Download P212C33

Electromagnetic Waves
Maxwell’s Equations
  Qenc
 E  dA 

 
 B  dA  0
 
d E
B

d
l


(
I


)
c

dt
 
d M
E

d
l



dt
  K o
  K m o
p212c33: 1
“Sourceless” Maxwell’s Equations
 
 E  dA 0
 
 B  dA 0
 
d E
d  
 B  dl  dt   dt  E  dA
 
d M
d  
 E  dl   dt   dt  B  dA
p212c33: 2
A simple Electromagnetic Wave
Pulse: E and B constant within a “sheet” moving
at velocity v
y
E
v
B
x
z
... need to verify consistency with
Maxwell’s Equations
p212c33: 3
 
 E  dA  0
 
 B  dA  0
Field lines continue forever: each field line which
enters (exits) a closed surface must also enter (exit), so
net number of field lines entering (exiting) a closed
surface must be zero.
p212c33: 4
 
 B  dl  BL
 
d (  E  dA)  ELvdt  0
d  
E  dA  ELv

dt
dA
dl
L
v dt
 
d  
 B  dl   dt  E  dA
BL  ELv
B  Ev
p212c33: 5
 
 E  dl   EL
dl
 
d (  B  dA)  BLvdt  0
d  
B  dA  BLv

dt
dA
L
v dt
 
d  
 E  dl   dt  B  dA
 EL   BLv
E  Bv
p212c33: 6
Pulse is consistent with Maxwell' s Equations iff
B  vE
E  Bv
v
1

c  speed of light in vacuum 
1
 o o
 2.99792458 108
m
s
other important features
  
v||E  B
E  Bv
p212c33: 7
y
General relations between (crossed) E
and B fields creating EM waves.
 
d  
 E  dl   dt  B  dA
 
 E  dl  ( E y ( x  x)  E y ( x))a
 
 B  dA  Bz ax
take
a
x
x
lim
x0
E y
B z


x
t
p212c33: 8
y
 
d  
 B  dl   dt  E  dA
 
 B  dl  ( Bz ( x  x)  Bz ( x))a
 
 E  dA  E y ax
take
x
x
a
lim
x 0
E y
B z

  
x
t
p212c33: 9
E y
 Bz
  
x
t
Start with
E y
B z

x
t
2

E

Ey

 B z
y



x x
x 2
x t
2
 E y  E y
1   Bz


2
t t
t
 t  x
 2Ey
 2Ey
 
2
x
t 2
 2Ey 1  2Ey
1

;
v

x 2 v 2 t 2

Classical Wave Equation!
p212c33: 10
Sinusoidal Electromagnetic Waves
Ey
Bz

E  yˆ Emax sin( kx   t )

B  zˆBmax sin( kx   t )
k
2
f 
Emax


x
  2f
v
k
 vBmax
p212c33: 11
Energy in an electromagnetic wave
1 2 1 2
u  E 
B
2
2
E
but B    E
v
so u   E 2
For a Harmonic Wave
u  E 2  Emax sin 2 (kx   t )
2
uav 
Emax 2
2
p212c33: 12
Energy in an electromagnetic wave
Energy :
dU  udV   E 2 ( Avdt )
Energy Flow :
dU
 2 EB
2
S
A  vE 
E 
dt


 1  
S  EB


S  Poynting Vector
S = Intensity (instantan eous)
p212c33: 13
Energy Flow and Harmonic Waves
 1  
S  EB

 xˆ
Emax Bmax

sin 2 (kx  
Emax Bmax Emax

S av 


2
2 v
2
t)
2
vEmax
 I  Intensity
2
p212c33: 14
Example: A radio station the surface of the earth emits 50 kW
sinusoidal waves. Determine the intensity, and the Electric and
Magnetic field amplitudes for an orbiting satellite at a distance of
100 km from the station.
p212c33: 15
Momentum in an electromagnetic wave
Momentum Density


p S
= 2
V c
pavg S avg
 2
V
c
Radiation Pressure :
F S av
F 2 S av
Pabs  
Pref  
A c
A
c
p212c33: 16
Example: Satellite in previous example has a 2m diameter antenna. What is the force
of the radiation on the antenna assuming perfect reflection?
p212c33: 17
Standing Waves:
Superposition of equal amplitude traveling waves
of opposite directions.
E   Emax sin( kx  t )  Emax sin(  kx  t )
 2 Emax sin kx cos t
B  Bmax sin( kx  t )  Bmax sin(  kx  t )
 2 Bmax cos kx sin t
nodal planes (for E)
 3

x  , , ,  n
2
2
2
Standing wave mode when L  n
2L
c
c
n  ; f n   n
n
n
2L

2
p212c33: 18
Example: EM standing waves are set up in a cavity used for electron spin resonance
studies. The cavity has two parallel conducting plates separated by 1.50 cm.
a) Calculate the longest wavelength and lowest frequency of EM standing waves
between the walls.
b) Where in the cavity is the maximum magnitude electric field and magnetic
field?
p212c33: 19
Electromagnetic Spectrum
(see graphic)
in vacuum, v = c = 2.99792458x108 m/s
f = v increasing frequency <=> decreasing wavelength
visible spectrum: 400 nm (violet) to 700 nm (red)
p212c33: 20
Radiation from a Dipole
Q
Q
p0 k 2 sin 
E
sin( kr  t )
4 r
p0 k 2 sin 
B
sin( kr  t )
4v r
 sin  
I 

 r 
2
p212c33: 21
p212c33: 22