Physics 202, Lecture 21 Parameters For A Sinusoidal Wave › undergrads › courses › ..." Nodes...
Transcript of Physics 202, Lecture 21 Parameters For A Sinusoidal Wave › undergrads › courses › ..." Nodes...
Physics 202, Lecture 21
Today’s Topics
! 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
Parameters For A Sinusoidal Wave " 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
Snapshot:Fixed t
Snapshot:Fixed x 2
Standing Waves " When two waves of the same amplitude, same
frequency but opposite direction meet standing waves occur.
!
y1 = Asin(kx "#t)
!
y2 = Asin(kx +"t)
!
y = y1 + y2 = 2Asin(kx)cos("t)" Points of destructive interference (nodes)
!
kx = n"; x =n"k
=n#2
; n =1,2,3...
" Points of constructive interference (antinodes)
!
kx = (2n "1) #2
; x = (2n "1) $4
; n =1,2,3...3
Standing Waves (cont)
" Nodes and antinodes will occur at the same positions, giving impression that wave is standing
4
Standing Waves (cont) " Standing waves with a string of given length L are
produced by waves of natural frequencies or resonant frequencies:
!
" =2Ln
; n =1,2,3...
!
f =v"
=nv2L
=Tµ
n2L
5
Demo: Hertz Experiment
In 1887, Heinrich Hertz first demonstrated that EM fields can transmit over space.
6
Review: Gauss’s Law / Coulomb’s Law " The relation between the electric flux through a
closed surface and the net charge q enclosed within that surface is given by the Gauss’s Law
!
E • dA =q"0
#
7
Gauss’s Law for Magnetism " The Gauss’s Law for the electric flux is a reflection
of the existence of electric charge. In nature we have not found the equivalent, a magnetic charge, or monopole
" We can express this result differently: if any closed surface as many lines enter the enclosed volume as they leave it
!
B• dA = 0"
8
Review: Faraday’s Law " The emf induced in a “circuit” is proportional to the
time rate of change of magnetic flux through the “circuit” or closed path.
" Since
" Then
A
B
'$
!
"B = B• dA#
nominal direction of ($
!
" = #d$B
dt
!
" = E • d!#
!
E • d! = "d#B
dt$9
Review: Ampere’s Law " A magnetic field is produced by an electric current is
given by the Ampere’s Law
But Ampere’s Law can be ambiguous: currents enclosed by surfaces 1 and 2 are different!
!
B• d! = µ0I"
10
But note time-dependent electric flux though surface 2.
Maxwell’s modification of Ampere’s Law
" A time-dependent electric flux induces a magnetic field (in analogy to Faraday’s law)
B!"• d#"= µ0 I + !0
d"E
dt#$%
&'($)
!
"E = E • dA#
11
Ampere’s law
“Maxwell’s displacement current”
Maxwell Equations
!
E • dA =q"0
#
!
E • d! = "d#B
dt$
!
F = qE + qv "B
!
B• d! = µ0I + "0µ0d#E
dt$!
B• dA = 0"
!Gauss’s Law/ Coulomb’s Law
!Faraday’s Law
!Gauss’s Law of Magnetism, no magnetic charge
!Ampere Maxwell Law
Also, Lorentz force Law !
These are the foundations of the electromagnetism 12
EM Fields in Space " Maxwell equations when there is no charge and current:
! =• 0AE d
!
E • d! = "d#B
dt$
!
B• d! = "0µ0d#E
dt$! =• 0AB d
tB
xE zy
!!
"=!
!
tE
xB yz
!
!"=
!!
00#µdifferential forms:
(single polarization)
x
y
z 2
2
002
2
tE
xE yy
!
!=
!
!"µ 2
2
002
2
tB
xB zz
!!
=!!
"µ
E
B
13
Linear Wave Equation " Linear wave equation
2
2
22
2 1ty
vxy
!!
=!!
certain physical quantity
Wave speed
" Sinusoidal wave
)22sin( !"#"
+$= ftxAy
A:Amplitude
f: frequency %:Phase
General wave: superposition of sinusoidal waves
v=!f k=2"/!$#=2"f
!:wavelength
14
Electromagnetic Waves " EM wave equations:
" Plane wave solutions: E= Emaxcos(kx-#t+%) B= Bmaxcos(kx-#t+%) " Properties:
# No medium is necessary. # E and B are normal to each other # 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: # E/B = Emax/Bmax=c # Transverse wave: two polarizations possible
2
2
002
2
tE
xE yy
!
!=
!
!"µ 2
2
002
2
tB
xB zz
!!
=!!
"µ
same %, set to be 0
smc /109972.21 8
00
!=="µ
x
y
z
E
B c
15
The EM Wave
x
y
z
E
B c
Two polarizations possible (showing one) 16