chapter 32: electromagnetic waves · chapter 32: electromagnetic waves maxwell’s equations...
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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
Phys 2435: Chap 32, Pg 3
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...
Phys 2435: Chap. 32, Pg 4
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.
Phys 2435: Chap. 32, Pg 5
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
%
!!
!!
!!
!!
Phys 2435: Chap 32, Pg 6
MaxwellMaxwell’’s Predictionss Predictions
J. C. Maxwell (1831 - 1879)J. C. Maxwell (1831 - 1879)
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
Phys 2435: Chap 32, Pg 7
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
Phys 2435: Chap. 32, Pg 8
Electromagnetic Waves inElectromagnetic Waves inFree SpaceFree Space
New Topic
Phys 2435: Chap 32, Pg 9
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?
Phys 2435: Chap 32, Pg 13
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
Phys 2435: Chap 32, Pg 14
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
Phys 2435: Chap 32, Pg 15
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.
Phys 2435: Chap 32, Pg 16
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
Phys 2435: Chap 32, Pg 17
MaxwellMaxwell’’s Predictionss Predictions
J. C. Maxwell (1831 - 1879)J. C. Maxwell (1831 - 1879)
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)
Phys 2435: Chap 32, Pg 18
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
Phys 2435: Chap 32, Pg 19
Electromagnetic SpectrumElectromagnetic Spectrum
New Topic
Phys 2435: Chap 32, Pg 20
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
Phys 2435: Chap 32, Pg 21
The electromagnetic spectrumThe electromagnetic spectrum
fc !=
Phys 2435: Chap 32, Pg 22
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
Phys 2435: Chap 32, Pg 23
Energy in EM WavesEnergy in EM Waves
New Topic
Phys 2435: Chap 32, Pg 24
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
!=
Phys 2435: Chap 32, Pg 25
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
Phys 2435: Chap 32, Pg 26
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
Phys 2435: Chap 32, Pg 27
AntennasAntennas
Phys 2435: Chap 32, Pg 28
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
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