Electromagnetic Waves

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Alternating Current Electromagnetic Waves
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Sinusoidal Function of Distance
A sinusoidal function of distance is
characterized by its amplitude, wavelength, and
phase constant
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Sinusoidal Function of Time
A sinusoidal function of time is characterized
by its amplitude, frequency, and phase constant
w is the angular frequency (angular speed)
radians per second. f is the frequency cycles
per second, or Hertz (Hz) w 2p f
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Alternating Voltages and Currents
When we plug a lamp into a wall socket, the
voltage and current supplied vary sinusoidally,
with a frequency f of 60 cycles per second
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Root Mean Square (rms) Values
The average value of I2 is ½ I2MAX
The root mean square (rms) value of the current
is defined as
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Alternating Voltages and Currents
The power dissipated in the resistor is
The average power dissipated in the resistor is
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• The AC generator has a maximum voltage of VMAX
  24 V
• and frequency f 60 Hz. The resistor has R 265
  ?.
• Find
• The rms voltage
• The rms current
• The average dissipated power
• The maximum instantaneous value of the dissipated
  power

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Transformers
A transformer is used to change the voltage in
an electric circuit
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Transformers
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Transformers
Since energy is conserved, the power in the
primary circuit equals the power in the secondary
circuit
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Bug Zapper
If the zapper operates at 4000 V, and the primary
coil plugs to a standard 120 V outlet, and has 27
turns How many turns does the secondary coil
have? What is the ratio of the current in the
primary, to the current in the secondary
circuit? (IP/IS)
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Plane Electromagnetic Waves
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Plane Electromagnetic Waves
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Plane Electromagnetic Waves
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Plane Electromagnetic Waves
Waves are in phase, but fields oriented at
900 Speed of wave is c At all times E c B
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• Moving wave
• F(x, t) FP sin (kx - ?t )
• ? 2? f
• ? angular frequency
• f frequency
• k 2? ? ?
• k wavenumber
• wavelength
• v ? / k f ?

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The distance between Earth and the Sun is
1.50x1011 m. How long does it take for the light
to cover this distance?
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Visible Region of the Electromagnetic Spectrum
Find the frequency of red light with wavelength
700 nm. Find the wavelength of light with
frequency 7.5x1014 Hz
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Plane Electromagnetic Wave

• What is the distance from a to b?
• How long does it take for the wavefront to move
  from a to b?
• What does an electric field sensor detect at
  point P
• as a function of time?

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The Electromagnetic Spectrum
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The Electromagnetic Spectrum
Ultraviolet 7.5x1014 1017 Hz Damaging to
skin/organisms. X-Rays 1017 1020 Hz Damaging.
Radiography. Gamma Rays ? 1020 Hz Damaging.
Used in therapy and sterilization.
Radio Waves 106 109 Hz Radio/TV, from
antennas. Microwaves 109 1012 Hz Radar,
telephone, cooking. Infrared 1012 1014
Hz Heat, remote controls. Visible 4.3x1014
7.5x1014 Hz Detected by our eyes.
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Energy in Electromagnetic Waves

• 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.

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Energy in Electromagnetic Waves
The energy density stored in electric and
magnetic fields uE ½ ?0 E2 electric field
energy density uB (1/?0) B2 magnetic field
energy density was calculated for static fields
(E?constant, B?constant)
Electromagnetic fields are constantly
changing and therefore it is more appropriate to
treat the energy in the field using average
values.
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Intensity of an Electromagnetic Wave
The amount of energy that a wave delivers per
unit area, per unit time, is referred to as the
Intensity of the wave
In the figure, a beam of light of cross-sectional
area A, shines on a surface. All the light energy
contained in the volume ?V A (c?t) strikes
the surface in the time ?t.
The energy in the volume ?V is ?U u ?V. Then,
the intensity of the wave is
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Intensity of an Electromagnetic Wave
The average intensity of an electromagnetic wave
is
Using E c B we have
Using Erms (1/?2) EMAX and Brms (1/?2) BMAX
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At a given instant in time, the electric field in
a beam of light has a magnitude of 510 N/C. What
is the magnitude of the magnetic field at that
same time?
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Since Intensity is energy per unit area per unit
time Intensity is power per unit area

• A garage is illuminated by a light bulb dangling
  from a wire.
• The bulb radiates uniformly in all directions and
  consumes 50 W.
• Calculate
• The average intensity of light 1 m from the bulb
• The rms values of E and B, 1 m from the bulb
• (assume 5 of the dissipated power is converted
  into light)

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• A small laser emits a cylindrical beam of light 1
  mm in diameter
• with and average power of 5 mW.
• Calculate the average intensity of the beam
• Compare with the 50 W bulb of the previous
  problem
• (Hint use a 1 m separation to compare)

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Wave Momentum and Radiation Pressure
An electromagnetic wave not only carries energy
U but also carries momentum p. The momentum of a
wave is related to its energy by p U /
c For an electromagnetic wave absorbed by an
area A in time ?t, the total energy received
is ?U uAV A c ?t Then, the momentum received
by the surface is ?p ?U/c uAV A ?t IAV
A ?t / c
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Wave Momentum and Radiation Pressure
The momentum received by the surface is ?p
IAV A ?t / c Then, the average force exerted by
the light on the surface is FAV ?p / ?t
IAV A / c And the average pressure on the
surface is
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• One a sunny day the average intensity of sunlight
  on earth is
• about 1.0x103 W/m2.
• Find
• The average radiation pressure due to sunlight
• The average force exerted on a beach towel that
  is
• 1 m by 2.5 m in size (assume the towel absorbs
  the light)

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Propagation of Electromagnetic Waves in Matter