EM Waves
Reminders on waves
This Lecture
!More on EM waves
!EM spectrum
!Polarization
!
Traveling waves on a string along x obey the wave equation:
" 2 y(x,t) 1 " 2 y(x,t)
= 2
"x 2
v
"t 2
y=wave function
General solution : y(x,t) = f1(x-vt) + f2(x+vt)
y = displacement
!
From previous Lecture
Displacement currents
!Maxwell’s equations
!EM Waves
!
1
Maxwell equations in vacuum
!
pulse traveling along +x
Traveling wave: superposition of sinusoidal waves (produced by a
source that oscillates with simple harmonic motion): y(x,t) = A sin(kx-"t)
y(x,t) = sin(kx+ "t)
A = amplitude
k = 2#/! = wave number ! = wavelength
v
f = frequency T = 1/f = period
" = 2#f=2#/T angular frequency
2
Solutions of these equations:
sinusoidal traveling transverse waves propagating along x
!
# E" ds = $
L
# SB" dA = 0
d%B
(Faraday - Henry)
dt
!
# B" ds = µ &
L
0 0
pulse traveling along -x
EM waves from Maxwell equations
in the absence of charges (q=0) and conduction
currents (I=0)
#SE" dA = 0 (Gauss' Law)
!
d%E
(Ampere - Maxwell law)
dt
E = Emax cos (kx – "t)
B = Bmax cos (kx – "t)
E x B direction of c
!
From these equations we get EM wave equations traveling in vacuum!
•E and B are perpendicular oscillating
vectors
!
E!B=0
4
Quick Quiz on EM Waves
E and B are orthogonal
!
An easy way to understand this:
3
2
1
$
E
I
Increasing B-field
!B = B A cos$
1. Max B flux $=0 => also
circular E is largest!
2. Less flux
3. Null flux $=90° =>
circular E smallest!
B parallel to area normal and E
perpendicular to circuit so E % B
$ E " ds = -
E orthogonal to B!
Shown below is the E-field of an EM wave broadcast at 30
MHz and traveling to the right.
What is the direction of the magnetic field during the first
!/2?
1) Into the page 2) out of the page
E
x
d#B
dt
!/2
What is the wave length?
E!B=0
!
5
1) 10 m
2) 5 m
Quick Quiz pn EM waves
Important Relation between E and B
Which orientation will have the largest induced emf?
E
y
x
z
B
loop in xz
plane
loop in xy
plane
A
B
C
yz
in
p
o
lo ane
pl
!
E = Emax cos (kx – "t)
!
B = Bmax cos (kx – "t)
!
First derivatives:
From:
"E
= #kE max sin(kx # $t)
"x
"B
= $Bmax sin(kx # $t)
"t
"E
"B
=#
"x
"t
This relation comes from
Maxwell’s equations!
!
EM Waves generators: Antennas
Source:
atoms and
molecules
Human eye
Visible range
from red (700
nm) to violet
(400 nm)
The EM Spectrum
Sources of EM waves: oscillating charges, accelerated/decellerated charges,
electron transitions between energy levels in atoms, nuclei and molecules
Gamma rays: !~ 10-14- 10-10 m
Source: radioactive nuclei, cause
serious damage to living tissues
X-rays: ~10-12 -10-8 m
source: deceleration of high-energy
electrons striking a metal target
Diagnostic tool in medicine
UV !~ 6 x 10-10 - 4 x 10-7 m
Most UV light from the sun is absorbed
in the stratosphere by ozone
2 rods connected to alternate current generator; charges oscillate between
the rods (a)
As oscillations continue, the rods become less charged, the field near the
charges decreases and the field produced at t = 0 moves away from the
rod (b)
The charges and field reverse (c)
The oscillations continue (d)
!
!
!
!
Microwaves: ! ~10-4 -0.3 m
sources: electronic devices
radar systems, MW ovens
Energy density of E and B field
Poynting vector
E/B=c
!
True for any geometry
!
!
!
Similarly for a solenoid with current:
EB E
=
µ0 cµ0
uB =
Magnitude is time dependent
! reaches a max at the same instant as E and B do
!
"0 A
d
1
1" A
U 1
U = CV 2 = 0 E 2 d 2 # uE =
= "0 E 2
2
2 d
Ad 2
Its direction is the direction of propagation of the EM wave
2
Magnitude:
S=
!
In a parallel plate capacitor: C =
This is the power per unit area (J/s.m2 = W/m2)
!
!
!
Rate at which energy flows through a unit area perpendicular
to direction of wave propagation
!
Radio:
! ~ 10 - 0.1 m
Sources:
charges
accelerating
through
conducting
wires
Radio and TV
Infrared: ! ~ 7 x 10-7-10-3 m
Sources: hot objects and molecules
B2
2µ0
True for any geometry
!
Energy carried by EM waves
!
!
Total instantaneous energy density of EM waves
u =uE + uB = 1/2 'o
!
Intensity and Poynting vector
E2
+
B2
/(2µo)
!
uE = uB
Since B = E/c and
Pav =
1
B2
E2
uE = "0 E 2 = uB =
= 2
2
2µ0 2c µ0
!
EM waves carry energy!
!
In a given volume, the energy is shared
equally by the two fields
!
Let’s consider a cylinder with axis along x of area A and length
L and the time for the wave to travel L is !t=L/c
The average power of the EM wave in the cylinder is:
U av uav AL
=
= uav Ac
"t
"t
The intensity is
I=
!
!
I= Sav
The average energy density over one or more cycles of
oscillations is:
1
2
since <sin2(kx - "t)> = 1/2
uav = 2uE ,av = "0 E 2 = "0 E max
E/B=c
Pav 1
E B
2
= "0cE max
= max max
A 2
2µ0
I & E2
!
average power per unit area (units W/m2)
ExB
2
!
!
Radiation pressure and momentum
!
dv dp
F = ma = m
=
dt dt
time
energy U
p = force " time = force " distance "
=
=
distance velocity c
Radiation momentum:
!
the radiation momentum is the radiation energy/velocity
Radiation pressure from the Sun
prad =
!
!
prad
!
!
!
Radiation Pressure
Power
F
p
U
Power I
Sav = I =
= =
=
=
=
A
A A"t cA"t
cA
c
Complete absorption on a surface:
total transferred momentum p = U / c and prad=Sav/c
!
Solar intensity at the Earth ( 1.5 x 108 km far from the Sun):
1350 W/m2
!
!
!
!
!
!
!
Direct sunlight pressure I/c ~4.5 x 10-6 N/m2 !
If the sail is a reflecting mirror prad = 2 x 4.5 x 10-6 N/m2
Can be used by spacecrafts for propulsion as wind for sailing
boats!
What is the sail area needed to accelerate a 104 kg spacecraft
at a = 0.01 m/s2 assuming perpendicular
incidence of the radiation on the sail?
A = F/prad = ma/prad = 107 m2
Perfectly reflecting surface: p = 2U/c and prad = 2Sav/c
Circular and elliptical polarization
Polarization of Light Waves (34.8)
!
S
F Power / A I
= av
=
=
A
c
c
c
Linearly polarized waves: E-field oscillates at all times in the
plane of polarization
Unpolarized light: E-field in random
directions. Superposition
of waves with E vibrating
in many different
directions
Linearly polarized
light: E-field has one
spatial orientation
!
Circularly polarized light: superposition of 2 waves of equal
amplitude with orthogonal linear polarizations, and 90˚ out of phase.
The tip of E describes a circle (counterclockwise = RH and
clockwise=LH depending on y component ahead or behind)
!
The electric field rotates in time with constant magnitude.
!
If amplitudes differ ( elliptical polarization
DEMO with MW generator and metal grid
Producing polarized light
!
Polarization by selective absorption: material that transmits
waves whose E-field vibrates in a plain parallel to a certain
direction and absorbs all others
pick up antenna connected to Ammeter
Metal grid
MW generator
This polarization
absorbed
This polarization
transmitted
transmission axis
Polaroid sheet
(E. Land 1928)
Long-chain hydrocarbon
molecules
If the wires of the grid are parallel to the plane of polarization the grid absorbs
the E-component (electrons oscillate in the wire).
The same thing happens to a polaroid: the component parallel
to the direction of the chains of hydrocarbons is absorbed.
If the grid is horizontal the Ammeter will measure a
This
not null current since the wave reaches the antenna
polarization
absorbed
pick-up
This polarization
transmitted
transmission axis
Relative orientation of polarizers
Polarizers and MALUS’ LAW
If linearly polarized light (plane of polrization indicated by red arrow)
of intensity I0 passes through a polarizing filter with transmission axis at
y
Polaroid sheet
an angle $ along y
$
Einc = E0sin$ i + E0 cos$ j
After the polarizer
Etransm = E0cos$ j
So the intensity transmitted is
Itransm = E02 cos2$ = )0cos2$
!
!
!
Transmitted amplitude is Eocos$
(component of polarization along polarizer axis)
Transmitted intensity is Iocos2$
( square of amplitude)
Perpendicular polarizers give zero intensity.
Unpolarized light on polarizers
Only I0/2 is transmitted of unpolarized light by a polarizer and it is
polarized along the transmission axis.
An analyzer rotated at an angle $ respect to the polarizer transmits
100% of the incident intensity when $ = 0 and zero when $ = 90°
Allowed component
parallel to analyzer axis
Polaroid sheets
!
x
transmission axis
E0cos$
Long-chain
hydrocarbon
molecules
Transmitted intensity: I = I0cos2$ I0 = intensity of polarized beam on analyzer
(Malus’ law)
this is called
Malus’ law