Videos
SP212
Ch. 33 - Electromagnetic Waves
Double Rainbow http://youtu.be/OQSNhk5ICTI
Internal Reflection http://youtu.be/2kBOqfS0nmE
Snells Law http://youtu.be/yfawFJCRDSE
Maj Jeremy Best USMC
Physics Department, U.S. Naval Academy
April 5, 2016
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
April 5, 2016
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Maxwell’s Rainbow
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Visible Range
Our eyes are most sensitive to yellow-green light:
∼ 550 nm
The reddest thing we can see is ∼ 720 nm
The purple limit of our vision is ∼ 360 nm
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SP212
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SP212
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Electromagnetic Waves
Electromagnetic Waves
An electromagnetic wave consists of a sinusoidally
varying electric field, and a sinusoidally varying magnetic
field , each inducing the other.
E = Em sin(kx − ωt)
B = Bm sin(kx − ωt)
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SP212
April 5, 2016
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Things to note about E&M
waves:
~ fields are perpendicular to each other
The ~E and B
~E × B
~ gives the direction of travel of the wave
The fields vary sinusoidally and are in phase with
one another
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SP212
April 5, 2016
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Electromagnetic Waves
Electromagnetic Waves also have one other very special
property, they always move at the speed of light,
c = 299 792 458 m/s ≈ 3 × 108 m/s, regardless of the
observer’s motion.
c=√
c=
1
µ0 0
E
B
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Let’s zoom in on a tiny portion of the wave and see
where this comes from. We start with Faraday’s Law:
I
~E · d~s = −dΦB
dt
First, let’s handle the left side
I
~E · d~s = (E + dE )h − Eh = h dE
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
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We equate our two results:
And the right side
h dE = −h dx
ΦB = Bh dx
dΦB
dB
= h dx
dt
dt
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
−dB
dE
=
dx
dt
kEm cos(kx − ωt) = −ωBm cos(kx − ωt)
ω
Em
=
k
Bm
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Electromagnetic Waves
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SP212
April 5, 2016
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The Poynting Vector
The direction of travel of an E&M wave is given by the
Poynting vector, ~S
A similar argument starting from Maxwell’s Law of
induction
I
~ · d~s = µ0 0 dΦE
B
dt
~S = 1 ~E × B
~
µ0
S has the dimensions of power/area . If we observe that
E = Bc we can re-write the magnitude of ~S
yields the numerical result for this speed
ω
Em
1
=
=√
= c = 299 792 458 m/s
k
Bm
µ0 0
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
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dB
dt
April 5, 2016
S=
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1 2
1
E = EB
cµ0
µ0
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SP212
April 5, 2016
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Average Intensity
Radiation Pressure
We stated that the intensity of a light ray depends on
the square of the electric field. True, but usually we
measure an average intensity over several cycles. We’re
thus more interested the average of E 2 . We define the
root-mean-squared
p=
Em
Erms = √
2
1 2
Iavg =
E
cµ0 rms
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
U
c
Where U is the energy of the light.
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Radiation Pressure
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Radiation Pressure
Starting with that expression for momentum , we
differentiate with respect to time , and divide by the
surface area of the object.
The previous derivation applied for the case of total
absorption. If the light is perfectly reflected however,
because of Newton’s 3rd Law, the radiation pressure is
twice as much.
dp
d U
=
dt
dt c
power
F =
c
power
F
= A
A
c
I
pressure =
c
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Light carries momentum as well as energy. A result
from special relativity tells us that the momentum of a
light ray is:
I
(Total absorption)
c
2I
pr =
(Total reflection)
c
pr =
Simpson Video Radiation Pressure
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Polarization
Polarized Light
We describe the plane of oscillation of the electric field
of a light ray to be the polarization of the ray. Most
common sources of light (the sun, a light bulb) are
non-polarized. That is, if you looked down the path of
the incoming light, the electric field vector would seem
to jump about randomly
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
April 5, 2016
If we pass light light through a polarizer however, we
filter out light whose magnetic field is not aligned with
the sheet.
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Intensity of Transmitted Light
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Intensity of Transmitted Light
If the incident light is already polarized however, we will
only pass the component of the incoming ray aligned
with the polarizer. In the figure, this is Ey = E cos θ .
since the intensity of light is proportional to the square
of the electric field magnitude, we can see that the
intensity of already polarized light passing through a
second polarizer is
When you shine unpolarized light on a polarizer, by
chance, half of it will be aligned with the polarizer and
therefore transmitted
I = (1/2)I0
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
(unpolarized light through a polarizer)
I = I0 cos2 θ
Where θ is the angle between the two polarizers.
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Reflection and Refraction
Reflection and Refraction
Light, unlike other waves we studied last semester, does
not require a medium though which to travel. However,
it can travel through many materials, which we call
transparent. When light passes from one medium to
another, it is both reflected and refracted at the
boundary.
First we define a normal to the surface
of the medium. All angles of
reflection/refraction are defined relative
to this normal. For reflected light,
θ1 = θ10 . Refracted light obeys Snell’s
Law
n1 sin θ1 = n2 sin θ2
Where n1 and n2 are called the indicies
of refraction for the media.
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Index of Refraction
The index of refraction for a material is usually a function
of color, with blue light bending more (n larger) than
red light. This causes the effect known as chromatic
dispersion, or in more technical language, Rainbows!
The index of refraction n is a property of all transparent
materials. A perfect vacuum has an index of refraction of
exactly 1. The index of refraction indicates the speed of
light in the material, n = c/v
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SP212
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Total Internal Reflection
Brewster’s Angle
For angles larger than the critical angle, θc light rays
will be totally internally reflected, trapped in a higher
index material. The critical angle is defined as that angle
which results in a 90◦ refraction.
n1 sin θc = n2 sin 90◦
n2
θc = sin−1
n1
Light reflecting from a surface is always
partially polarized. By light incident at
a particular angle, called Brewster’s
angle, θB , is completely polarized
perpendicular to the plane of incidence.
We find that Bewster’s angle is the
angle such that the reflected ray is
perpendicular to the refracted ray. We
can easily solve for this angle with basic
trig:
θB = tan−1
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n2
n1
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