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1P30-
Class 30: Outline
Hour 1:Traveling & Standing Waves
Hour 2:Electromagnetic (EM) Waves
2P30-
Last Time:Traveling Waves
3P30-
Traveling Sine WaveNow consider f(x) = y = y0sin(kx):
x
Amplitude (y0)2Wavelength ( )
wavenumber ( )kπλ =
What is g(x,t) = f(x+vt)? Travels to left at velocity vy = y0sin(k(x+vt)) = y0sin(kx+kvt)
4P30-
Traveling Sine Wave( )0 siny y kx kvt= +
0 0sin( ) sin( )y y kvt y tω= ≡At x=0, just a function of time:
Amplitude (y0)
1Period ( )frequency ( )
2angular frequency ( )
Tfπ
ω
=
=
5P30-
Traveling Sine Wave
0 sin( )y y kx tω= −Wavelength: Frequency :
2Wave Number:
Angular Frequency: 21 2Period:
Speed of Propagation:
Direction of Propagation:
f
k
f
Tf
v fk
x
λ
πλω π
πω
ω λ
=
=
= =
= =
+
ii
i
i
i
i
i
6P30-
This Time:Standing Waves
7P30-
Standing WavesWhat happens if two waves headed in opposite directions are allowed to interfere?
1 0 sin( )E E kx tω= − 2 0 sin( )E E kx tω= +
1 2 0Superposition: 2 sin( ) cos( )E E E E kx tω= + =
8P30-
Standing Waves: Who Cares?Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
02 sin( ) cos( )E E kx tω=
9P30-
Standing Waves: Bridge
Tacoma Narrows Bridge Oscillation:http://www.pbs.org/wgbh/nova/bridge/tacoma3.html
10P30-
Group Work: Standing WavesDo Problem 2
1 0 sin( )E E kx tω= − 2 0 sin( )E E kx tω= +
1 2 0Superposition: 2 sin( ) cos( )E E E E kx tω= + =
11P30-
Last Time:Maxwell’s Equations
12P30-
Maxwell’s Equations
0
0 0 0
(Gauss's Law)
(Faraday's Law)
0 (Magnetic Gauss's Law)
(Ampere-Maxwell Law)
( (Lorentz force Law)
in
S
B
C
S
Eenc
C
Qd
dddt
d
dd Idt
q
ε
µ µ ε
⋅ =
Φ⋅ = −
⋅ =
Φ⋅ = +
= + ×
∫∫
∫
∫∫
∫
E A
E s
B A
B s
F E v B)
13P30-
Which Leads To…EM Waves
14P30-
Electromagnetic Radiation: Plane Waves
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html
15P30-
Traveling E & B Waves
0ˆ sin( )E kx tω= −E EWavelength:
Frequency : 2Wave Number:
Angular Frequency: 21 2Period:
Speed of Propagation:
Direction of Propagation:
f
k
f
Tf
v fk
x
λ
πλω π
πω
ω λ
=
=
= =
= =
+
ii
i
i
i
i
i
16P30-
Properties of EM Waves
8
0 0
1 3 10 mv csµ ε
= = = ×
0
0
EE cB B= =
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):
Direction of propagation = Direction of ×E B
17P30-
PRS Questions:Direction of Propagation
18P30-
How Do Maxwell’s Equations Lead to EM Waves?
Derive Wave Equation
19P30-
Wave Equation0 0
C
dd ddt
µ ε⋅ = ⋅∫ ∫B s E AStart with Ampere-Maxwell Eq:
20P30-
Wave Equation
( , ) ( , )z zC
d B x t l B x dx t l⋅ = − +∫ B s
Apply it to red rectangle:
0 0 0 0yEd d l dx
dt tµ ε µ ε
∂⎛ ⎞⋅ = ⎜ ⎟∂⎝ ⎠
∫E A
0 0C
dd ddt
µ ε⋅ = ⋅∫ ∫B s E AStart with Ampere-Maxwell Eq:
0 0( , ) ( , ) yz z
EB x dx t B x tdx t
µ ε∂+ −
− =∂
0 0yz
EBx t
µ ε∂∂
− =∂ ∂So in the limit that dx is very small:
21P30-
Wave Equation
C
dd ddt
⋅ = − ⋅∫ ∫E s B ANow go to Faraday’s Law
22P30-
Wave Equation
C
dd ddt
⋅ = − ⋅∫ ∫E s B A
( , ) ( , )y yC
d E x dx t l E x t l⋅ = + −∫ E s
zBd d ldxdt t
∂− ⋅ = −
∂∫B A
Faraday’s Law:
Apply it to red rectangle:
( , ) ( , )y y zE x dx t E x t B
dx t+ − ∂
= −∂
y zE Bx t
∂ ∂= −
∂ ∂So in the limit that dx is very small:
23P30-
1D Wave Equation for E
0 0 y yz zE EB Bx t x t
µ ε∂ ∂∂ ∂
= − − =∂ ∂ ∂ ∂
Take x-derivative of 1st and use the 2nd equation2 2
0 02 2y y yz z
E E EB Bx x x x t t x t
µ ε∂ ∂ ∂⎛ ⎞ ∂ ∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞= = − = − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2
0 02 2y yE E
x tµ ε
∂ ∂=
∂ ∂
24P30-
1D Wave Equation for E2 2
0 02 2y yE E
x tµ ε
∂ ∂=
∂ ∂( )yE f x vt= −This is an equation for a wave. Let:
( )
( )
2
2
22
2
''
''
y
y
Ef x vt
xE
v f x vtt
∂= −
∂∂
= −∂
2
0 0
1vµ ε
=
25P30-
1D Wave Equation for B
0 0 y yz zE EB B
t x x tµ ε
∂ ∂∂ ∂= − = −
∂ ∂ ∂ ∂
Take x-derivative of 1st and use the 2nd equation2 2
2 20 0
1y yz z zE EB B B
t t t t x x t xµ ε∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂ ∂ ∂⎛ ⎞ = = − = − =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠
2 2
0 02 2z zB B
x tµ ε∂ ∂
=∂ ∂
26P30-
Electromagnetic RadiationBoth E & B travel like waves:2 2 2 2
0 0 0 02 2 2 2y y z z
E E B Bx t x t
µ ε µ ε∂ ∂ ∂ ∂
= =∂ ∂ ∂ ∂
0 0y yz z
E EB Bt x x t
µ ε∂ ∂∂ ∂
= − = −∂ ∂ ∂ ∂
But there are strict relations between them:
Here, Ey and Bz are “the same,” traveling along x axis
27P30-
Amplitudes of E & B
( ) ( )0 0Let ;y zE E f x vt B B f x vt= − = −
yzEB
t x∂∂
= −∂ ∂ ( ) ( )0 0' 'vB f x vt E f x vt⇒ − − = − −
0 0vB E⇒ =
Ey and Bz are “the same,” just different amplitudes
28P30-
Group Problem: EM Standing Waves
Consider EM Wave approaching a perfect conductor:
If the conductor fills the XY plane at Z=0 then the wave will reflect and add to the incident wave1. What must the total E field (Einc+Eref) at Z=0 be?2. What is Ereflected for this to be the case?3. What are the accompanying B fields? (Binc & Bref)4. What are Etotal and Btotal? What is B(Z=0)?5. What current must exist at Z=0 to reflect the
wave? Give magnitude and direction.
incident 0ˆ cos( )xE kz tω= −E
( ) ( ) ( ) ( ) ( )Recall: cos cos cos sin sinA B A B A B+ = −
29P30-
Next Time: How Do We Generate Plane Waves?
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/09-planewaveapp/09-planewaveapp320.html