w13d2: maxwell ’ s equations and electromagnetic waves
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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. - PowerPoint PPT PresentationTRANSCRIPT
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W13D2:Maxwell’s Equations and Electromagnetic Waves
Today’s Reading Course Notes: Sections 13.5-13.7
AnnouncementsNo 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
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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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Maxwell’s Equations in Vacua
0
0
No charges or currents
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Wave Equations: Summary
Electric & magnetic fields travel like waves satisfying:
2 Ey
x2
1
c2
2 Ey
t2
2 Bz
x2
1
c2
2 Bz
t2
with speed
But there are strict relations between them:
c 1
0
0
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Understanding Traveling Wave Solutions to Wave Equation
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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
y(x, t) y0e (x vt )2 /a2
y(x vt) y0e (x )2 /a2
y(x vt) y0e (x (2 m))2 /a2
y(x vt) y0e (x (4 m))2 /a2
y(x vt) y0e (x vt )2 /a2
y(x vt)
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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
y(x, t) y0e (xvt )2 /a2
y(x vt) y0e (x )2 /a2
y(x vt) y0e (x(2 m))2 /a2
y(x vt) y0e (x(4 m))2 /a2
y(x vt) y0e (xvt )2 /a2
y(x vt)
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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
2 2/0( ) u ay u y e
2 2
2 2
y y u y y f f u f yf
x u x u x x u x u u
and
u x vt
u
x1 and
u
t v
y
t
y
u
u
t v
y
u vf and
2 y
t2 v
f
t v
f
u
u
tv2 f
u
2 y
u2
2y
x2
1
v2
2y
t 2 y(x,t) satisfies the wave equation!
2 2/0( ) ( , ) x vt ay u y x t y e
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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
y(x vt) y(x vt)
1
v2
2 y(x,t)
t2
2 y(x,t)
x2
y(x vt)
y(x vt)
( )y u
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Group Problem: Traveling Sine Wave
Let ,
where .
Show that
satisfies .
1
v2
2 y(x,t)
t2
2 y(x,t)
x2
y(x,t) y(x vt) y0sin(k(x vt))
y(u) y0sin(ku)
u x vt
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Wavelength and Wave Number: Spatial Periodicity
Fix t 0 : y(x,0) y0sin(kx)
When x k 2 k 2 /
Consider y(x,t) y0sin(k(x vt))
is called the wavelength, k is called the wave number
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Concept Question: Wave Number
The graph shows a plot of the function
The value of k is
y(x,0) cos(kx)
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Concept Q. Answer: Wave Number
Wavelength is 4 m so wave number is
Answer: 4.
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Period: Temporal Periodicity
Fix x 0 : y(0,t) y0sin( kvt) y
0sin(kvt)
When t T kvT 2 2vT / 2
Consider y(x,t) y0sin(k(x vt))
T is called the period
T / v
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Do Problem 1In this Java Applet
http://web.mit.edu/8.02t/www/applets/superposition.htm
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Traveling Sinusoidal Wave: Summary
y(x,t) y0sin(k(x vt))
Spatial period : Wavelength ; Temporal period T .
Two periodicities:
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Traveling Sinusoidal Wave
Wave Number : k 2 /
Angular Frequency : 2 / T
Dispersion Relation : vT kv
Frequency : f 1 / T v f
y(x,t) y0sin(k(x vt)) y
0sin(kx t)
Alternative form:
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Electromagnetic Waves: Plane Sinusoidal Waves
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)
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Electromagnetic Spectrum
Wavelength and frequency are related by:
f c
Hz
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Traveling Plane Sinusoidal Electromagnetic Waves
are special solutions to the 1-dim wave equations
2Ey
x2
1
c2
2Ey
t2
2Bz
x2
1
c2
2Bz
t2
where
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Group Problem: 1 Dim’l Sinusoidal EM Waves
Show that in order for the fields
to satisfy either condition below
then
B0E
0/ c
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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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Electromagnetic Radiation: Plane Waves
http://youtu.be/3IvZF_LXzcc
Magnetic field vector uniform on infinite plane.
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Direction of Propagation
Special case generalizes
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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
1. +x direction
2. –x direction
3. +z direction
4. –z direction
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Concept Question Answer: Propagation
The propagation direction is given by the (Yellow x Blue)
Answer: 4. The wave is moving in the –z direction
Properties of 1 Dim’l EM Waves
c 1
0
0
3.0 108 m
s
E0/ B
0c
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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Concept Question: Traveling Wave
The B field of a plane EM wave isThe electric field of this wave is given by
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Concept Q. Ans.: Traveling Wave
From the argument of the , we know the wave propagates in the positive y-direction.
Answer: 4.
sin(ky t)
Concept Question EM Wave
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The magnetic field of this wave is given by:
The electric field of a plane wave is:
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Concept Q. Ans.: EM Wave
From the argument of the , we know the wave propagates in the negative z-direction.
Answer: 1.
sin(kz t)
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Energy in EM Waves:The Poynting Vector
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Energy in EM Waves
u
E
1
2
0E2 , u
B
1
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B2Energy densities:
Consider cylinder:
dU (uE u
B)Adz
1
2
0E2
B2
0
Acdt
What is rate of energy flow per unit area?
c
2
0cEB
EB
c0
20 0
0
12
EBc
EB
0
1 dUS
A dt
c
2
0E2
B2
0
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Poynting Vector and Intensity
units: Joules per square meter per sec
Direction of energy flow = direction of wave propagation
Intensity I:
I S
E0B
0
20
E
02
20c
cB
02
20
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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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Appendix AStanding Waves
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Standing Waves
What happens if two waves headed in opposite directions are allowed to interfere?
E2E
0sin(kx t)
Superposition : E E1 E
22E
0sin(kx)cos(t)
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Standing Waves
E2E
0sin(kx t)
Superposition :
E E1 E
2
E 2E0sin(kx)cos(t)
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Standing Waves
Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
E 2E0sin(kx)cos(t)
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Standing Waves Do Problem 2 In the Java Applet
http://web.mit.edu/8.02t/www/applets/superposition.htm
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Appendix BRadiation Pressure
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Momentum & Radiation PressureEM waves transport energy:
P
F
A
1
A
dp
dt
1
cA
dU
dt
S
c
This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled, as follows:
Momentum transfer: p
2U
c; Radiation pressure: P
2S
c
They also transport momentum:
p
U
c
And exert a pressure:
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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 kmSpeed of light c = 3 x 108 m/s