Parameters For A Sinusoidal Wave
Physics 202, Lecture 21
!  More Wave Review
!  Standing waves
!  Electromagnetic Waves (EM Waves)
!  The Hertz Experiment
!  Review of the Laws of Electromagnetism
!  Maxwell’s equations
!  Propagation of E and B
!  The Linear Wave Equation
1
Standing Waves
Snapshot:Fixed t
Snapshot:Fixed x
2
Standing Waves (cont)
" When two waves of the same amplitude, same
frequency but opposite direction meet standing
waves occur.
y1 = Asin(kx " #t)
Snapshot with fixed t:
wavevector = k
wavelength !=2"/k
"  Snapshot with fixed x:
angular frequency = !
frequency f =#/2"$
Period T=1/f
Amplitude =A$
"  Wave Speed v=#/k
! v=!f, or
! v=!/T
"  Phase angle difference
between two positions
"% =-k&x
"
Today’s Topics
" Nodes and antinodes will occur at the same
positions, giving impression that wave is standing
y 2 = Asin(kx + "t)
y = y1 + y 2 = 2Asin(kx)cos("t)
" Points of destructive interference (nodes)
!
!
!
kx = n" ; x =
n" n#
=
; n = 1,2,3...
k
2
" Points of constructive interference (antinodes)
!
#
$
kx = (2n "1) ; x = (2n "1) ; n = 1,2,3...
2
4
3
4
Standing Waves (cont)
Demo: Hertz Experiment
" Standing waves with a string of given length L are
produced by waves of natural frequencies or
resonant frequencies:
"=
!
f =
2L
; n = 1,2,3...
n
v nv
T n
=
=
" 2L
µ 2L
5
!
6
Review: Gauss’s Law / Coulomb’s Law
Gauss’s Law for Magnetism
" The relation between the electric flux through a
closed surface and the net charge q enclosed within
that surface is given by the Gauss’s Law
" The Gauss’s Law for the electric flux is a reflection
of the existence of electric charge. In nature we have
not found the equivalent, a magnetic charge, or
monopole
" We can express this result differently: if any closed
surface as many lines enter the enclosed volume as
they leave it
#
!
In 1887, Heinrich Hertz first demonstrated that EM fields
can transmit over space.
E • dA =
q
"0
" B• dA = 0
7
!
8
Review: Faraday’s Law
Review: Ampere’s Law
" The emf induced in a “circuit” is proportional to the
time rate of change of magnetic flux through the
“circuit” or closed path.
" A magnetic field is produced by an electric current is
given by the Ampere’s Law
nominal
" = # d$B
dt
" Since
" Then
!
"=
direction of
# E • d!
$ E • d! = "
($
'$
d#B
dt
" B• d! = µ I
0
B
!
A
But Ampere’s Law can be
ambiguous: currents enclosed
by surfaces 1 and 2 are
different!
But note time-dependent
electric flux though surface 2.
"B =
# B• dA
!
9
10
!
!
Maxwell’s modification of Ampere’s Law
Maxwell Equations
q
# E • dA = "
" A time-dependent electric flux
induces a magnetic field
(in analogy to Faraday’s law)
!" "
d" E &
#
B
•
d#
=
µ
I
+
!
(
0%
0
$)
$
dt '
Ampere’s
law
!
“Maxwell’s
displacement
current”
!Gauss’s Law of Magnetism,
no magnetic charge
" B• dA = 0
!
"E = # E • dA
!Gauss’s Law/ Coulomb’s Law
0
!
$ E • d! = "
d#B
dt
!Faraday’s Law
$ B• d! = µ I + " µ
0
0
0
d#E
dt
!Ampere Maxwell Law
!
Also, Lorentz force Law !
!
F = qE + qv " B
These are the foundations of the electromagnetism
11
12
!
EM Fields in Space
Linear Wave Equation
"  Maxwell equations when there is no charge and current:
! E • dA = 0
$ E • d! = "
! B • dA = 0
differential forms:
!
(single polarization)
y
E
d#B
dt
$ B• d! = " µ
0
!E y
!E
!Bz
= " µ0# 0 y
!x
!t
!B
=" z
!
!x
!t
!2Ey
x
z
!x
2
= µ0" 0
!2Ey
!t
2
! 2 Bz
! 2 Bz
=
µ
"
0 0
!x 2
!t 2 13
Electromagnetic Waves
"  EM wave equations:
!x
2
" Sinusoidal wave
certain
physical quantity
!2 y 1 !2 y
=
!x 2 v 2 !t 2
f: frequency
y = A sin(
= µ0" 0
!2Ey
!t
2
2
2
! Bz
! B
= µ0" 0 2 z
2
!x
!t
E
B
y
c
z
" Plane wave solutions:
E= Emaxcos(kx-#t+%)
B= Bmaxcos(kx-#t+%)
" Properties:
#  No medium is necessary.
same %,
#  E and B are normal to each other
set to be 0
#  E and B are in phase
#  Direction of wave is normal to both E and B
(EM waves are transverse waves)
#  Speed of EM wave: c = 1 = 2.9972 ! 108 m / s
µ0" 0
#  E/B = Emax/Bmax=c
#  Transverse wave: two polarizations possible
%:Phase
2"
x $ 2"ft + ! )
#
A:Amplitude
Wave speed
B
!2Ey
d#E
0
dt
" Linear wave equation
!:wavelength
v=!f
k=2"/!$
#=2"f
General wave: superposition of sinusoidal waves
14
The EM Wave
x
E
B
y
c
z
x
Two polarizations possible (showing one)
15
16