Chapter 23 – Electromagnetic Waves by ewghwehws

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◦ Quiz Today
◦ Review Exam
   Begin Chapter 23 – Electromagnetic Waves
   No 10:30 Office Hours Today. (Sorry)
   Next Week … More of the same.
   Watch for still another MP Assignment
◦ Will they ever stop??? (No)

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What do we learn from this?
6

 Some of you studied.
 Some of you didn’t.
 If you didn’t, do.
Or         take my Studio Class in the Spring!

Electromagnetic Waves
   Electric Fields and Potential
   Magnetic Fields
   The interactions between E & M
   E&M Oscillations (AC Circuits/Resonance)

   James Clerk Maxwell related all of this
together is a form called Maxwell’s Equations.

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James Clerk Maxwell
   1831 – 1879
   Electricity and magnetism
were originally thought to
be unrelated
   In 1865, James Clerk
Maxwell provided a
mathematical theory that
showed a close relationship
between all electric and
magnetic phenomena
   Electromagnetic theory of
light

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Maxwell Equations

closed surface     enclosed charge          closed surface               no mag. charge

• Conservation of energy                 • Conservation of charge

Lorentz force law

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   When an E or B field is changing in time, a
wave is created that travels away at a speed c
given by:
1
c             3  108 m / s
 0 0

   This is the experimental value for the speed
of light. This suggested that Light is an
Electromagnetic Disturbance,
   In depth experimental substantiation
followed.

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   Can travel through empty space or through
some solid materials.
   The electric field and the magnetic field are
found to be orthogonal to each other and
both are orthogonal to the direction of travel
of the wave.
   EM waves of this sort are sinusoidal in nature.
◦ Picture a sine wave traveling through space.

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Hertz’s Confirmation of
Maxwell’s Predictions
   1857 – 1894
   First to generate and
detect electromagnetic
waves in a laboratory
setting
could be reflected,
refracted and diffracted
(later)
   The unit Hz is named for
him

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   An induction coil is
connected to two large
spheres forming a
capacitor
   Oscillations are initiated
by short voltage pulses
   The oscillating current
(accelerating charges)
generates EM waves
   Several meters away
from the transmitter is
◦ This consisted of a single
loop of wire connected to
two spheres

   When the oscillation frequency of the
transfer occurred between them
   Hertz hypothesized the energy transfer was in
the form of waves
◦ These are now known to be electromagnetic waves
   Hertz confirmed Maxwell’s theory by showing
the waves existed and had all the properties of
light waves (e.g., reflection, refraction,
diffraction)
◦ They had different frequencies and wavelengths which
obeyed the relationship v = f λ for waves
◦ v was very close to 3 x 108 m/s, the known speed of
light
   Two rods are connected to an oscillating source, 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)
   Because the oscillating charges in
the rod produce a current, there is
also a magnetic field generated
   As the current changes, the
the antenna
   The magnetic field is
perpendicular to the electric field
 A changing magnetic field produces an
electric field
 A changing electric field produces a magnetic
field
 These fields are in phase
At any point, both fields reach their maximum value
at the same time
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Was It Magic?
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• The waves are transverse:
electric to magnetic and both
to the direction of
propagation.
•The ratio of electric to
magnetic magnitude is E=cB.
•The wave(s) travel in vacuum
at c.
•Unlike other mechanical
waves, there is no need for a
medium to propagate.
   The old RH-Rule
◦ turn E into B and
you get the
direction of
propagation c.
◦ Rotate c into E and
get B.
◦ Rotate B into c and
get E.

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l

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   Seeing in the UV, for example, steers insects to
pollen that humans could not see.

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Two types of waves

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LikeSound Waves,the equation
for a moving EM wave is :
t x
E  E m axSin(2    )  Em axSin(t  kx)
T l 
t x
B  Bm axSin(2    )  Bm axSin(t  kx)
T l 
2
k
l
Em ax  cBm ax
c
l
f

Suggestion – Look again at the
chapter on sound to solidify
this stuff.
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V  c t

How much Energy
is in this volume?

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Light carries

Energy

and

Momentum

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&

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Energy 1       1
u        0E 
2
B 2
Volume 2      20

From before
E  cB

E    E
B                   E  0 0
c  1
 0 0
B 2   0 0 E 2

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So.
B 2   0 0 E 2
ng
Substituti into the previous,
1         1
u  0E 2
 0 0 E 2
2        20
or
1     1
u  0E  0E2  0E2
2
2     2

Energy stored in the B and B fields are the same!

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• Electric and magnetic fields contain energy,
potential energy stored in the field: uE and uB
uE: ½ 0 E2 electric field energy density
uB: (1/0) B2 magnetic field energy density

•The energy is put into the oscillating fields by
the sources that generate them.

•This energy can then propagate to locations
far away, at the velocity of light.
Energy per unit volume is                          B

dx
u = uE + uB    1 (  0 E 2  1 B2 )
2             0            area
E
A
Thus the energy, dU, in a box of
area A and length dx is
1
dU  ( 0 E 2  1 B2 ) Adx      c         propagation
2         0                        direction

Let the length dx equal cdt. Then all of this energy leav
the box in time dt. Thus energy flows at the rate
dU 1          1 2
 ( 0 E 
2
B ) Ac
dt 2         0
B
Rate of energy flow:
dx
dU c           1 2
 ( 0 E 
2
B )A
dt  2         0                      area
A
E

We define the intensity S, as the
rate of energy flow per unit area:

S   c (  E 2  1 B2 )        c      propagation
2    0
0                   direction

Rearranging by substituting E=cB and B=E/c,
we get,
S  c (  cEB  1 EB )  1 (   c 2  1 )EB  EB
2 0            0c       20 0 0               0
B
In general, we find:
dx
S = (1/0) EB                     area
E
A

S is a vector that points in the
direction of propagation of the 
wave and represents the rate of
S
propagation
energy flow per unit area.                   direction
We call this the “Poynting vector”.

Units of S are Jm-2 s-1, or
Watts/m2.
The Inverse-Square Dependence of S
out in all directions:                   Power, P, flowing
through sphere
Source                           is same for any

P
S 
4r 2

Source
r
Area  r 2
1
S 2
r
Intensity of light at a distance r is           S= P / 4r2
P       1 2
I               Erms
4r 2   0 c

P0 c   ( 250W )( 4 107 H / m )( 3108 m / s )
.
 Erms          
4r 2                 4 ( 18m )2
.

 Erms  48V / m

Erms    48V / m
B                8m / s
 0.16 T
c      .
310
• When present in
large flux, photons
can exert
measurable force on
objects.
• Massive photon
flux from excimer
lasers can slow
molecules to a
complete stop in a
phenomenon called
“laser cooling”.
Momentum and energy of a wave
are related by, p = U / c.

Now,    Force = d p /dt = (dU/dt)/c

pressure (radiation) = Force / unit area

P = (dU/dt) / (A c) = S / c

c
Polarization
The direction of polarization of a wave is
the direction of the electric field. Most light
is randomly polarized, which means it
contains a mixture of waves of different
polarizations.
Ey

B                           Polarization
z                           direction

x
Polarization
A polarizer lets through light of only
one polarization:
E0              E                Transmitted light
q                                    has its E in the
direction of the
polarizer’s
transmission axis.

E = E0 cosq

hence,   S = S0 cos2q

- Malus’s Law
At least of this
chapter.

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