Download Ch 33 Electromagnetic Waves I

Ch 33
Electromagnetic
Waves I
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The information age is based almost entirely on the
physics of electromagnetic (EM) waves. We are globally
connected by TV, telephones, and the Web, and
constantly immersed ( 沉 浸 在 ) in those signals (EM
waves).
The starting point in this chapter is understanding the
basic physics of EM waves, which come in so many
different types that they are said to form Maxwell’s
rainbow.
Ch33: part I → Electromagnetic wave
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part II → Optics
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Maxwell
The crowning achievement ( 最 高 成 就 ) of James Clerk
Maxwell was to show that a beam of light is a traveling
wave of electric and magnetic fields — an electromagnetic
(EM) wave — and thus that optics, the study of visible
light, is a branch of electromagnetism.
h “Optics” is becoming most important study for the
development of industry h光電產業(LCD, LED, solar cell)
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In Maxwell’s time (the mid 1800s), the visible, infrared,
and UV forms of light were the only EM waves known.
Spurred on (激勵) by Maxwell’s work, Hertz discovered the
radio waves and verified that they move through the
laboratory at the same speed as visible light (光速c).
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Maxwell’s rainbow
常用可見雷射光
He-Ne red laser: 632.8nm
λ↓, ν↑ (c=λ/T=λν)
Ar+ laser: 488nm, 514.5nm
光波能量 E=hν
Microwave
p.6
p.7
p.8
p.9
p.10
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Figure 33-2: relative sensitivity (感光度) of the human
eye to light of various wavelengths. The center of the
visible region is about 555 nm, which produces the
sensation that we call yellow-green.
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[Serway] The various types of EM waves are listed in
Figure 33.1, which shows the EM spectrum. No sharp
dividing point exists between one type of wave and the
next. Remember that all forms of the various types of radiation
are produced by the same phenomenon — accelerating
charges (see §33-3).
Radio waves: 104m<λ<0.1m, the result of charges
accelerating through conducting wires (天線). They are
generated by electronic devices as LC oscillators and are
used in radio and TV communication systems.
Microwaves: 0.3m<λ<10-4m, also generated by
electronic devices. Because of their short wavelengths,
they are well suited for radar systems and for studying
the atomic and molecular properties of matter.
Microwave ovens (微波爐) are an interesting application.
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Infrared waves: 10-3m<λ<7×10-7m. These waves,
produced by molecules and room-temperature objects,
are readily absorbed by most materials. The infrared
(IR) energy absorbed by a substance appears as
internal energy because the energy agitates (攪動)
the atoms of the object, increasing their vibrational or
translational motion, which results in a temperature
increase.
IR radiation has practical and scientific applications
in many areas, including physical therapy (物理治療),
IR photography ( 照 相 術 ), and vibrational
spectroscopy (振動頻譜儀).
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Visible light: produced by the rearrangement of
electrons in atoms and molecules. The various
wavelengths of visible light, which correspond to
different colors, range from red (7×10-7m) to violet
(4×10-7m). The sensitivity of the human eye is a
function of λ, being a maximum at a λ~5.5×10-7m (Fig.
33-2). With this in mind, why do you suppose tennis
balls often have a yellow-green color?
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Ultraviolet waves: 4×10-7m<λ<6×10-10m. The Sun is
an important source of UV light, which is the main
cause of sunburn (曬傷皮膚). Sunscreen lotions (防曬油)
are transparent to visible light but absorb most UV light.
The higher a sunscreen’s solar protection factor (SPF),
the greater the percentage of UV light absorbed. UV
rays have also been implicated in the formation of
cataracts (白內障), a clouding of the lens inside the eye.8
Most of UV light from the Sun is absorbed by ozone (臭
氧) (O3) molecules in the Earth’s upper atmosphere, in
a layer called the stratosphere (同溫層、平流層). This
ozone shield ( 護 罩 ) converts lethal ( 致 命 の ) highenergy UV radiation to IR radiation, which in turn
warms the stratosphere.
X-rays: 10-8m<λ<10-12m. The most common source of
x-rays is the stopping of high-energy electrons upon
bombarding a metal target. X-rays are used as a
diagnostic (診斷) tool in medicine and as a treatment
(治療) for certain forms of cancer. X-rays are also used
in the study of crystal structure because x-ray
wavelengths ~ atomic separation distances in solids
(~0.1nm).
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Gamma rays: EM waves emitted by radioactive (放射
性) nuclei [such as 60Co (鈷) and 137Cs (銫)] and during
certain nuclear reactions. High-energy gamma rays are
a component of cosmic rays (宇宙射線) that enter the
Earth’s atmosphere from space. They have
wavelengths ranging from 10-10m~10-14m. They are
highly penetrating and produce serious damage when
absorbed by living tissues.
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Some EM waves, including x rays, γ rays, and visible
light, are radiated (emitted) from sources that are of
atomic or nuclear size, where quantum physics rules.
We restrict ourselves to that region of the spectrum
(λ~1m, ν~108Hz) in which the source of the radiation
(the emitted waves) is both macroscopic (巨觀の) and
of manageable (可控制の) dimensions.
Figure 33-3: generation of such waves. The LC
oscillator establishes an angular frequency ω
[=1/(LC)1/2]. Charges and currents in this circuit vary
sinusoidally at this frequency.
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+q
-q
The LC oscillator is coupled by a transformer (變壓器)
and a transmission line to an antenna (天線), which
consists essentially of two thin, solid, conducting rods.
Through this coupling, the sinusoidally varying current
i(t) in the oscillator causes charge to oscillate
sinusoidally along the antenna at the angular frequency
ω of the LC oscillator.
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The current i(t) in the antenna associated with this
movement of q also varies sinusoidally, in magnitude
and direction, at angular frequency ω. The antenna has
the effect of an electric dipole whose electric dipole
moment p(t)=q0dcosωt along the antenna.
Because p(t) varies in magnitude and direction, the
electric field E(t) produced by the dipole varies in
magnitude and direction. Also, because the current
varies sinusoidally i(t), the magnetic field B(t) produced
by the current varies in magnitude and direction.
Together the changing fields form an EM wave that
travels away from the antenna at speed c. The angular
frequency of this wave is ω, the same as that of the LC
oscillator.
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Electric dipole radiation
[ 如 圖 ] First consider the B-field. Because it varies
sinusoidally [because i(t) varies sinusoidally in the
antenna], it induces (via Faraday’s law of induction) a
perpendicular E-field that also varies sinusoidally. However,
because that E-field is varying sinusoidally, it induces
(via Maxwell’s law of induction) a perpendicular B-field that
also varies sinusoidally.
The two intercrossing fields (E- & B-field) continuously
create each other via induction, and the resulting
sinusoidally variable fields travel as a wave — EM wave.
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Without this amazing result, we could not see; indeed,
because we need EM waves from the Sun to maintain
Earth’s temperature, without this result we could not even
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exist (could not live).
p固定不變: 有靜電場，無磁場
v
v
∂E
∇ × B = μ0ε0
∂t
p=p(t) 遠
麥克斯威爾感應定
場 律(微分形式)
p(t)=q0d0cos ωt
波谷
p(t)
近場
v
v
∂B
∇× E = −
∂t
法拉第感應定律
(微分形式)
波峰
可推導出
p(t)
λ
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遠
場
Wave Equation
v
v 2 v 1 ∂2E
∇ E= 2 2
c ∂t
v
v 2 v 1 ∂2B
∇ B= 2 2
c ∂t 15
輻射越遠越近似平面波
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Figure 33-4 shows how the E-field and the B-field change
with time as one wavelength of the wave sweeps past the
distant point P of Fig. 33-3; in each part of Fig. 33-4, the
wave is traveling directly out of the page (k: 電磁波傳播
方向).
EM wave’s properties:
1. E- & B-fields are always ⊥ the traveling direction (k) of
the EM wave. Thus, the wave is a transverse wave (橫波).
2. E-field is always ⊥ B-field.
3. The cross product E×B always gives the traveling
direction of the EM-wave (k) — obey right-hand rule.
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4. E- & B-fields always vary sinusoidally. Moreover, B- &
E-fields vary with the same frequency (ω) and in phase
(in step: 初相位相等) with each other. Liquid Crystal Photonics16LAB
B
€k
P
k
k: EM-wave傳播方向
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k
Fig. 33-5
(a) An EM wave represented with a ray and two wavefronts; the wavefronts are
separated by one wavelength λ.
(b) The wave components E and B are in phase, perpendicular to each other, and
perpendicular to the wave’s direction (k) of travel.
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[Fig. 33-5] We assume that the EM wave is traveling
toward P in the +x direction, that the E-field is parallel
to the y axis, and that the B-field is parallel to the z
axis. Then we can write the E- and B-fields as
sinusoidal functions of position x and time t:
where
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From Ch. 16, we know that the speed of the wave is
v=ω/k. However, because this is an EM wave, its speed
(in vacuum) is given the symbol c rather than v.
The wave speed c and the amplitudes of the E- and Bfields are related by
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The magnitudes of the fields at every instant and at
any point are related by
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The waves we discussed in Chs. 16-17 require a medium
via which or along which to travel. We had waves
traveling along a string (string wave) and through the air
(sound wave).
However, an EM wave is curiously different in that it
requires no medium for its travel. It can, indeed, travel
through a medium (ε, μ) such as air or glass, but it can
also travel through the vacuum of space (ε0, μ0) between
a star and us.
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Once the special theory of relativity ( 狹 義 相 對 論 )
became accepted (Einstein, 1905) the speed of light
waves was realized to be special. One reason is that light
has the same speed regardless of the frame of reference
from which it is measured.
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If you send a beam of light along an axis and ask
several observers to measure its speed while they
move at different speeds along that axis, either in the
direction of the light or opposite it, they will all
measure the same speed for the light. The speed of
light (any EM wave) in vacuum has the exact value
c = 299,792,458 m/s ~ 3×108 m/s
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(略)
Derive Eqs. 33-3 and 33-4 yourselves from Faraday’s
law of induction and Maxwell’s law of induction:
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All sunbathers (日光浴者) know that an EM wave can
transport energy and deliver it to a body on which the
wave falls.
The rate of energy transport per unit area in such a
wave is described by a vector S, called the Poynting
vector (波印廷向量).
Its magnitude S is related to the rate at which energy is
transported by a wave across a unit area at any instant (單
位時間垂直通過單位面積的瞬間能量):
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Because E and B are perpendicular to each other in an EM
wave, the magnitude of E×B is EB. Then the magnitude of
S is
in which S, E, and B are instantaneous values.
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Most instruments (儀器設備) for detecting EM waves deal
with the electric component of the wave rather than the
magnetic component. Using B=E/c from Eq. 33-5, we can
rewrite Eq. 33-21 as
¨ S ∝ E2
By substituting E=Emsin(kx-ωt) into Eq. 33-22, we could
obtain an equation for the energy transport rate as a
function of time. ¨ S(t) ∝ Em2sin2(kx-ωt)
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More useful in practice is the average energy transported
over time [因為ㄧ般光偵測器最快反應時間為~ns，無法測
到瞬間值S(t) (可見光振盪週期~10-15s)，故實際上所測得的
皆屬平均值]; for that, we need to find the time-averaged
value of S(t), written Savg and also called the intensity (強
度) I of the wave. Thus from Eq. 33-20, the intensity I is
From Eq. 33-22 (S=E2/cμ0), we find
(½)
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where the average value of sin2(kx-ωt) is ½ over a full cycle
(參考 § 31-10).
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We define a new quantity Erms, the root-mean-square
value of the electric field, as
We can then rewrite I as
Because E=cB and c is such a very large number, you
might conclude that the energy associated with the E-field
is much greater than that associated with the B-field. That
conclusion is incorrect: the two energies are exactly equal.
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<Proof> Eq. 25-25 gives the energy density uE=ε0E2/2
within an electric field, and substitute cB for E
If we now substitute for c with Eq. 33-3, we get
However, we know that B2/2μ0 is the energy density uB
of a magnetic field B; so we see that uB=uE everywhere
along an EM wave.
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How intensity varies with distance from a real source of
EM radiation is often complex. However, in some situations
we can assume that the source is a point source that emits
the light isotropically ( 等 向 性 の ) — that is, with equal
intensity in all directions. The spherical wavefronts
spreading from such an isotropic point source S at a
particular instant are shown in cross section in Fig. 33-8.
imaginary sphere
Fig. 33-8
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Assume that the energy of the waves is
conserved as they spread from this source.
Let us also center an imaginary sphere of
radius r on the source.
All the energy emitted by the source must pass through
the imaginary sphere (radius=r). The rate at which energy
passes through the imaginary sphere via the radiation =
the rate at which energy is emitted by the source — that is,
the source power PS. The intensity I at the sphere must
then be, from Eq. 33-23,
where 4πr2 is the area of the sphere.
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EM waves have and transport linear momentum and
energy. This means that we can exert a pressure — a
radiation pressure (輻射壓力) — on an object by shining
light on it.
However, the pressure must be very small because, e.g.
you do not feel a camera flash when it is used to take your
photograph.
To find an expression for the radiation pressure, let us
shine a beam of EM radiation on an object for a time
interval Δt. Further, let us assume that the radiation is X
totally absorbed by the object. This means that during the
interval Δt, the object gains an energy ΔU from the
radiation.
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Maxwell showed that the object also gains linear
momentum. The magnitude Δp of the momentum change
of the object is related to the energy change ΔU by
Instead of being absorbed, the radiation can be Y reflected
by the object; that is, the radiation can be sent off in a
new direction as if it bounced off the object. If the
radiation is entirely reflected back along its original path,
the magnitude of the momentum change of the object is
twice that given above, or
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If the incident radiation is Z partly absorbed and partly
reflected, the momentum change of the object is between
ΔU/c and 2ΔU/c.
From Newton’s 2nd-law in its linear momentum form, we
know that a change in momentum is related to a force by
To find expressions for F exerted by EM-radiation in
terms of the intensity I of the radiation, we first note that
intensity is
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EM-radiation
A
Next, suppose that a flat surface of area A, perpendicular
to the path of the radiation, intercepts the radiation. In
time interval Δt, the energy intercepted by area A is
If the energy is X completely absorbed, then Eq. 33-28
tells us that Δp=IAΔt/c, and, from Eq. 33-30, the
magnitude of the force on the area A is
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If the radiation is Y totally reflected back along its original
path, Eq. 33-29 tells Δp=2IAΔt/c and, from Eq. 33-30,
If the radiation is Z partly absorbed and partly reflected,
the magnitude of the force on area A is between the
values of IA/c and 2IA/c.
The force per unit area on an object due to radiation is
the radiation pressure pr. We can find it for the situations
of Eqs. 33-32 and 33-33 by dividing both sides of each
equation by A. We obtain
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and
The development of laser technology has permitted (允
許 ) researchers to achieve radiation pressures much
greater than, say, that due to a camera flashlamp. This
comes about because a beam of laser light — unlike a
beam of light from a small lamp filament — can be focused
to a tiny spot. This permits the delivery of great amounts
of energy to small objects placed at that spot.
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Example of radiation pressure:
The tiny starlike speck is a minute
(1/1000 inch diameter) transparent
glass sphere suspended in midair
on an upward 250-mW laserbeam
(綠色箭頭).
Radiation force = Gravitational force
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Homework
Ans: (a)
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(b)
(c)
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Ans: (a)
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(b)
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Ans: (a)
(e)
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(h)
(b)
(c)
(f)
(d)
(g)
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Ans:
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Ans: (a) P =
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