. to One-Dim’l Wave Eq. Consider any function of a single variable, for example Change variables. Let then and Now take partial derivatives using the chain rule Similarly Therefore y(x,t)satisfies the wave equation!

10. Generalization Take any function of a single variable , where Then or (or a linear combination) is a solution of the one-dimensional wave equation corresponds to a wave traveling in the positive x-direction with speed v and corresponds to a wave traveling in the negative x-direction with speed v

11. Group Problem: Traveling Sine Wave Let , where . Show that satisfies .

12. Wavelength and Wave Number: Spatial Periodicity

13. Concept Question: Wave Number The graph shows a plot of the function The value of k is

14. Concept Q. Answer: Wave Number Answer: 4. Wavelength is 4 m so wave number is

15. Period: Temporal Periodicity

16. Do Problem 1In this Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm

17. Traveling Sinusoidal Wave: Summary Two periodicities:

18. Traveling Sinusoidal Wave Alternative form:

19. Plane Electromagnetic Waves http://youtu.be/3IvZF_LXzcc

20. Electromagnetic Waves: Plane Sinusoidal 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 electric field (gold) and magnetic field (blue) http://youtu.be/3IvZF_LXzcc

21. Hz Electromagnetic Spectrum Wavelength and frequency are related by:

22. Traveling Plane Sinusoidal Electromagnetic Waves are special solutions to the 1-dim wave equations where

23. Show that in order for the fields Group Problem: 1 Dim’l Sinusoidal EM Waves to satisfy either condition below then

24. 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?”

25. Electromagnetic Radiation: Plane Waves Magnetic field vector uniform on infinite plane. http://youtu.be/3IvZF_LXzcc

26. Direction of Propagation Special case generalizes

27. Concept Question: Direction of Propagation The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the • +x direction • –x direction • +z direction • –z direction

28. Concept Question Answer: Propagation Answer: 4. The wave is moving in the –z direction The propagation direction is given by the (Yellow x Blue)

29. Properties of 1 Dim’l EM Waves 1. Travel (through vacuum) with speed of light 2. At every point in the wave and any instant of time, electric and magnetic fields are in phase with one another, amplitudes obey 3. Electric and magnetic fields are perpendicular to one another, and to the direction of propagation (they are transverse):

30. Concept Question: Traveling Wave The B field of a plane EM wave is The electric field of this wave is given by

31. Concept Q. Ans.: Traveling Wave Answer: 4. From the argument of the , we know the wave propagates in the positive y-direction.

32. Concept Question EM Wave The electric field of a plane wave is: The magnetic field of this wave is given by:

33. Concept Q. Ans.: EM Wave Answer: 1. From the argument of the , we know the wave propagates in the negative z-direction.

34. Energy in EM Waves:The Poynting Vector

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

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

37. An electric field of a plane wave is given by the expression Find the Poynting vector associated with this plane wave. Group Problem: Poynting Vector

38. Appendix AStanding Waves

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

40. Standing Waves

41. Standing Waves Most commonly seen in resonating systems: Musical Instruments, Microwave Ovens

42. Standing Waves Do Problem 2 In the Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm

43. Appendix BRadiation Pressure

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

45. Problem: Catchin’ Rays As you lie on a beach in the bright midday sun, approximately what force does the light exert on you? The sun: Total power output ~ 4 x 1026 Watts Distance from Earth 1 AU ~ 150 x 106 km Speed of light c = 3 x 108 m/s