Download April 1 - University of Utah Physics

24.4 The Energy Carried by Electromagnetic Waves
A related quantity: Intensity of EM waves (How bright is the light hitting you?)
Analogous to sound intensity (how loud is the sound wave arriving at you)?
Intensity: S (unit: W/m2)
S=
x
z
(EM energy density)(Volume)
(Area) ∆t
=
y
area
A
We compute the intensity by finding the
energy contained the gray box (of length
c∆t, and area A) passing through the
yellow frame (area A) in time ∆t.
∆x = c∆t
u=
Average intensity of
an electromagnetic
wave:
Power Total EM energy
=
(Area) ∆t
Area
S=
1
1
2
B0
ε o E0 2 =
2
2 µo
1
2
S = c ⋅ ε o E0
2
u ⋅ (c∆t ⋅ A)
= c⋅u
A∆t
→ S = c⋅u
and
S = c⋅
1
2 µo
B0
2
1
Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS
value is Erms = 720 N/C. Find
(a) the average intensity of this electromagnetic wave and
(b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u.
(c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u.
*** a.u. = astronomical unit.
2
Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS
value is Erms = 720 N/C. Find
(a) the average intensity of this electromagnetic wave and
(b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u.
(c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u.
*** a.u. = astronomical unit.
3
Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS
value is Erms = 720 N/C. Find
(a) the average intensity of this electromagnetic wave and
(b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u.
(c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u.
*** a.u. = astronomical unit.
C 2 
N
1

2
2
8 m 
−12
3 W

720
1
.
38
10
(a ) S = cε o E0 = cε o E rms =  3.00 × 10
=
×
 8.85 × 10


s 
N ⋅ m 2 
C
m2
2

E
E
720 N/C
( b) B0 = 0 → Brms = rms =
= 2.40 × 10 −6 T
8
c
3.00 × 10 m/s
c
2
−2
PSUN
PSUN
SV rE2  rV 
rV 0.723 a.u.


=
→
=
=
(c) SV =
and
,
=
= 0.723
S
E
4π rV2
4π rE2
1.00 a.u.
S E rV2  rE 
rE
r
SV =  V
 rE
S = c⋅
−2

SE
1.38 × 108 W/m
8
2
 S E =
=
=
2
.
63
×
10
W/m
(0.723) 2
(rV / rE )2

1
2 µo
B0 =
→ Brms =
c
2
µo
µo S
c
Brms
=
2
( 4π × 10 −7 T ⋅ m/A)(2.63 × 10 3 W/m 2 )
= 3.32 × 10 -6 T
8
3.00 × 10 m/s
4
24.6 Polarization
POLARIZED ELECTROMAGNETIC WAVES
In (linearly) polarized light, the electric field
oscillates along a single transverse direction.
In unpolarized light, the electric field
oscillates along random transverse
direction at a given moment in time.
5
24.6 Polarization
Polarized light may be produced from
unpolarized light with the aid of
polarizing material.
Average Intensity
of light:
S=
1
cε 0 E02
2
E0 ' = E0 cos θ
Absorption and
retransmission by
antennae in polarizer Primed = after polarizer
Unprimed = before polarizer
1
1
2
2
cε 0 (E0 ' ) = cε 0 (E0 cosθ )
2
2
1
2
S ' =  cε 0 E 0  cos 2 θ = S cos 2 θ
2

S'=
E0’=E0cosθ
E0
θ
Malus’ Law
For unpolarized incident light
Random Input polarization:
cos 2θ → cos 2θ =
(
)
S ' = S cos 2θ =
1
S
2
1
2
6
24.6 Polarization
MALUS’ LAW (general case)
E0
θ = angle between
polarization before and
after analyzer
E0cosθ
S = S o cos 2 θ
intensity after
analyzer
The textbook’s
notational convention
intensity before
analyzer
7
24.6 Polarization
Example Using Polarizers and Analyzers
What value of θ should be used so the average intensity of the polarized light
reaching the photocell is one-tenth the average intensity of the unpolarized light?
1
10
S o = (12 S o )cos 2 θ
1
5
= cos 2 θ
cos θ =
1
5
θ = 63.4
8
24.6 Polarization
THE OCCURANCE OF POLARIZED LIGHT IN NATURE
9
Chapter 25
The Reflection of
Light: Mirrors
We will start to treat light in terms of “rays” -technical for the poetic “beam” of light.
This works as long as the “width” of our rays are
significantly larger than the wavelength of the light
10
25.1 Wave Fronts and Rays
A hemispherical view of a sound wave emitted
by a small pulsating sphere (i.e. a point-like
source).
The rays are perpendicular to the wave
fronts (locations where the wave is at the
same phase angle, e.g. at the maxima or
compressions.
At large distances from the source, the
wave fronts become less and less curved.
They become plane-waves described by:


ψ ( x, t ) = ψ 0 sin 2πft −
2πx 

λ 
11
25.2 The Reflection of Light
LAW OF REFLECTION
2 types of reflecting surfaces:
(a) smooth surface
The incident ray, the reflected ray, and
the normal to the surface all lie in the
same plane, and the angle of incidence
equals the angle of reflection.
Specular reflection: the reflected
rays are parallel to each other.
(b) rough surface
Diffuse
reflection
Plane mirror
Law of reflection still applies locally, where the normal is the
direction perpendicular to the tangent plane
12
25.4 Spherical Mirrors
Mirrors cut from segments of a
sphere (Spherical mirrors) are
used in many optical instruments.
If the inside surface of the
spherical mirror is silvered or
polished, it is a concave mirror.
B
If the outside surface is silvered or
polished, it is a convex mirror.
B
The principal axis of the
mirror is a straight line drawn
through the center (C) and the
midpoint of the mirror (B).
Just as it does for a plane mirror,
the Law of reflection still applies
locally, where the normal is the
direction perpendicular to the
tangent plane
13
Applications of
spherical mirrors
8.2
m diameter
(largest single
Large
radius-of-curvature
(R) piece)
reflector (concave
concave
mirrors tomirror***)
enlarge of the
SUBARU
Telescope
nearby
object.
National Astronomical Observatory of
e.g. compact
Japan
(Mauna mirror
Kea Hawaii)
Wide-angle rear-view convex
mirror found on many vehicles.
*** actually the SUBARU main
mirror is closer to an hyperboloid
14