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