Electromagnetic waves

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12-1

• Electromagnetic waves

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Chapter 32 Electromagnetic Fields and Waves
Goals - complete Maxwells Equations - waves
in general - Electromagnetic waves light,
radio, etc - connection to Maxwells
Equations - properties of light
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Meta-comments - the first unified theory
(electricity and magnetism) - focal point of
course completion of development - new physics
waves of E and B
I wish that I could do more!

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Two more parts to Maxwells equations
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Recall Gauss Law
Look at electric fields going through surfaces
flux Can tell information about enclosed charges
- Gauss Law
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And recall Magnetic Flux - same as electric
flux but with B
constant B general
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Gausss Law for magnetism
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One more part to Maxwells equations
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A capacitor in an RC circuit is fully charged, In
the region of space between the plates of the
capacitor, there is 1) an electric field but no
magnetic field 2) a magnetic field but no
electric field 3) both electric and magnetic
fields 4) no field of any type
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Modification to Amperes Law
-Amperes law must be wrong! - it depends on
what enclosed means Surface S1 encloses a
current Surface S2 does not! What if we moved
S1 into the gap? How can we modify the rule to
handle all situations?
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Maxwells solution
- modify Amperes law by displacement current
archaic term, but
?E is the electric flux
FE EA cos?

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Why does this work - Maxwells solution
FE E A (s/?0) A Q/e0 then
consistent!
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Important conclusion Time varying electric
fields produce magnetic fields!
Recall Faradays Law time varying B fields
produce E fields
Here the reverse
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A capacitor in an RC circuit begins to
discharge, During the discharge, in the region of
space between the plates of the capacitor, there
is 1) an electric field but no magnetic field 2)
a magnetic field but no electric field 3) both
electric and magnetic fields 4) no field of any
type
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The electric field in four identical capacitors
is shown as a function of time. Rank in order,
from largest to smallest, the magnetic field
strength at the outer edge of the capacitor at
time T.
1. Ba Bb gt Bc Bd 2. Ba gt Bb gt Bc gt Bd 3.
Ba Ba gt Bc gt Bd 4. Bc gt Ba gt Bd gt Bb 5. Bd
gt Bc gt Ba Bb
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Finally, Maxwells equations
Now, let there be light
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Basic idea of light waves no need for charges
Q 0, I0
Changing B generates E Changing E generates
B Self-consistent wave of E and B light
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Waves key ideas
Sinusoidal waves - waves on string, pond or
sound waves or light. Let y(x,t) be the
property that is oscillating (e.g. height on
string) Take snapshot at t0
A amplitude ? wavelength
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Note First peak at x ?/4
Now consider other times
Peak at (x-vt) ?/4 or x ?/4
vt ?Wave moves at velocity v
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Period and Frequency
Wave repeats after T ?/v T period of
oscillation
Frequency
f 1/T of oscillations per second Units
of f Hz sec-1 Hertz cycles/sec
Velocity of wave
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Alternate notation/names
wavenumber
(in radians/sec)
angular frequency
Then
f in 1/sec Hz
Frequency
Speed
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Electromagnetic waves - results
- connected to Maxwells Equations soon
Basic idea - changing B fields generate E
fields - changing E fields generate B fields
- solution with both E and B changing together
Properties
Amplitudes
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• If the electric field in a plane
  electromagnetic wave is 100 V/m then the
  amplitude of the magnetic field is
• 1. 3.3?10-7 T
• 2. 6.7?10-7 T
• 3. 0.27 T
• 4. 8.0?107 T
• 5. 3.0?1010 T

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A microwave employs radiation at a frequency
of 1010 Hz. The wavelength of these waves is
about 1. kilometers 2. meters 3. centimeters 4.
micrometers
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The Electromagnetic Spectrum
Any frequency is possible for EM
waves Different frequencies have diff. names and
effects
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The visible spectrum
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Antennas
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Polarization
E and B are perpendicular to direction of
motion B is also perpendicular to E Two possible
choices - for motion in z direction 1) E in x
direction, B in y direction 2) E in y direction,
B in x direction
These are called polarizations
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Most light is unpolarized -random mixture
of all directions A single polarization can be
made using a polarizer
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Reflected light is also partially polarized
Light becomes partially polarized horizontally
when it reflects from a road or a lake.
Sunglasses made from (vertically oriented)
polarizers can reduce the reflected glare.
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Unpolarized light of equal intensity is incident
on four pairs of polarizing filters. Rank in
order, from largest to smallest, the intensities
Ia to Id transmitted through the second polarizer
of each pair.
1. Ia Id gt Ib Ic 2. Ib Ic gt Ia Id 3.
Ib Ic gt Ia gt Id 4. Id gt Ia gt Ib gt Ic 5. Id
gt Ia gt Ib Ic
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The Poynting Vector
Electromagnetic waves always travel in the
direction of (use the vector cross
product). The Poynting vector gives the
direction and magnitude of energy flow
x
z
y
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Intensity
I Power/Area Save S EB/µ0 E2/cµ0 Save
Smax /2
I E02 / 2cµ0
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• An electromagnetic wave is transporting energy
  in the negative y direction. At one point and
  one instant the magnetic field is in the positive
  x direction. The electric field at that point
  and instant is in the direction
• 1. y
• 2. -y
• 3. z
• 4. -z
• 5. -x

z
y
x
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An electromagnetic wave is propagating in the
positive x-direction. At this instant of time,
what is the direction of at the center of the
rectangle?
1. In the positive x-direction 2. In the
negative x-direction 3. In the positive
y-direction 4. In the positive z-direction 5. In
the negative z-direction
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Connection to Maxwells Equations
-in empty space no charges, no currents
Derivation in book is clumsy, ugly (but at OK
math level) - I will give elegant derivation
-but a bit too advanced in math level for
the course - this is for general
interest
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Need new math identity (multi-variable calculus)
Recall usual identity
boundary
inside
New identity (Stokes theorem)
inside
boundary
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Then we have
If this is true for any dA, then
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Look for wave in z direction - E in x
direction, B in -y direction
Solve like simultaneous equations
likewise for B
Solution
?waves
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Which type of waves are NOT predicted by
Maxwells Equations 1) Gamma rays 2) X rays 3)
Light 4) TV signals 5) Sound waves 6) Radio
waves 7) Microwaves 8) More than one of the
above 9) All are predicted
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E
i
B
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E
i
B
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A charge oscillates up and down as shown. Which
of the field patterns shown could correspond to
the fields due to the oscillating charge?
E
E
B
B
b
a
B
B
E
E
B
B
E
E
d
c
E
E
B
B
B
E
E
e
B
E
B
7) a and b 8) d and e 9) b and c
4) d only 5) a, d, e 6) c and d

• a only
• b only
• c only

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t0
tT/4
tT/2
t3T/4
tT
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ym
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l
ym
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A waveform is described by y ym sina(btz).
The velocity of the wave is

• 1/b x
• -1/b x
• b x
• 1/ab x
• ab x
• 1/b z
• -1/b z
• 1/ab z
• -1/ab z
• None of the above

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• B (Eo/c)cos(kz - wt)y
• B (Eo/c)cos(ky - wt)z
• B (Eo/c)cos(ky - wt)x
• B (Eo/c)cos(kz - wt)z
• B Eocos(kz - wt)y
• B Eocos(ky - wt)z
• B Eocos(ky - wt)x
• B Eocos(kz - wt)z
• B cEocos(ky - wt)y
• 0) none of the above

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For a plane electromagnetic wave, which of the
following statements is true about the energy
density uE and uB in the electric and magnetic
fields, respectively, at a given point?

• uE gt uB always
• uE lt uB always
• uE uB always
• uE gt uB at some times, smaller at other times,
  as the electric and magnetic fields become larger
  then smaller
• uE can be gt, lt, or uB at a given point, but
  the relationship wont change with time
• none of the above

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ltsin2q??? ltcos2q???????
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At a distance 20 km from the transmitter of a
radio station, the electric field of the radio
wave has a maximum value of Em. At a distance 40
km from the transmitter, the maximum value of the
electric field is

• 2 Em
• Em
• Em /2
• Em /4
• none of the above

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A polaroid sheet has rows of elongated molecules
along which electrons can conduct, as shown below
(in an schematic close-up).

• The component of an E-field parallel to the row
  is transmitted.
• The component of an E-field parallel to the row
  is absorbed.
• All E-fields are absorbed unless the field aligns
  with the rows completely.

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A polaroid sheet has rows of elongated molecules
along which electrons can conduct, as shown below
(in an schematic close-up).
Polarization axis

• The component of an E-field parallel to the row
  is transmitted.
• The component of an E-field parallel to the row
  is absorbed.
• All E-fields are absorbed unless the field aligns
  with the rows completely.

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gets through
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An electromagnetic wave in the form of light from
a flash bulb is made to reflect off a metal
plate. Which of the following happens to the
plate?

• The plate rebounds noticeably due to the momentum
  transferred from the light.
• The plate rebounds unnoticeably but with an
  audible ping due to the momentum transferred from
  the light.
• The plate rebounds unnoticeably because light has
  no mass and therefore no momentum
• Nothing (its a lame demonstration)

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A plane electromagnetic wave of intensity I is
completely reflected, directly backwards off a
metal plate. What is the radiation pressure Pr
(the force per unit area) exerted by the wave?

• I / c
• 2I / c
• 4I / c
• I / 2c
• I / 4c
• zero

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A plane electromagnetic wave of intensity I is
20 absorbed and 80 reflected, directly
backwards off a metal plate. What is the
radiation pressure Pr (the force per unit area)
exerted by the wave?
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• 0.2 I / c
• 0.8 I / c
• 1.6 I / c
• 1.8 I / c
• 2.0 I / c
• None of the above

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A plane electromagnetic wave of intensity I
incident at an angle q to a metal plate is
completely reflected. What is the radiation
pressure Pr (the force per unit area) exerted by
the wave?
q
6) 2Icosq / c 7) 2Icos2q / c 8) 2Isinq / c 9)
2Isin2q / c 0) None of the above

• Icosq / c
• Icos2q / c
• Isinq / c
• Isin2q / c
• Icosq / csinq

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Chapter 32 Summary
We completed Maxwells Equations
Gauss Law for magnetism
The Displacement Current
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Chapter 32 Summary(cont.)
Maxwells Equations predict electromagnetic waves
Ex E0 sin(kz - ?t) By B0 sin(kz - ?t) with
E0/B0 ?/k (µ0e0)-1/2 c 3.0e8 m/s
The sin function also has the form
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Properties of electromagnetic waves

• E and B are perpendicular to the velocity, and to
  each other.
• The waves travel at velocity
• EcB in magnitude
• The Poynting vector gives the direction and
  energy flow
• The intensity (power/area) is
• The polarization specifies the direction of the E
  field