chapter 32: electromagnetic waves · chapter 32: electromagnetic waves maxwell’s equations...

Phys 2435: Chap 32, Pg 1

Chapter 32: Electromagnetic WavesChapter 32: Electromagnetic Waves

MaxwellMaxwell’’s equationss equations Electromagnetic waves in free spaceElectromagnetic waves in free space The electromagnetic spectrumThe electromagnetic spectrum Energy in electromagnetic wavesEnergy in electromagnetic waves

The The Poynting Poynting vectorvector Radiation pressureRadiation pressure Radio and TelevisionRadio and Television

Phys 2435: Chap. 32, Pg 2

MaxwellMaxwell’’s Equationss Equations

New Topic


MaxwellMaxwell’’s Insights Insight

Faraday: A changing B field creates a E field

Maxwell: A changing E field creates a B field

Maxwell looked at this table and, appealing to symmetry, postulated...

E Field produced by B Field produced by

electric charge moving electric charge

changing B field changing E field

It doesn’t just induce a current,it produces an E field!

So let’s summarize what we know so far...


MaxwellMaxwell’’s Equationss Equations

Gauss’s law for electric field:electric charges produceelectric fields.0

!

QAdE ="#!!

0=!" AdB!!

dt

dldE

B!

"=#$!!

dt

dIldB

E!

+="# 000$µµ

!!

J. C. Maxwell (1831 - 1879)J. C. Maxwell (1831 - 1879)

Gauss’s law for magnetic field: but there’re no magnetic charges.

Faraday’s law: changing B produces E.

Ampere’s law as modified by Maxwell:electric current or changing Eproduces B.

All of electromagnetism is contained in this set of four equations.


Perfect Symmetry ?Perfect Symmetry ?

E Field produced by B Field produced by

electric charge magnetic charge

moving magnetic charge moving electric charge

changing B field changing E field

dt

dIldB

dt

dldE

AdB

QAdE

E

B

e

!+="

!#="

="

="

$

$

$

$

000

0

0

%µµ

%

!!

!!

!!

!!

What would Maxwell’s Equations look like if there wereisolated magnetic charges?

dt

d

dt

dQldB

dt

d

dt

dQldE

QAdB

QAdE

Ee

Bm

m

e

!+="

!#="

="

="

$

$

$

$

000

0

0

0

%µµ

µ

µ

%

!!

!!

!!

!!


MaxwellMaxwell’’s Predictionss Predictions


dt

dIldB

dt

dldE

AdB

QAdE

E

B

e

!+="

!#="

="

="

$

$

$

$

000

0

0

%µµ

%

!!

!!

!!

!!

The rest was history.telegraph, radio, television, cell-phone, other wirelesscommunications,…

There exist electromagnetic waves (EMwaves) that can travel in empty space

EM waves travel at the speed of light Light is an EM wave


ConcepTestConcepTest 32.1 32.1 (Post) MaxwellMaxwell’’s Equationss Equations

The Maxwell modificationThe Maxwell modificationof Ampereof Ampere’’s law describings law describingthe creation of a magneticthe creation of a magneticfield is the analog offield is the analog of

1) Gauss’s law on electric fields and charges

2) Gauss’s law on magnetic fields and poles

3) the Lorentz equation

4) Faraday’s Law


Electromagnetic Waves inElectromagnetic Waves inFree SpaceFree Space

New Topic


LetLet’’s put them both together: s put them both together: we obtain changing electricwe obtain changing electricand magnetic fields that continuously produce each other!and magnetic fields that continuously produce each other!

Electromagnetic wavesElectromagnetic waves

oscillating E field oscillating B field oscillating charge

The production and propagation ofThe production and propagation ofelectromagnetic waveselectromagnetic waves

Phys 2435: Chap. 32, Pg 10

Brief Review of Wave PropertiesBrief Review of Wave Properties

h

x

λA

A = amplitudeλ = wavelengthf = frequencyv = speedk = wave number

The one-dimensional wave equation:

has a general solution of the form:

where h1 represents a wave traveling in the +x direction and h2 represents a wavetraveling in the -x direction. The wave velocity is given by v.

A specific solution for harmonic waves traveling in the +x direction is:

h x t h x vt h x vt( , ) ( ) ( )= − + +1 2

( ) ( )tkxAtxh ω−= sin,

k = 2πλ

ω π π= =2 2fT

v fk

= =λ ω

Phys 2435: Chap. 32, Pg 11

WavesWaves are traveling disturbances that transport energy, not matter.

are produced by oscillations. have a speed of the wave depends on the properties of the

medium and not on the wavelength or frequency come in two basic flavors:

Transverse wave

Longitudinal wave

Phys 2435: Chap. 32, Pg 12

Properties of EM Waves:Properties of EM Waves: The E and B fields at any point are

perpendicular to each other, and to thedirection of wave propagation

The E and B fields are in phase. Far away from the source, the EM waves

are approximately plane waves

How does an electromagnetic wave propagate?How does an electromagnetic wave propagate?


MaxwellMaxwell’’s Equations in Differential Forms Equations in Differential Form

Integral forms

! E ! d! A " =

Q

#0

! B ! d! A " = 0

! E ! d!

l " = $d%B

dt! B ! d!

l " = µ0I + µ

0#0

d%E

dt

! "! E =

#

$0

! "! B = 0

! %! E = &

'! B

't

! %! B = µ

0

! J + µ

0$0

'! E

't

Differential forms

ρ is charge density and J is current density

Q is charge and I is current


Electromagnetic Waves in Free Space (Q=0, I=0)Electromagnetic Waves in Free Space (Q=0, I=0)

!B

!x= "µ

0#0

!E

!t

Differential forms

!E

!x= "

!B

!t

!2B

!t!x= "µ

0#0

!2E

!t2

!2E

!x2

= "!2B

!x!t

Take partial derivatives

so !

2E

!x2

= µ0"

0

!2E

!t2

Read off wave speed from the wave equation:

v =1

µ0!

0

= 3.00 "108 m/s = c

! "! E = 0

! "! B = 0

! #! E = $

%! B

%t

! #! B = µ

0&0

%! E

%t


Electromagnetic Waves in Free SpaceElectromagnetic Waves in Free Space

Plane-wave solution

!2E

!x2

=1

c2

!2E

!t2

E = Ey = E0 sin(kx !"t)

B = Bz = B0 sin(kx !"t)

k =2!

",# = 2!f ,c =

#

k= "f

where

!2B

!x2

=1

c2

!2B

!t2

Wave equations

•E and B are in phase.

•The directions of E and B andwave travel form a right-hand-rule.


Relation between magnitudes of E and BRelation between magnitudes of E and B

Apply to plane-wave solution

!E

!x= "

!B

!t

E = Ey = E0 sin(kx !"t)

B = Bz = B0 sin(kx !"t)

Faraday’s law in differential form

E0k cos(kx !"t) = B

0" cos(kx !"t)

E

B= c

or E0k = B

0!

But ω/k=c and E and B in phase, so


MaxwellMaxwell’’s Predictionss Predictions


Physics that changed the world:telegraph, radio, television, cell-phone, satellite, electric power, ….

There exist electromagnetic waves (EM waves)that can travel in vacuum

EM waves travel at the speed of light

E and B are in phase (E / B = c)

Light is an EM wave

c =1

µ0!

0

= 3.00 "108 m/s

! E !

! B = direction of wave travel (right - hand - rule)


ConcepTestConcepTest 32.2 32.2 (Post) EM WaveEM Wave

An electromagnetic wave with itsAn electromagnetic wave with itselectric field in the positive yelectric field in the positive ydirection propagates in thedirection propagates in thenegative z direction. What is thenegative z direction. What is thedirection of the magnetic field?direction of the magnetic field?

1) +x2) -y3) -x4) +z5) -z

x

y

z


Electromagnetic SpectrumElectromagnetic Spectrum

New Topic


A bit of historyA bit of history Long before J.C. Maxwell (1831-1879) predicted EM waves that

can travel in vacuum, light was known to behave like a wave,but what kind of wave was it? What’s oscillating in a light wave? It’s E and B fields according to Maxwell.

Eight years after Maxwell’s death, H. Hertz (1857-1894) firstgenerated and detected EM waves in his laboratory. Spark-gap for generation, wire loop for detection 109 Hz EM waves, but not visible to the eye. Strong confirmation of Maxwell’s theory.

The wavelengths of visible light were measured long beforeanyone imagined that light was an EM wave. 400 nm to 750 nm, or by c = λ f 4.0x1014 Hz to 7.5x1014 Hz But visible light is not the only EM waves

EM waves exist at all frequencies: the electromagnetic spectrum


The electromagnetic spectrumThe electromagnetic spectrum

fc !=


ConcepTestConcepTest 32.3 32.3(Post) EM wavesEM waves Since Superman is from the planetSince Superman is from the planet

Krypton his eyes are sensitive to theKrypton his eyes are sensitive to theentire electromagnetic spectrum. Doesentire electromagnetic spectrum. Doesthat mean he can use x-ray vision tothat mean he can use x-ray vision tosee that Lois Lane is being kidnappedsee that Lois Lane is being kidnappedin the other room?in the other room?

(1) Yes, no problem(1) Yes, no problem

(2) Nope, he can(2) Nope, he can’’tt

(3) Need more information(3) Need more information


Energy in EM WavesEnergy in EM Waves

New Topic


Energy in EM WavesEnergy in EM Waves

EBB

EB

Eu

0

0

0

2

2

0

0

2

2

0

22

1

µ

!

µ!

µ! ===+=

EM waves carry energy from one region of space toanother. The energy density

Introduce the Poynting vector: the energy transportedby the EM wave per unit time per unit area (W/m2).

BES!!!

!=0

1

µ

which also defines the direction of the wave.

is shared equally between E and B fields.

00

00

0

2

02

00

222

1

µµµ!

rmsrmsBEBEcB

cES ====

Recall intensity:2

4

Power

rS

!=


Radiation PressureRadiation Pressure

For example: Radiation from the sun that reaches theEarth’s surface transports energy at a rate of about1000 W/m2.. So the pressure

When an EM wave encounters the surface of amaterial, it exerts pressure on the surface.

Absorbedc

SP =

Reflectedc

SP

2=

26

8N/m 103.3

103

1000 !"=

"=#

c

SP

which is a force on your hand of

N106.60.02 103.386 !!

"=""== PAF


How do Radio and Television Work?How do Radio and Television Work?transmitter receiver

AM (amplitude modulation)530 kHz to 1600 kHz

FM (frequency modulation)88 MHz to 108 Mhz


AntennasAntennas


ConcepTestConcepTest 32.4 32.4 (Post) EM WaveEM Wave

Which gives the largestWhich gives the largestaverage intensity at theaverage intensity at thedistance specified anddistance specified andthus, at least qualitatively,thus, at least qualitatively,the best illumination?the best illumination?

1) a 50-W source at a distance R

2) a 100-W source at a distance 2R

3) a 200-W source at a distance 4R

chapter 32: electromagnetic waves · chapter 32: electromagnetic waves maxwell’s equations...

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