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W13D2: Maxwell ’ s Equations and Electromagnetic Waves. Today ’ s Reading Course Notes: Sections 13.5-13.7. 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.
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W13D2:Maxwell’s Equations and Electromagnetic Waves
Today’s Reading Course Notes: Sections 13.5-13.7
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
Maxwell’s Equations and the Wave Equation
Understanding Traveling Waves
Electromagnetic Waves
Plane Waves
Energy Flow and the Poynting Vector
0
0
No charges or currents
Electric & magnetic fields travel like waves satisfying:
with speed
But there are strict relations between them:
Understanding Traveling Wave Solutions to Wave Equation
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
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
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!
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
Let ,
where .
Show that
satisfies .
The graph shows a plot of the function
The value of k is
Answer: 4.
Wavelength is 4 m so wave number is
Do Problem 1In this Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm
Two periodicities:
Alternative form:
http://youtu.be/3IvZF_LXzcc
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
Hz
Wavelength and frequency are related by:
Traveling Plane Sinusoidal Electromagnetic Waves
are special solutions to the 1-dim wave equations
where
Show that in order for the fields
Group Problem: 1 Dim’l Sinusoidal EM Waves
to satisfy either condition below
then
1)Plot E, B at each of the ten points pictured for t = 0
2)Why is this a “plane wave?”
Electromagnetic Radiation: Plane Waves
Magnetic field vector uniform on infinite plane.
http://youtu.be/3IvZF_LXzcc
Special case generalizes
The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the
Answer: 4. The wave is moving in the –z direction
The propagation direction is given by the
(Yellow x Blue)
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):
The B field of a plane EM wave is
The electric field of this wave is given by
Answer: 4.
From the argument of the , we know the wave propagates in the positive y-direction.
The electric field of a plane wave is:
The magnetic field of this wave is given by:
Answer: 1.
From the argument of the , we know the wave propagates in the negative z-direction.
Energy in EM Waves:The Poynting Vector
Energy densities:
Consider cylinder:
What is rate of energy flow per unit area?
Direction of energy flow = direction of wave propagation
units: Joules per square meter per sec
Intensity I:
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
Appendix AStanding Waves
What happens if two waves headed in opposite directions are allowed to interfere?
Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
Standing Waves Do Problem 2 In the Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm
Appendix BRadiation 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:
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