1. W13D2: Maxwell’s Equations and Electromagnetic Waves
Today’s Reading Course Notes: Sections
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2. Announcements No Math Review next week
PS 10 due Week 14 Tuesday May 7 at 9 pm in boxes outside or
Next Reading Assignment W13D3 Course Notes: Sections 13.9, 13.11, 13.12
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3. Outline Maxwell’s Equations and the Wave Equation
Understanding Traveling Waves
Electromagnetic Waves
Plane Waves
Energy Flow and the Poynting Vector
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4. Maxwell’s Equations in Vacua
No charges or currents
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5. Wave Equations: Summary
Electric & magnetic fields travel like waves satisfying:
with speed
But there are strict relations between them:
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6. Understanding Traveling Wave Solutions to Wave Equation
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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
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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
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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!
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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
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11. Group Problem: Traveling Sine Wave
Let ,
where
Show that
satisfies
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12. Wavelength and Wave Number: Spatial Periodicity
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13. Concept Question: Wave Number
The graph shows a plot of the function
The value of k is
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14. Concept Q. Answer: Wave Number
Wavelength is 4 m so wave number is
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15. Period: Temporal Periodicity
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16. Do Problem 1 In this Java Applet http://web. mit. edu/8
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17. Traveling Sinusoidal Wave: Summary
Two periodicities:
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18. Traveling Sinusoidal Wave
Alternative form:
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19. Plane Electromagnetic Waves
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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)
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21. Electromagnetic SpectrumHz
Wavelength and frequency are related by:
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22. Traveling Plane Sinusoidal Electromagnetic Waves
are special solutions to the 1-dim wave equations
where
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23. Group Problem: 1 Dim’l Sinusoidal EM Waves
Show that in order for the fields
to satisfy either condition below
then
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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?”
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25. Electromagnetic Radiation: Plane Waves
Magnetic field vector uniform on infinite plane.
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26. Direction of Propagation
Special case generalizes
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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
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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)
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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):
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30. Concept Question: Traveling Wave
The B field of a plane EM wave is
The electric field of this wave is given by
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31. Concept Q. Ans.: Traveling Wave
Answer: 4.
From the argument of the , we know the wave propagates in the positive y-direction.
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32. Concept Question EM Wave
The electric field of a plane wave is:
The magnetic field of this wave is given by:
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33. Concept Q. Ans.: EM Wave Answer: 1.
Week 13, Day 2
Concept Q. Ans.: EM Wave
Answer: 1.
From the argument of the , we know the wave propagates in the negative z-direction.
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34. Energy in EM Waves: The Poynting Vector
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35. Energy in EM Waves Energy densities: Consider cylinder:
Week 13, Day 1
Energy in EM Waves
Energy densities:
Consider cylinder:
What is rate of energy flow per unit area?
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36. Poynting Vector and Intensity
Week 13, Day 1
Poynting Vector and Intensity
Direction of energy flow = direction of wave propagation
units: Joules per square meter per sec
Intensity I:
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37. Group Problem: Poynting Vector
An electric field of a plane wave is given by the expression
Find the Poynting vector associated with this plane wave.
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38. Appendix A Standing Waves
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39. Standing Waves
What happens if two waves headed in opposite directions are allowed to interfere?
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40. Standing Waves
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41. Standing Waves Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
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42. Standing Waves Do Problem 2 In the Java Applet http://web. mit. edu/8
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43. Appendix B Radiation Pressure
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44. Momentum & Radiation Pressure
Week 13, Day 1
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:
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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