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Lecture 21-1
Resonance
For given peak, R, L, and C, the current amplitude Ipeak will be
at the maximum when the impedance Z is at the minimum.
 peak  I peak R   X L  X C 
2
2
Z
X L  XC
res L 
1
resC
i.e., load purely resistive
This is called resonance.
Resonance angular
frequency:
1
res 
LC
, Z  R, and I peak 
 peak
R
ε and I in phase
Lecture 21-2
Transformer
• AC voltage can be stepped up or
down by using a transformer.
• AC current in the primary coil
creates a time-varying magnetic
flux through the secondary coil
via the iron core. This induces
EMF in the secondary circuit.
Ideal transformer (no losses and magnetic
flux per turn is the same on primary and
secondary).
(With no load)
V1   L  0
d  B V1 V2
 turn 
 
dt
N1 N 2
N1  N 2  V1  V2
N1  N 2  V1  V2
step-up
step-down
With resistive load R in secondary, current I2 flows in secondary by the induced
EMF. This then induces opposing EMF back in the primary. The latter EMF must
somehow be exactly cancelled because  is a defined voltage source. This occurs by
another current I1 which is induced on the primary side due to I2.
Lecture 21-3
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 21-4
Parallel-Plate Capacitor Revisited

-Q
S
B d l  0 I S
Q
E
E 0
dQ
dV  0 A dV
C

dt
dt
d dt
dE
dE
 0 A
 0
dt
dt
I
dE
Id  0
dt
will work.
Lecture 21-5
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 21-6
Maxwell’s Equations

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!
Lecture 21-7
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
EB k
c
Lecture 21-8
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
Lecture 21-9
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 21-10
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
EB k
• Transverse wave
• Plane wave (points of
given phase form a plane)
x • Linearly polarized (fixed
plane contains E)
Lecture 21-11
Non-scored Test Quiz
Electromagnetic wave travel in space where E is electric field, B is magnetic
field. Which of the following diagram is true?
z
z
(a).
(b).
E
y
x
travel
direction
travel
direction
B
E
y
x
B
z
travel
direction
z
(c).
(d).
E
x
B
E
travel
direction
y
B
x
y
Lecture 21-12
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 21-13
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

 u c
A
Erms Brms
1

EB 
0
Define Poynting vector
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 21-14
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 t , the energy flux through
area A is U =IA  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:
p  2U / c  2 IA / c  t
2 IA
F
, pr  2 I / c
c
Lecture 21-15
Maxwell’s Rainbow
Light is an
Electromagnetic
Wave
f   c
Lecture 21-16
Physics 241 –Quiz 18b – March 27, 2008
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 +y direction at a certain instant
in time. Which direction is this wave traveling in?
a) +x
b) x
c) y
d) z
e) None of the above
Lecture 21-17
Physics 241 –Quiz 18c – March 27, 2008
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?
a) z
b) x
c) y
d) +x
e) None of the above