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Transcript
Ch 33 Electromagnetic Waves I 1 1 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 2 part II → Optics 2 Liquid Crystal Photonics LAB 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) 3 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). 3 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 4 4 Liquid Crystal Photonics LAB 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. 5 5 Liquid Crystal Photonics LAB [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. 6 6 Liquid Crystal Photonics LAB 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). 7 7 Liquid Crystal Photonics LAB 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? 8 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). 9 9 Liquid Crystal Photonics LAB 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. 10 Liquid Crystal Photonics10 LAB 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. 11 Liquid Crystal Photonics11 LAB +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. 12 Liquid Crystal Photonics12 LAB 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. 13 Liquid Crystal Photonics13 LAB 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. 14 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 14 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 15 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 Liquid Crystal Photonics LAB 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. 16 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 17 Liquid Crystal Photonics17 LAB 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. 18 Liquid Crystal Photonics18 LAB [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 19 Liquid Crystal Photonics19 LAB 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 20 Liquid Crystal 20 The magnitudes of the fields at every instant and at any point are related by 21 Liquid Crystal Photonics21 LAB 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. 22 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. Liquid Crystal Photonics22 LAB 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 23 Liquid Crystal Photonics23 LAB (略) Derive Eqs. 33-3 and 33-4 yourselves from Faraday’s law of induction and Maxwell’s law of induction: 24 Liquid Crystal Photonics24 LAB 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): 25 Liquid Crystal Photonics25 LAB 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. 26 Liquid Crystal Photonics26 LAB 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) 27 Liquid Crystal Photonics LAB 27 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) 28 where the average value of sin2(kx-ωt) is ½ over a full cycle (參考 § 31-10). Liquid Crystal Photonics28 LAB 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. 29 Liquid Crystal Photonics29 LAB <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. 30 Liquid Crystal Photonics30 LAB 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 31 Liquid Crystal Photonics31 LAB 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. 32 Liquid Crystal Photonics32 LAB 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. 33 Liquid Crystal Photonics33 LAB 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 34 Liquid Crystal Photonics34 LAB 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 35 Liquid Crystal Photonics35 LAB 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 36 Liquid Crystal Photonics36 LAB 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 37 Liquid Crystal Photonics37 LAB 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. 38 Liquid Crystal Photonics38 LAB 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 39 Liquid Crystal Photonics39 LAB Homework Ans: (a) 40 (b) (c) 40 Ans: (a) 41 (b) 41 Ans: (a) (e) 42 (h) (b) (c) (f) (d) (g) 42 Ans: 43 43 Ans: (a) P = 44 44