Class 30: Outline
Hour 1:
Traveling & Standing Waves
Hour 2:
Electromagnetic (EM) Waves
P30- 1
Last Time:
Traveling Waves
P30- 2
Traveling Sine Wave
Now consider f(x) = y = y0sin(kx):
Amplitude (y0)
2π
Wavelength (λ ) =
wavenumber (k )
x
What is g(x,t) = f(x+vt)? Travels to left at velocity v
y = y0sin(k(x+vt)) = y0sin(kx+kvt)
P30- 3
Traveling Sine Wave
y = y0 sin ( kx + kvt )
At x=0, just a function of time: y = y0 sin( kvt ) ≡ y0 sin(ω t )
Amplitude (y0)
1
Period (T ) =
frequency (f )
2π
=
angular frequency (ω )
P30- 4
Traveling Sine Wave
i Wavelength: λ
i Frequency : f
i Wave Number: k =
y = y0 sin(kx − ω t )
2π
λ
i Angular Frequency: ω = 2π f
1 2π
i Period: T = =
ω
f
ω
i Speed of Propagation: v =
=λf
k
i Direction of Propagation: + x
P30- 5
This Time:
Standing Waves
P30- 6
Standing Waves
What happens if two waves headed in opposite
directions are allowed to interfere?
E1 = E0 sin(kx − ω t )
E2 = E0 sin(kx + ω t )
Superposition: E = E1 + E2 = 2 E0 sin( kx) cos(ω t )
P30- 7
Standing Waves: Who Cares?
Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
E = 2 E0 sin(kx) cos(ω t )
P30- 8
Standing Waves: Bridge
Tacoma Narrows Bridge Oscillation:
http://www.pbs.org/wgbh/nova/bridge/tacoma3.html
P30- 9
Group Work: Standing Waves
Do Problem 2
E1 = E0 sin(kx − ω t )
E2 = E0 sin(kx + ω t )
Superposition: E = E1 + E2 = 2 E0 sin( kx) cos(ω t )
P30- 10
Last Time:
Maxwell’s Equations
P30- 11
Maxwell’s Equations
Qin
∫∫ E ⋅ dA = ε
S
(Gauss's Law)
0
dΦB
∫C E ⋅ d s = − dt
(Faraday's Law)
∫∫ B ⋅ dA = 0
(Magnetic Gauss's Law)
dΦE
∫C B ⋅ d s = µ0 I enc + µ0ε 0 dt
(Ampere-Maxwell Law)
F = q (E + v × B)
(Lorentz force Law)
S
P30- 12
Which Leads To…
EM Waves
P30- 13
Electromagnetic Radiation:
Plane Waves
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html
P30- 14
Traveling E & B Waves
i Wavelength: λ
i Frequency : f
ˆ E sin( kx − ω t )
E=E
0
i Wave Number: k =
2π
λ
i Angular Frequency: ω = 2π f
1 2π
i Period: T = =
ω
f
ω
i Speed of Propagation: v =
=λf
k
i Direction of Propagation: + x
P30- 15
Properties of EM Waves
Travel (through vacuum) with
speed of light
v=c=
1
m
= 3 × 10
s
µ 0ε 0
8
At every point in the wave and any instant of time,
E and B are in phase with one another, with
E E0
=
=c
B B0
E and B fields perpendicular to one another, and to
the direction of propagation (they are transverse):
Direction of propagation = Direction of E × B
P30- 16
PRS Questions:
Direction of Propagation
P30- 17
How Do Maxwell’s Equations
Lead to EM Waves?
Derive Wave Equation
P30- 18
Wave Equation
d
E ⋅ dA
Start with Ampere-Maxwell Eq: ∫ B ⋅ d s = µ0ε 0
∫
dt
C
P30- 19
Wave Equation
d
E ⋅ dA
Start with Ampere-Maxwell Eq: ∫ B ⋅ d s = µ0ε 0
∫
dt
C
Apply it to red rectangle:
∫ B ⋅ d s = B ( x, t )l − B ( x + dx, t )l
z
z
C
∂E y ⎞
⎛
d
µ0ε 0 ∫ E ⋅ dA = µ0ε 0 ⎜ l dx
⎟
dt
∂t ⎠
⎝
∂E y
Bz ( x + dx, t ) − Bz ( x, t )
−
= µ 0ε 0
∂t
dx
So in the limit that dx is very small:
∂E y
∂Bz
−
= µ 0ε 0
∂x
∂t
P30- 20
Wave Equation
Now go to Faraday’s Law
d
∫C E ⋅ d s = − dt ∫ B ⋅ dA
P30- 21
Wave Equation
d
∫C E ⋅ d s = − dt ∫ B ⋅ dA
Faraday’s Law:
Apply it to red rectangle:
∫ E ⋅ d s = E ( x + dx, t )l − E ( x, t )l
y
y
C
∂Bz
d
− ∫ B ⋅ dA = −ldx
∂t
dt
E y ( x + dx, t ) − E y ( x, t )
dx
∂Bz
=−
∂t
So in the limit that dx is very small:
∂E y
∂Bz
=−
∂x
∂t
P30- 22
1D Wave Equation for E
∂E y
∂Bz
=−
∂x
∂t
∂E y
∂Bz
−
= µ 0ε 0
∂x
∂t
Take x-derivative of 1st and use the 2nd equation
∂ ⎛ ∂E y
⎜
∂x ⎝ ∂x
∂2 Ey
⎞
∂ ⎛ ∂Bz
= ⎜−
⎟=
2
∂x ⎝ ∂t
⎠ ∂x
∂ Ey
∂ ⎛ ∂Bz
⎞
⎟=− ⎜
∂t ⎝ ∂x
⎠
2
∂x
2
∂2 Ey
⎞
⎟ = µ 0ε 0
2
∂t
⎠
∂ Ey
2
= µ 0ε 0
∂t
2
P30- 23
1D Wave Equation for E
∂ Ey
2
∂x
2
∂ Ey
2
= µ 0ε 0
∂t
This is an equation for a wave. Let:
∂ Ey
2
∂x
∂2 Ey
2
∂t
2
= f '' ( x − vt )
= v f '' ( x − vt )
E y = f ( x − vt )
v =
2
2
2
1
µ 0ε 0
P30- 24
1D Wave Equation for B
∂E y
∂Bz
=−
∂t
∂x
∂E y
∂Bz
= − µ 0ε 0
∂x
∂t
Take x-derivative of 1st and use the 2nd equation
∂ ⎛ ∂Bz ⎞ ∂ Bz ∂ ⎛ ∂E y
⎜
⎟ = 2 = ⎜−
∂t ⎝ ∂t ⎠ ∂t
∂t ⎝ ∂x
2
⎞
∂ ⎛ ∂E y
⎟=− ⎜
∂x ⎝ ∂t
⎠
2
⎞
1 ∂ Bz
⎟=
2
∂
x
µ
ε
⎠
0 0
∂ Bz
∂ Bz
µ
ε
=
0 0
2
2
∂x
∂t
2
2
P30- 25
Electromagnetic Radiation
Both E & B travel like waves:
∂ Ey
2
∂x
2
∂ Ey
2
= µ 0ε 0
∂t
2
∂ Bz
∂ Bz
µ
ε
=
0
0
2
2
∂x
∂t
2
2
But there are strict relations between them:
∂E y
∂Bz
=−
∂t
∂x
∂E y
∂Bz
= − µ 0ε 0
∂x
∂t
Here, Ey and Bz are “the same,” traveling along x axis
P30- 26
Amplitudes of E & B
Let E y = E0 f ( x − vt ) ; Bz = B0 f ( x − vt )
∂E y
∂Bz
=−
⇒ −vB0 f ' ( x − vt ) = − E0 f ' ( x − vt )
∂t
∂x
⇒ vB0 = E0
Ey and Bz are “the same,” just different amplitudes
P30- 27
Group Problem:
EM Standing Waves
Consider EM Wave approaching a perfect conductor:
ˆ 0 cos(kz − ωt )
Eincident = xE
If the conductor fills the XY plane at Z=0 then the
wave will reflect and add to the incident wave
1. 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.
Recall: cos ( A + B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B )
P30- 28
Next Time: How Do We
Generate Plane Waves?
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/09-planewaveapp/09planewaveapp320.html
P30- 29