# Electromagnetic Waves

Electromagnetic Waves
Ey = Ey0 sin [2π(x  vt)/],
Bz = Bz0 sin [2π(x  vt)/]
Bzo = Ey0/v
v=
()-1/2
 is the wavelength of the
wave and its frequency is
f = v/.

v
The intensity
(energy/time/area) of the
wave  Ey02.
 and  (and therefore v) are properties of the material in which the wave is traveling [how
much the material affects E and B].
In vacuum, the speed of EM waves is v = (00)-1/2  c = 2.99725 x 108 m/s  3 x 108 m/s
(and air speed is very close to vacuum speed!)
Radio waves, microwaves, infrared waves, visible light, ultraviolet rays, x-rays, and gamma
rays are all electromagnetic waves of different frequencies (f) and wavelengths (in
vacuum/air):  = c/f.
c = 2.99725 x 108 m/s  3 x 108 m/s
D
If there was a tunnel through the earth, it would take an
electromagnetic wave
t = D/c = 1.3 x 104 m / (3 x 108 m/s) = 4.3 x 10-5 s to pass
through the earth
earth
Rem
Earth to moon:
t = Rem/c = 3.8 x 108 m / (3 x 108 m/s) = 1.3 s
earth
moon
Rse
Sun to earth:
t = Rse/c = 1.5 x 1011 m / (3 x 108 m/s) = 500 s
 8.3 minutes
sun
earth
Since the speed of electromagnetic waves is so large (seems instantaneous in
everyday applications), it cannot be measured using conventional techniques
(e.g. meter stick and stop watch). Fizeau (1849) had first successful terrestrial
measurement:
light
F
Time for light to travel between
wheel and mirror and back
t = 2d/c. If the wheel has N
spokes, the light will be blocked by
the next spoke (and not reach the
detector) if it has turned
 =2/2N in time t, i.e. if
 = 2F = /t = (/N)/ (2d/c)
 c = 4dNF
eye
(detector)
e.g. Fizeau: N = 720, d = 9500 m (~ 6 miles)
 F = 11/sec
Electromagnetic Spectrum
The boundaries between different
types of waves are “fuzzy” – the
different names mostly refer to how
the waves are generated.
Different colors of light correspond to
different wavelengths/frequencies.
White light is a combination of visible
light of all frequencies.
Visible light (what we can see) only
occupies less than one octave (factor
of two) in frequency and wavelength,
but this is an octave where there is a
lot of sunlight.
Unless otherwise stated, wavelength means wavelength in vacuum:  = c/f
Consider two electromagnetic waves from the same source with the same wavelength
that travel different distances (x) before coming back together:
E1 = E0 sin[2π(x - x/2 vt)/] and E2 = E0 sin[2(x + x/2  vt)/]
E1 = E0 sin(-/2) and E2 = E0 sin (+/2)
where  = 2 x / .
After they come together, the total electric field is
E = E1 + E2 = E0 [sin(-/2) + sin(+/2)]
E = [2E0 cos(/2)]sin() = [2E0 cos(/2)] sin(2(xvt)/]
Intensity I  4E02 cos2(/2) = 4I0 cos2(/2) = 4I0 cos2(x/)
I0 = intensity of each wave alone
4
constructive interference
I / I0
3
2
1
destructive interference
0
0.0
0.5
1.0
x/
1.5
2.0
2.5
An interesting instrument which uses interference: Michelson Interferometer
(pp. 1147-1148)
(or detector)
x = 2(L1 – L2)
If |x| = N  constructive interference
If |x| = (N+1/2) destructive int.
Rings (alternating constructive and destructive interference) appear because
rays of light not going exactly to center travel different distances.
If use white light, different colors (different wavelengths)
have constructive interference at different places.
E = E0 sin [2π(x  vt)/]
Since E and B only depend on x and t, they are the same for all y and z  surfaces
of constant phase (wave fronts) are planes: this is called a “plane wave”.
E
y,z
Rays
x
The electromagnetic wave is traveling in the direction perpendicular to
the wavefronts along imaginary lines called rays.
A point source creates a spherical wave,
where the wave fronts are spherical surfaces
A line source (e.g. antenna) creates a
cylindrical wave, where the wave
fronts are cylindrical surfaces and the
If you block most of a spherical or
cylindrical wave, the part that gets
through is approximately a plane
wave.
Consider a hole (or obstacle) or size d. If  &gt;&gt; d or even   d, a wave will spread
out when passing through the hole (or around an obstacle). This effect is called
diffraction and the study of the optics when one must be concerned with large  is
called physical optics. It will be studied in PHY232.
If  &lt;&lt; d, the rays will continue traveling in straight lines, until they bounce (reflect) off
a surface or change direction (refract) when passing through a second material.
Therefore, in this small  limit (geometric optics), the ray approximation is good and light
passing through holes or around obstacles, will cast sharp spots or shadows.
1905: Einstein pointed out that
electromagnetic waves are not “continuous”
but that their energy travels in bundles (now
called photons); each photon has energy
E = hf where f is the frequency of the wave
and Planck’s constant h = 6.63 x 10-34 J&middot;s.
However, if the power emitted or absorbed is
much larger than the energy of one
photon/period, P &gt;&gt; (hf)/(1/f) = hf2, photons
will overlap and the wave can usually be
treated as a continuous, classical wave -“quantum” effects can be ignored.
For visible light, with f ~ 5 x 1014 Hz, this
classical regime corresponds to P &gt;&gt; 0.2 mW.
Quantum effects will be studied in PHY 361.
Electromagnetic waves slow down in materials, i.e.
have speed v &lt; c. One can view this in the
following way. When a wave of frequency f hits
molecule A, it is absorbed, causing the charges in A
to vibrate at frequency f. A then emits an EM wave
at the same frequency, which is then absorbed by
B, and the pattern continues. Although the EM
wave travels at speed c between A and B, the delay
while A is absorbing and re-emitting the wave
causes the average speed of the wave while
passing through the material to be less than c.
Note that the frequency of the wave doesn’t
change when passing through the material, but its
wavelength does change:
material = v/f &lt; c/f = vacuum:
material /  = v/c &lt; 1.
[Note: v/c will depend on the density of the material, the
types of atoms, and the frequency.]
Unless otherwise stated, wavelength means
wavelength in vacuum:   vacuum.