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Lecture 21-1
Maxwell’s Equations (so far)
Gauss’s law

Gauss’ law for magnetism

S
E dA
S
Qinside
0
B dA0
Faraday’s law
dB
 C E d l   dt
Ampere’s law*

C
B d l  0 I
Lecture 21-2
Parallel-Plate Capacitor Revisited

S
B d l  0 I S
-Q
For surface S1, Is = I,
but for surface S2, Is = 0
E
Wait, LHS is the same
(because C is the same)!
B=0 ? Not
experimentally!
Q
E 0
dQ
dV  0 A dV
I
C

dt
dt
d dt
dE
dE
 0 A
 0
dt
dt
??
You could make this work if a fictitious
current Id is added to Is in such a way that
Id is zero for S1 but is equal to I for S2.
dE
Id  0
dt
will work.
Lecture 21-3
Displacement Current
James Clerk Maxwell proposed that a changing electric field
induces a magnetic field, in analogy to Faraday’s law: A
changing magnetic field induces an electric field.
Ampere’s law is revised to become Ampere-Maxwell law
dE
 C B d l  0 ( I  I d )  0 I  0 0 dt
where
dE
Id  0
dt
is the displacement current.
Lecture 21-4
MAXWELL’S EQUATIONS “COMPLETED”

S
E dA
Qinside
0
dB
 C E d l   dt

S
B dA0
dE
 C B d l  0 I  0 0 dt
Basis for electromagnetic waves!
The equations are often written in slightly different
(and more convenient) forms when dielectric and/or
magnetic materials are present.
Lecture 21-5
©2008 by W.H. Freeman and Company
Lecture 21-6
READING QUIZ 1
The extended Amperes law involves a displacement current which was
added to the real currents as a source of the magnetic (B) fields.
Which of the following statements is correct?
A| the displacement current has a constant magnitude ie. not a frequency
dependent magnitude.
B| The displacement current does not involve the constant εO= 1/4πk.
C| The displacement current is proportional to εO.
D| The displacement current was introduced by Faraday.
Lecture 21-7
7B-20 TRANSVERSE ELECTROMAGNETIC WAVE
Lecture 21-8
Electromagnetic Waves From Faraday’s Law
dB
C E dl   dt
E y
Bz

x
t
B  Bm sin(kx  t ) z
E  Em sin(kx  t ) y
Em 

Bm k
c
EB k
f   c
Lecture 21-9
ELECTOMAGNETIC WAVE FARADAY INDUCTION PART
©2008 by W.H. Freeman and Company
Lecture 21-10
Electromagnetic Waves From Ampère’s Law
dE
C B dl  0 0 dt
E y
Bz

 0 0
x
t
E  Em sin(kx  t ) y
B  Bm sin(kx  t ) z
Em k / 

c
Bm 0 0
EB k
f   c
Lecture 21-11
TRANSVERSE ELECTROMAGNETIC WAVE
AMPERE LAW PART
©2008 by W.H. Freeman and Company
Lecture 21-12
Electric Dipole Radiation
Lecture 21-13
©2008 by W.H. Freeman and Company
Lecture 21-14
WARM UP QUIZ 2
Light is an electromagnetic wave where the
wavelength λ in meters times the frequency f in
Hz is equal to the velocity of light c = 3 x 108
meters/second in vacuum. Which of the following
statements is correct?
(a) λ = 100 meters, f = 1x106 Hz
(b) λ = 105 meters, f = 3x104 Hz
(c) λ = 10-6 meters, f = 3x1014 Hz
(d) λ = 1 meter,
f = 3/2 x 108 Hz
Lecture 21-15
Electromagnetic Wave Propagation in Free Space
So, again we have a traveling electromagnetic wave
Em 
 c
Bm k
c
Em
1

Bm 0 0c
0  4  107 (T  m / A)
1
0 0
speed of light
in vacuum
 0  8.85  1012 (C 2 / N  m2 )
B
1 E
 2
x
c t
E
B

x
t
Ampere’s Law
Faraday’s Law
 2 B 1  2 B Wave Equation
 2 2
2
x
c t
c  3.00  108 (m / s)
Speed of light in vacuum is
currently defined rather than
measured (thus defining meter and
also the vacuum permittivity).
Lecture 21-16
Plane Electromagnetic Waves
2B 1 2B
 2 2
2
x
c t
2E 1 2E
 2 2
2
x
c t
where
B  Bm sin(kx  t ) z
E  Em sin(kx  t ) y
c
Em
1

c
Bm 0 0  / k 
EB k
• Transverse wave
• Plane wave (points of
given phase form a plane)
x • Linearly polarized (fixed
plane contains E)
Lecture 21-17
Energy Density of Electromagnetic Waves
• Electromagnetic waves contain energy. We know already
expressions for the energy density stored in E and B fields:
EM wave
2
1B
1
2
uE   0 E , uB 
2 0
2
E2
1
2
uB 


E
 uE
0
2
2 0c 2
Bm  Em / c
B  E/c
• So Total energy density is
EB
0
u  uE  uB   0 E 


EB
0 0c
0
B2
2
 u  0 E
2
 0E
2
rms

B2
0

2
Brms
0
EB
Erms Brms


0 c
0 c
Lecture 21-18
Energy Propagation in Electromagnetic Waves
• Energy flux density
= Energy transmitted through unit time per unit area
• Intensity I = Average energy flux density (W/m2)


P
A
1
0
 u c
EB 
Define Poynting vector
Erms Brms
0
S
1
0
EB
 Direction is that of wave propagation
 average magnitude is the intensity
S  I  c 0 E 2
1
1 Bm2 Em Bm
2
 c 0 Em  c

2
2 0
2 0
Lecture 21-19
Radiation Pressure
Electromagnetic waves carry momentum as well as energy.
In terms of total energy of a wave U, the momentum is U/c.
During a time interval Δt , the energy flux through
area A is ΔU =IA Δ t .
 If radiation is totally absorbed:
p  U / c  IA / c  t
p IA
F 

t
c
momentum imparted
pr  F / A  I / c  B /(2 0 )
2
m
radiation pressure EXERTED
 If radiation is totally reflected: x2
p  2U / c  2 IA / c  t
2 IA
F
, pr  2 I / c
c
Lecture 21-20
Maxwell’s Rainbow
Light is an
Electromagnetic
Wave
f   c
Lecture 21-21
Physics 241 – 10:30 QUIZ 3, November 8, 2011
An electromagnetic wave is traveling in +y direction and the
magnetic field at a particular point on the y-axis points in the
+z direction at a certain instant in time. At this same point
and instant, what is the direction of the electric field?
z
a) -z
B
b) -x
y
c) -y
d) +x
e) None of the above
x
The direction of travel
is that of E x B.
Lecture 21-22
Physics 241 -- 11:30 QUIZ 3, November 8, 2011
An electromagnetic wave is traveling in +x direction and the
electric field at a particular point on the x-axis points in the
+z direction at a certain instant in time. At this same point
and instant, what is the direction of the magnetic field?
a) -z
z
b) -x
E
c) -y
y
d) +y
e) None of the above
x
The direction of travel
is that of E x B.
Lecture 21-23
Physics 241 -- 11:30 QUIZ 3, March 31, 2011
An electromagnetic wave is traveling through a particular
point in space where the direction of the electric field is
along the +z direction and that of the magnetic field is along
the +y direction at a certain instant in time. Which direction
is this wave traveling?
z
a) +x
E
y
b) -x
c) -y
d) -z
e) None of the above
B
x
The direction of travel
is that of E x B.