1P30-

Workshop: Using Visualization in Teaching Introductory E&M

AAPT National Summer Meeting, Edmonton, Alberta, Canada.

Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

P30- 2

MIT Class: Electromagnetic Waves

P30- 3

Maxwell’s Equations

0

0 0 0

(Gauss's Law)

0 (Magnetic Gauss's Law)

(Faraday's Law)

(Ampere-Maxwell Law)

in

S

S

B

C

Eenc

C

Qd

d

dd

dt

dd I

dt

E A

B A

E s

B s

0

0

Solve in free space (no charge/current) to get…

P30- 4

Electromagnetic Radiation

P30- 5

A Question of Time…

P30- 6

Electromagnetic Waves: Plane Waves

P30- 7

Traveling Waves

Consider f(x) =

x=0

What is g(x,t) = f(x-vt)?

x=0

t=0

x=vt0

t=t0

x=2vt0

t=2t0

f(x-vt) is traveling wave moving to the right!

P30- 8

Traveling Sine Wave: Space

Look at t = 0: g(x,0) = y = y0sin(kx):

x

Amplitude (y0)2

Wavelength ( )wavenumber ( )k

What is g(x,t) = f(x+vt)? Travels to left at velocity v

y = y0sin(k(x+vt)) = y0sin(kx+kvt)

P30- 9

Traveling Sine Wave: Time

Amplitude (y0)

1Period ( )

frequency ( )

2

angular frequency ( )

Tf

0( , ) sing x t y k x vt

0 0(0, ) sin( ) sin( )g t y kvt y t Look at x=0:

P30-10

Traveling Sine Wave

0 sin( )y y kx t Wavelength: Frequency :

2Wave Number:

Angular Frequency: 21 2

Period:

Speed of Propagation:

Direction of Propagation:

f

k

f

Tf

v fkx

P30-11

Electromagnetic Waves

Remember: f c

Hz

P30-12

Electromagnetic Radiation: Plane Waves

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 E (gold) and B (blue)

P30-13

Traveling EM Wave: Space

At a fixed time (e.g. t=0):

x

Amplitude (E0)2

Wavelength ( )wavenumber ( )k

0 0sin sinE E k x ct E kx t

0 sin( )E E kx

P30-14

Traveling EM Wave: Time

Amplitude (E0)

1Period ( )

frequency ( )

2

angular frequency ( )

Tf

0 0sin( ) sin( )E E kvt E t At x=0, just a function of time:

0 0sin sinE E k x ct E kx t

P30-15

Traveling E & B Waves

0ˆ sin( )E kx t E E

Wavelength: Frequency :

2Wave Number:

Angular Frequency: 21 2

Period:

Speed of Propagation:

Direction of Propagation:

f

k

f

Tf

v fkx

P30-16

PRS Question:Wave

P30-17

PRS: Wave

The graph shows a plot of the function y = cos(k x). The value of k is

(m)

0%

0%

0%

0%

0%

:00

1. ½ m-1 2. ¼ m-1

3. m-1

4. /2 m-1

5. I don’t know

P30-18

PRS Answer: Wave

= 4 m k = 2/ = /2 m-1

y = cos (x /2) is 1 at x = –4 m, 0 m, 4 m, etc.

Answer: 4. k = /2 m-1

(m)

P30-19

Properties of EM Waves

8

0 0

13 10

mv c

s

0

0

EEc

B 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

P30-20

Direction of Propagation 0 0

ˆ ˆˆ ˆsin( ); sin( )E k t B k t E E p r B B p r

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆˆ

ˆ ˆˆ

ˆ ˆ ˆ

ˆ ˆˆ

ˆ ˆˆ

z

x

y

z

x

y

E B p p r

i j k

j k i

k i j

j i k

k j i

i k j

ˆ ˆ ˆ E B p

P30-21

PRS Question:Direction of Propagation

P30-22

PRS: Direction of Propagation

The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the

0

1. +x direction

2. –x direction

3. +z direction

4. –z direction

5. I don’t know

P30-23

PRS Answer: Propagation

The propagation direction is given by the direction of E x B (Yellow x Blue)

Answer: 4. The wave is moving in the –z direction

P30-24

PRS Questions:Traveling Wave

P30-

0%

0%

0%

0%

0%

25

PRS: Traveling Wave

The B field of a plane EM wave isThe electric field of this wave is given by

1. 2. 3. 4. 5. I don’t know

0ˆ( , ) sin( )z t B ky t B k

0ˆ( , ) sin( )z t E ky t E j

0ˆ( , ) sin( )z t E ky t E j

0ˆ( , ) sin( )z t E ky t E i

0ˆ( , ) sin( )z t E ky t E i

:20

P30-26

PRS Answer: Traveling Wave

From the argument of the sin(ky - t), we know the wave propagates in the +y direction.

Answer: 4. 0ˆ( , ) sin( )z t E ky t E i

ˆˆ ˆ ˆSo we have ?

ˆˆ

E B k j

E i

P30-27

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?”

,0

2 ˆ( , , , ) sin ( )yx y z t E x ct

E j

,0

1 2 ˆ( , , , ) sin ( )yx y z t E x ctc

B k

P30-28

Electromagnetic Radiation

0 0y yz zE EB B

t x x t

Both E & B travel like waves:2 2 2 2

0 0 0 02 2 2 2

y y z zE E B B

x t x t

But there are strict relations between them:

Here, Ey and Bz are “the same,” traveling along x axis

P30-29

Amplitudes of E & B

yzEB

t x

0 0Let ;y zE E f x vt B B f x vt

0 0' 'vB f x vt E f x vt

0 0vB E

Ey and Bz are “the same,” just different amplitudes

P30-30

Electromagnetic Radiation: Plane Waves

P30-31

How Do Maxwell’s Equations Lead to EM Waves?

Derive Wave EquationOptional

P30-32

Wave Equation

0 0

C

dd d

dt B s E A

Start with Ampere-Maxwell Eq:

P30-33

Wave Equation

( , ) ( , )z z

C

d B x t l B x dx t l B s

0 0

( , ) ( , ) yz zEB x dx t B x t

dx t

0 0

yzEB

x t

So in the limit that dx is very small:

0 0 0 0yEd

d l dxdt t

E A

Apply it to red rectangle:

0 0

C

dd d

dt B s E A

Start with Ampere-Maxwell Eq:

P30-34

Wave Equation

C

dd d

dt E s B A

Now go to Faraday’s Law

P30-35

Wave Equation

C

dd d

dt E s B A

( , ) ( , )y y

C

d E x dx t l E x t l E s

( , ) ( , )y y zE x dx t E x t B

dx t

y zE B

x t

zBdd ldx

dt t

B A

Faraday’s Law:

So in the limit that dx is very small:

Apply it to red rectangle:

P30-36

1D Wave Equation for E

0 0 y yz zE EB B

x t x t

Take x-derivative of 1st and use the 2nd equation

2 2

0 02 2

y y yz zE E EB B

x x x x t t x t

2 2

0 02 2

y yE E

x t

P30-37

1D Wave Equation for E

2 2

0 02 2

y yE E

x t

2

0 0

1v

This is an equation for a wave. Let: yE f x vt

2

2

22

2

''

''

y

y

Ef x vt

x

Ev f x vt

t

P30-38

1D Wave Equation for B

0 0 y yz zE EB B

t x x t

2 2

2 20 0

1y yz z zE EB B B

t t t t x x t x

2 2

0 02 2z zB B

x t

Take x-derivative of 1st and use the 2nd equation

P30-39

Electromagnetic Waves

0 0y yz zE EB B

t x x t

Both E & B travel like waves:2 2 2 2

0 0 0 02 2 2 2

y y z zE E B B

x t x t

But there are strict relations between them:

Here, Ey and Bz are “the same,” traveling along x axis

P30-40

Energy in EM Waves2 2

00

1 1,

2 2E Bu E u B

Energy densities:

Consider cylinder:2

20

0

1( )

2E B

BdU u u Adz E Acdt

What is rate of energy flow per unit area?

002

c EBcEB

c

20 0

0

12

EBc

0

EB

1 dUS

A dt

22

002

c BE

P30-41

Poynting Vector and Intensity

0

: Poynting vector

E B

S

units: Joules per square meter per sec

Direction of energy flow = direction of wave propagation

Intensity I: 2 20 0 0 0

0 0 02 2 2

E B E cBI S

c

P30-42

Momentum & Radiation PressureEM waves transport energy:

1 1F dp dU SP

A A dt cA dt c

This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled:

2 2Momentum transfer: ; Radiation pressure:

U Sp P

c c

0

E B

S

They also transport momentum:U

pc

And exert a pressure:

P30-43

Standing Waves

P30-44

Standing WavesWhat happens if two waves headed in opposite directions are allowed to interfere?

1 0 sin( )E E kx t 2 0 sin( )E E kx t

1 2 0Superposition: 2 sin( ) cos( )E E E E kx t

P30-45

Standing WavesMost commonly seen in resonating systems:

Musical Instruments, Microwave Ovens

02 sin( )cos( )E E kx t

P30-46

Group Work: Standing Waves

1 0 sin( )E E kx t 2 0 sin( )E E kx t

1 2 0Superposition: 2 sin( ) cos( )E E E E kx t

47P30-

Generating Plane Electromagnetic Radiation

P30-

Shake a Sheet of Charge

generating plane wave radiation applet

P30-

2) If sheet position is

What is B(x,t)?

What is E(x,t)? What Direction?

Group Problem: B Field Generation•Sheet (blue) has uniform charge density •Starting time T ago pulled down at velocity v

•1) What is B field?•(HINT: Change drawing perspective)

0( ) siny t y tsheet

P30-

You Made a Plane Wave!

generating plane wave

P30-

How to Think About E-Field

• E-Field lines like strings tied to plane

This is the fieldyou calculated &that propagates

52P30-

Group Problem: Energy in Wave

1 0 2B v You Found:

1) What is total power per unit area radiated away?

2) Where is that energy coming from?

3) Calculate power generated to see efficiency

53P30-

Generating Electric Dipole Electromagnetic Waves

P30-

Quarter-Wavelength AntennaAccelerated charges are the source of EM waves. Most common example: Electric Dipole Radiation.

t = 0 t = T/4 t = T/2 t = T

4

4

P30-

Why are Radio Towers Tall?AM Radio stations have frequencies 535 – 1605 kHz. WLW 700 Cincinnati is at 700 kHz.

8

3

3 10 m/s429m

700 10 Hz

/ 4 107m 350ft

c

f

Tower is 747 ft tall

P30-

Quarter-Wavelength Antenna

P30-

Quarter-Wavelength Antenna

P30-

Spark Gap Transmitter

59P30-

Spark Gap Generator:An LC Oscillator

60P30-

Our spark gap antenna

6 12 4(4.5 10 )(33 10 F) 1.5 10 sτ RC

rad

1

4

cf

T l

103 10 cm/s

12.4cm

1) Charge gap (RC)

2) Breakdown! (LC)

92.4 10 Hz 2.4GHz 3) Repeat

P30-

Spark Gap Transmitter

P30-

PRS Question:Spark Gap Antenna

P30-

Spark Gap Antenna

P30-

Spark Gap Antenna

P30-

Demonstration:Antenna

P30-

Polarization

P30-

Polarization of TV EM Waves

Why oriented as shown?

Why different lengths?

P30-

Demonstration:Microwave Polarization

P30-

Experiment 8:Microwaves

P30-

Standing WavesWhat happens if two waves headed in opposite directions are allowed to interfere?

1 0 sin( )E E kx t 2 0 sin( )E E kx t

1 2 0Superposition: 2 sin( ) cos( )E E E E kx t

P30-

PRS Questions:Angular Distribution &

Polarization of Radiation