28 Maxwell's equation and-2
最伟大的数学公式排名
最伟大的数学公式排名数学公式是数学的基础，它们用于描述各种概念、关系和规律。
但是，要评价哪个数学公式最伟大是主观的，因为每个公式都有其独特的背景和重要性。
以下是一些被广泛认为是非常重要和有影响的数学公式。
1. 欧拉公式（Euler's formula）：这个公式将三角函数、指数函数和复数关联起来，是数学中最重要的公式之一。
2. 牛顿-莱布尼兹公式（Newton-Leibniz formula）：这个公式用于计算定积分，是微积分的基础。
3. 泰勒公式（Taylor series）：这个公式用于展开函数，是分析数学的重要工具。
4. 欧拉-麦克劳林公式（Euler-Maclaurin formula）：这个公式将欧拉和麦克劳林两种求和的方法结合起来，用于求和和积分。
5. 贝叶斯定理（Bayes' theorem）：这个公式用于概率推理和统计推断，是贝叶斯方法的基石。
6. 傅里叶变换（Fourier transform）：这个公式将时间和空间域上的函数转换为频率域上的函数，是信号处理和图像处理领域的基础。
7. 拉普拉斯变换（Laplace transform）：这个公式将时间域上的函数转换为复平面上的函数，用于求解偏微分方程。
8. 欧拉-雅可比公式（Euler-Jacobi formula）：这个公式用于求解变分问题，是优化理论和力学中的重要工具。
9. 麦克斯韦方程组（Maxwell's equations）：这个方程组描述了电磁场的运动规律，是电动力学的基础。
10. 薛定谔方程（Schrödinger equation）：这个方程描述了量子力学中微观粒子的运动状态，是量子力学的基础。
这些公式都是数学中的重要工具，对于不同的领域有着不同的影响和应用。
因此，无法简单地评价哪个公式最伟大，它们各自在数学和科学中扮演着重要的角色。
麦克斯韦电磁方程
麦克斯韦电磁方程
麦克斯韦电磁方程，又称麦克斯韦方程，是20世纪初期热激励学派物理学家查尔斯特·麦克斯韦于1864年在他的著作《223课中对磁力学部分的建构》中提出的电磁学基本原理，它是电磁学中重要的基本公式，是把电磁学所有基础理论连接在一起的基础。
麦克斯韦电磁方程具有四个分支，分别是相对论电磁交互方程（Maxwell’s equations of the relativistic Electromagnetism）、非相对论电磁交互方程（Maxwell’s equations of the non-relativistic Electromagnetism）、伽马方程（the equation of gamma）以及静电场方程（the equation of electrostatic field）。
其中，四个麦克斯韦方程可以用一个总的式子描述：
\oint_\mathrm{c} \vec E \cdot d\vec l= - \frac{1}{c} \frac{\partial}{\partial t}\int_\mathrm{S}\vec B \cdot d\vec S 它由C积分表示，即分别表示某一点处的电场强度\vec E ，磁场强度\vec B ，磁环曲率c 以及时间t。
式中，只要存在选定曲线C 以及面积S，就能求出E 和B 的变化关系，即可求出相应空间处的传播速度c（光速）。
麦克斯韦方程表明电磁场和电流之间的联系，只要一定环境下存在电流，就会在相应空间形成电磁场；只要存在电磁场，就能在相应空间产生电流。
这也给把电磁理论应用到日常生活（例如电力、电磁机械）中提供了重要的理论基础，并且在磁波通信、高能物理学和天文学等许多领域有着深远的影响。
世界上最伟大的十大公式
几年前英国科学期刊《物理世界》曾让读者投票评选了“最伟大的公式”，最终榜上有名的十个公式既有无人不知的1+1=2，又有著名的E=mc^2;既有简单的-圆周公式，又有复杂的欧拉公式……这些公式不仅仅是数学家和物理学家的智慧结晶，更是人类文明的集中体现。
每一个公式都深深影响了人类社会的变革，甚至塑造了人类的思想。
这些公式中有些你会很熟悉，有些你也许不那么熟悉。
作为人类的我们有必要了解这些公式，了解人类的思想历程。
对于电子工程师来说，傅里叶变换、麦克斯韦方程都是最应该了解和掌握的。
今天我们从网络收集整理了这10著名公式，分享给大家:No.10 圆的周长公式(The Length of the Circumference of a Circle)创立者:古人意义:自然界之美的数学表达。
这公式贼牛逼了，初中学到现在。
目前，人类已经能得到圆周率的2061亿位精度。
还是挺无聊的。
现代科技领域使用的-圆周率值，有十几位已经足够了。
如果用35位精度的-圆周率值，来计算一个能把太阳系包起来的一个圆的周长，误差还不到质子直径的百万分之一。
现在的人计算圆周率，多数是为了验证计算机的计算能力，还有就是为了兴趣。
No.9 傅立叶变换(The Fourier Transform)创立者:让·巴普蒂斯·约瑟夫·傅立叶意义:任何不规则的信号都可以表示为规则的正弦波无限叠加。
它是数字信号处理领域的很重要的方法。
这个挺专业的，一般人完全不明白。
不多作解释。
简要地说没有这个式子没有今天的电子计算机，所以你能在这里上网除了感谢党感谢政府还要感谢这个完全看不懂的式子。
另外傅立叶虽然姓傅，但是法国人。
No.8 德布罗意方程组(The de Broglie Relations)创立者:路易·维克多·德布罗意意义:德布罗意认为，任何物质既有粒子性，又有波动性，或者说，任何物质也可以看成是一种波，包括人本身。
天线的性能外文文献翻译、中英文翻译
目录PROPERTIES OF ANTENNAS . (1)1.1ANTENNA RADIATION (2)1.2GAIN . (4)1.3EFFECTIVE AREA (6)1.4PATH LOSS (7)1.5RADAR RANGE EQUATION AND CROSS SECTION (9)1.6WHY USE AN ANTENNA? (11)天线的性能. (12)1.1天线辐射 (13)1.2增益. (14)1.3有效面积 (16)1.4路径损耗 (17)1.5雷达距离方程和截面 (19)1.6为什么要使用一个天线？ (20)英文原文：PROPERTIES OF ANTENNASOne approach to an antenna book starts with a discussion of how antennas radiate. Beginning with Maxwell 'es quations, we derive electromagnetic waves. After that lengthy discussion, which contains a lot of mathematics, we discuss how these waves excite currents on conductors. The second half of the story is that currents radiate and produce electromagnetic waves. You may already have studied that subject, or if you wish to further your background, consult books on electromagnetics.The study of electromagnetics gives insight into the mathematics describing antenna radiation and provides the rigor to prevent mistakes. We skip the discussion of those equations and move directly to practical aspects.It is important to realize that antennas radiate from currents. Design consists of controlling currents to produce the desired radiation distribution, called its pattern .In many situations the problem is how to prevent radiation from currents, such as in circuits. Whenever a current becomes separated in distance from its return current, it radiates. Simply stated, we design to keep the two currents close together, to reduce radiation. Some discussions will ignore the current distribution and instead, consider derived quantities, such as fields in an aperture or magnetic currents in a slot or around the edges of a microstrip patch. You will discover that we use any concept that provides insight or simplifies the mathematics.An antenna converts bound circuit fields into propagating electromagnetic waves and, by reciprocity, collects power from passing electromagnetic waves. Maxwell ' s equations p that any time-varying electric or magnetic field produces the opposite field and forms an electromagnetic wave. The wave has its two fields oriented orthogonally, and it propagates in the direction normal to the plane defined by the perpendicular electric and magnetic fields. The electric field, the magnetic field, and the direction of propagation form a right-handed coordinate system. The propagating wave field intensity decreasesb y 1/R away from the source, whereas a static field drops off by 1/R2. Any circuit with time-varying fields has the capability of radiating to some extent.We consider only time-harmonic fields and use phasor notation with time dependence e jwt. An outward-propagating wave is given by e j(kR wt), where k, the wave number, is given by 2 π / λ . λ is the wavelength of the wave given by c/f , where c is the velocity of light (38×10 m/s in free space) and f is the frequency. Increasing the distance from the source decreases thephase of the wave.Consider a two-wire transmission line with fields bound to it. The currents on a single wire will radiate, but as long as the ground return path is near, its radiation will nearly cancel the other line 's radiation becauseo tahree t1w80 ou°t of phase and the waves travel about thesame distance. As the lines become farther and farther apart, in terms of wavelengths, the fields produced by the two currents will no longer cancel in all directions. In some directions the phase delay is different for radiation from the current on each line, and power escapes from the line. We keep circuits from radiating by providing close ground returns. Hence, high-speed logic requires ground planes to reduce radiation and its unwanted crosstalk.1.1ANTENNA RADIATIONAntennas radiate spherical waves that propagate in the radial direction for a coordinate system centered on the antenna. At large distances, spherical waves can be approximated by plane waves. Plane waves are useful because they simplify the problem. They are not physical, however, because they require infinite power.The Poynting vector describes both the direction of propagation and the power density of the electromagnetic wave. It is found from the vector cross product of the electric and magnetic fields and is denoted S:2S = E H×* W/m2Root mean square (RMS) values are used to express the magnitude of the fields. H* is the complex conjugate of the magnetic field phasor. The magnetic field is proportional to the electric field in the far field. The constant of proportion is η , the impedance of free 376.73 Ω ):E 2 2S S W/ m2(1.1)Because the Poynting vector is the vector product of the two fields, it is orthogonal to both fields and the triplet defines a right-handed coordinate system: (E, H, S).Consider a pair of concentric spheres centered on the antenna. The fields around the 23antenna decrease as 1/R, 1R/ , 1/R , and so on. Constant-order terms would require that the power radiated grow with distance and power would not be conserved. For field terms 23proportional to 1/ R , 1/R , and higher, the power density decreases with distance faster thanthe area increases. The energy on the inner sphere is larger than that on the outer sphere. The energies are not radiated but are instead concentrated around the antenna; they are near-field 2terms. Only the 1/R term of the Poynting vector (1/R field terms) represents radiated power2because the sphere area grows as R and gives a constant product. All the radiated power flowingthrough the inner sphere will propagate to the outer sphere. The sign of the input reactance depends on the near-field predominance of field type: electric (capacitive) or magnetic (inductive). At resonance (zero reactance) the stored energies due to the near fields are equal. Increasing the stored fields increases the circuit Q and narrows the impedance bandwidth.Far from the antenna we consider only the radiated fields and power density. The powerflow is the same through concentric spheres:2The average power density is proportional to 1/R . Consider differential areas on the two spheres at the same coordinate angles. The antenna radiates only in the radial direction; therefore, no power may travel in the θ or φ direction. Power travels in flux tubes between areas, and it follows that not only the average Poynting vector but also every part of the power2density is proportional to 1/R :22S 1R 1 sin d d S 2R 2 sin d dSince in a radiated wave S is proportional to 1R / 2 , E is proportional to 1/R. It is convenient to define radiation intensity to remove the 1/R 2dependence: U(θ, φ ) = S(R, R θ2, φW) /solid angle Radiation intensity depends only on the direction of radiation and remains the same at all distances. A probe antenna measures the relative radiation intensity (pattern) by moving in a circle (constant R) around the antenna. Often, of course, the antenna rotates and the probe is stationary.Some patterns have established names. Patterns along constant angles of the spherical coordinates are called either conical (constant θ ) or great circle (constantφ ). Thcuts when φ = 0 ° or φ = 90 ° are the principal plane patterns. Other named cuts are also used, but their names depend on the particular measurement positioner, and it is necessary to annotate these patterns carefully to avoid confusion between people measuring patterns on different positioners. Patterns are measured by using three scales: (1) linear (power), (2)2 4 R 12S 1 ,avg 4 R 2 S 2, avgsquare root (field intensity), and (3) decibels (dB). The dB scale is used the most because it reveals more of the low-level responses (sidelobes).Figure 1.1 demonstrates many characteristics of patterns. The half-power beamwidth is sometimes called just the beamwidth. The tenth-power and null beamwidths are used in some applications. This pattern comes from a parabolic reflector whose feed is moved off the axis. The vestigial lobe occurs when the first sidelobe becomes joined to the main beam and forms a shoulder. For a feed located on theaxis of the parabola, the first sidelobes are equal. 1.2 GAINGain is a measure of the ability of the antenna to direct the input power into radiation in aparticular direction and is measured at the peak radiation intensity. Consider the power2 densityradiated by an isotropic antenna with input power Po at a distance R: S = Po/4 R . An isotropic antenna radiates equally in all directions, and its radiated power density S is found by dividing the radiated power by the area of thes phere 4 R π2. The isotropic radiator is considered to be 100% efficient. The gain of an actual antenna increases the power density in the direction of the peak radiation:Gain is achieved by directing the radiation away from other parts of the radiation sphere. Ingeneral, gain is defined as the gain-biased pattern of the antenna:SP 0GS 2 4 R 2 or (1.2)P 0G4 R 2 power densityP 0G , 4radiation intensity (1.3)1 4where Pr is the radiated power. Material losses in the antenna or reflected power due to poor impedance match reduce the radiated power. In this book, integrals in the equation above and those that follow express concepts more than operations we perform during design. Only for theoretical simplifications of the real world can we find closed-form solutions that would call for actual integration. We solve most integrals by using numerical methods that involve breaking the integrand into small segments and performing a weighted sum. However, it is helpful that integrals using measured values reduce the random errors by averaging, which improves the result.In a system the transmitter output impedance or the receiver input impedance may not match the antenna input impedance. Peak gain occurs for a receiver impedance conjugate matched to the antenna, which means that the resistive parts are the same and the reactive parts are the same magnitude but have opposite signs. Precision gain measurements require atuner between the antenna and receiver to conjugate-match the two. Alternatively, theP rP 0 2 G , sin d d 0 0 4 efficiencyFIGURE 1.1 Antenna pattern characteristics.The surface integral of the radiation intensity over the radiation sphere divided by the input power Po is a measure of the relative power radiated by the antenna, or the antennamismatch loss must be removed by calculation after the measurement. Either the effect ofmismatches is considered separately for a given system, or the antennas are measured into the system impedance and mismatch loss is considered to be part of the efficiency.Example Compute the peak power density at 10 km of an antenna with an input power of 3 W and a gain of 15 dB.1510First convert dB gain to a ratio: G = 10 10= 31.62. The power spreads over the sphere area with radius 10 km or an area of 4π(104 )2 m 2. The power density is 3W (31.62) 75.5nW /m 2We calculate the electric field intensity using Eq. (1-2):ES 75.5 10 9376.7 5333 V /m Although gain is usually relative to an isotropic antenna, some antenna gains are referred to a λ /2 dipole with an isotropic gain of 2.14 dB.If we approximate the antenna as a point source, we compute the electric field radiated by using Eq. (1.2):e jkRP 0GR4 This requires only that the antenna be small compared to the radial distance R. Equation (1.4) ignores the direction of the electric field, which we define as polarization. The units of the electric field are volts/meter. We determine the far-field pattern by multiplying Eq. (1.4) by R and removing the phase term e jkRsince phase has meaning only when referred to another point in the far field. The far-field electric field E ff unit is volts:During analysis, we often normalize input power to 1 W and can compute gain easily from the electric field by multiplying by a constant 4 = 0.1826374.1.3 EFFECTIVE AREAAntennas capture power from passing waves and deliver some of it to the terminals. Given the power density of the incident wave and the effective area of the antenna, the power 4 108m 2(1.4)E ffP 0G or G 1 E ff P 0 (1.5)delivered to the terminals is the product.(1.6)For an aperture antenna such as a horn, parabolic reflector, or flat-plate array, effective area is physical area multiplied by aperture efficiency. In general, losses due to material, distribution, and mismatch reduce the ratio of the effective area to the physical area. Typical estimatedaperture efficiency for a parabolic reflector is 55%. Even antennas with infinitesimal physical areas, such as dipoles, have effective areas because they remove power from passing waves.1.4 PATH LOSSWe combine the gain of the transmitting antenna with the effective area of the receivingantenna to determine delivered power and path loss. The power density at the receivingantenna is given by Eq. (1.3), and the received power is given by Eq. (1.6). By combining the two, we obtain the path loss:P d A 2G 1 ,P t 4 R 2Antenna 1 transmits, and antenna 2 receives. If the materials in the antennas are linearand isotropic, the transmitting and receiving patterns are identical (reciprocal) [2, p. 116]. When we consider antenna 2 as the transmitting antenna and antenna 1 as the receivingantenna, the path loss isP d A 1G 2 , 2 P t4 R 2Since the responses are reciprocal, the path losses are equal and we can gather and eliminate terms:G 1 G 2= constantA 1 A 2 Because the antennas were arbitrary, this quotient must equal a constant. This constant was found by considering the radiation between two large apertures [3]:(1.7)We substitute this equation into path loss to express it in terms of the gains or effective areas:P d GG 2A 1A 2 P t G 1G 2 4 R 2R 2P d SA eff(1.8)We make quick evaluations of path loss for various units of distance R and for frequency f in megahertz using the formula. path loss(dB)=K U 20log fR G 1 dB G 2 dB (1.9) where Ku depends on the length units:Example Compute the gain of a 3-m-diameter parabolic reflector at 4 GHz assuming 55% aperture efficiency.Gain is related to effective area by Eq. (1.7):4A2 We calculate the area of a circular aperture byA 2 D/2 . By combining these equations,we have Df c where D is the diameter and a is the aperture efficiency. On substituting the values above, we obtain the gain: 923 4 109G 9 0.55 8685 (39.4dB) 0.3 10 G D 2 a a (1.10) Example Calculate the path loss of a 50-km communication link at 2.2 GHz using a transmitter antenna with a gain of 25 dB and a receiver antenna with a gain of 20 dB. Path loss = 32.45 + 20 log[2200(50)] - 25 - 20 = 88.3 dBWhat happens to transmission between two apertures as the frequency is increased? If we assume that the effective area remains constant, as in a parabolic reflector, the transmission increases as the square of frequency: P d A 1A 2 1 P t R 2 2 A 1A 2 f 2 R 2Bf 2where B is a constant for a fixed range. The receiving aperture captures the same power regardless of frequency, but the gain of the transmitting antenna increases as the square of frequency. Hence, the received power also increases as frequency squared. Only for antennas, whose gain is a fixed value when frequency changes, does the path loss increase as the square of frequency.1.5 RADAR RANGE EQUATION AND CROSS SECTIONRadar operates using a double path loss. The radar transmitting antenna radiates a field that illuminates a target. These incident fields excite surface currents that also radiate to produce a second field. These fields propagate to the receiving antenna, where they arecollected. Most radars use the same antenna both to transmit the field and to collect the signal returned, called a monostatic system, whereas we use separate antennas for bistatic radar. The receiving system cannot be detected in a bistatic system because it does not transmit and has greater survivability in a military application.We determine the power density illuminating the target at a rangeR T by using Eq. (1.2):The target ' s radar creocstsio sn (RCS), the scattering area of the object, is expressed in square2meters or dB m : 10 log(square meters). The RCS depends on both the incident and reflected wave directions. We multiply the power collected by the target with its receiving pattern by the gain of the effective antenna due to the currents induced:In a communication system we call Ps the equivalent isotropic radiated power (EIRP), which equals the product of the input power and the antenna gain. The target becomes thetransmitting source and we apply Eq. (1.2) to find the power density at the receiving antenna at a range R R from the target. Finally, the receiving antenna collects the power density with (1.11)S inc P T G T4 R T 2RCS power reflected P s r , r , i , ipowerdensityincident P T G T 4 R T 2 (1.12)an effective areaA R . We combine these ideas to obtain the power delivered to the receiver: We apply Eq. (1.7) to eliminate the effective area of the receiving antenna and gather terms to determine the bistatic radar range equation:P rec G T G Rr , r , i , i P T 4 3 R T 2R R 2 We reduce Eq. (1.13) and collect terms for monostatic radar, where the same antenna is used for both transmitting and receiving:42 Radar received power is proportional to 1/Rand to G . We find the approximate RCS of a flat plate by considering the plate as an antenna with an effective area. Equation (1.11) gives the power density incident on the plate that collects this power over an areaA R :The power scattered by the plate is the power collected,P C , times the gain of the plate as an antenna,G P :This scattered power is the effective radiated power in a particular direction, which in an antenna is the product of the input power and the gain in a particular direction. We calculate the plate gain by using the effective area and find the scattered power in terms of area:P P T G T 4 A RP s 4 R T 2 2We determine the RCS σ Ebqy . (1 .12), the scattered power divided by the incident power density:22P s 4 A R G R i , i G R r , r (1.14)P T G T s 4 R T 2 2 R R i i 4R r r (1.14)The right expression of Eq. (1.14) divides the gain into two pieces for bistatic scattering, where the scattered direction is different from the incident direction. Monostatic scattering uses the same incident and reflected directions. We can substitute any object for the flat plate and use the idea of an effective area and its associated antenna gain. An antenna is an objectP rec S R A RA R P T G T4 R T 2 4 R R 2(1.13)P rec G 2 2P T344 3 R 4 P CP T G T , 4 R T 2 A RP s P C G PP T G T i , i 4 R T 2 A R G P rrwith a unique RCS characteristic because part of the power received will be delivered to the antenna terminals. If we provide a good impedance match to this signal, it will not reradiate and the RCS is reduced. When we illuminate an antenna from an arbitrary direction, some of the incident power density will be scattered by the structure and not delivered to the antenna terminals. This leads to the division of antenna RCS into the antenna mode of reradiated signals caused by terminal mismatch and the structural mode, the fields reflected off the structure for incident power density not delivered to the terminals.1.6WHY USE AN ANTENNA?We use antennas to transfer signals when no other way is possible, such ascommunication with a missile or over rugged mountain terrain. Cables are expensive and take a long time to install. Are there times when we would use antennas over level ground? The large path losses of antenna systems lead us to believe that cable runs are better.中文译文：天线的性能天线的方法一书是从天线辐射的讨论开始。
【原创1.11】南京邮电大学专业英语词汇释义精选
medium // 介质 metallic // 金属的 method of moments // 矩 量法
MF/ // 中频 MIC/ // 微波集成电路 microstrip // 微带 minority carrier // 少数载 流子
第1页共2页
南京邮电大学 电磁场与无线技术 专业英语词汇释义 精选
net power // 净功率 newsletter // 通讯 N-LOS/ / 非视距 nonconductive // 不导电 nonresonant // 非谐振 nonthermal // 非热的 nonuniform plane // 非均 匀平面
ohm // 欧姆 ohmic // 阻值 Ohm's law // 欧姆定律 open-circuited // 开路 optics // 光学 orientation // 定向 orthonormal bases // 正交 基
oscillator // 振荡器 oscilloscope // 示波器
C
cable // 电缆 capacitance // 电容 cathode // 阴极 cavity // 空穴 空腔 ceiver // 接收器 cellular // 蜂窝网 channelization // 信道化
charge // 电荷 circuit // 电路 circulator // 环行器 coaxial probe // 同轴探针 coil // 线圈 cold cathode // 冷阴极管 compatibility // 兼容性 component // 元件 compound // 化合物的 conductance // 电导 conductivity // 电导率 conductor // 导体 conformal // 共形 conjugate // 共轭 constitutive relationship // 本构关系
麦克斯韦方程 讲义说明书
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Sources of electromagnetic fields
Differential form of Maxwell’s equation
Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation
Some clarifications on all four equations
)
Moving charge creates magnetic field
－ Typically charge and current densities are utilized in Maxwell’s equations
to quantify the effects of fields:
1. Uniform field
Gauss: Integration over closed surface
= ∙ = ∙ = ∙ ∙ []
- angle between field and normal vector
to surface matters
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ρ = electric charge density – total electric charge per unit volume V
(or =
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=
)
()
lim electric
→0
(or =
current density – total electric current per unit area S
= 0
µ0 =permeability of free space
高等电动力学
第一页，共61页。
Chapter 1 Basic Electromagnetic Theory
• Experimental laws in electromagnetism • Maxwell’s hypothesis of displacement current • Basic Maxwell’s equations in integral form • Basic Maxwell’s equations in differential form • Maxwell’s equations in material media • Complex Maxwell’s equations • Boundary conditions • Wave equations and Helmholtz’s equations • Poynting’s theorem
Faraday's concept of field (1831)
: magneic field strength, instantaneous vector function
Gauss's law for magnetic fields
Later, people knew that in Columb’s experiment, magnetic poles of magnet are the effects of molecular current loops, which can be recognized as charge pairs or magnetic dipoles, but not the effects of single magnetic charges.
第二页，共61页。
激光专业英语
2011年技术物理学院08级（激光方向）专业英语翻译重点！！！作者：邵晨宇Electromagnetic电磁的principle原则principal主要的macroscopic宏观的microscopic微观的differential微分vector矢量scalar标量permittivity介电常数photons光子oscillation振动density of states态密度dimensionality维数transverse wave横波dipole moment偶极矩diode 二极管mono-chromatic单色temporal时间的spatial空间的velocity速度wave packet波包be perpendicular to线垂直be nomal to线面垂直isotropic各向同性的anistropic各向异性的vacuum真空assumption假设semiconductor半导体nonmagnetic非磁性的considerable大量的ultraviolet紫外的diamagnetic抗磁的paramagnetic顺磁的antiparamagnetic反铁磁的ferro-magnetic铁磁的negligible可忽略的conductivity电导率intrinsic本征的inequality不等式infrared红外的weakly doped弱掺杂heavily doped重掺杂a second derivative in time对时间二阶导数vanish消失tensor张量refractive index折射率crucial主要的quantum mechanics 量子力学transition probability跃迁几率delve研究infinite无限的relevant相关的thermodynamic equilibrium热力学平衡（动态热平衡）fermions费米子bosons波色子potential barrier势垒standing wave驻波travelling wave行波degeneracy简并converge收敛diverge发散phonons声子singularity奇点（奇异值）vector potential向量式partical-wave dualism波粒二象性homogeneous均匀的elliptic椭圆的reasonable公平的合理的reflector反射器characteristic特性prerequisite必要条件quadratic二次的predominantly最重要的gaussian beams高斯光束azimuth方位角evolve推到spot size光斑尺寸radius of curvature曲率半径convention管理hyperbole双曲线hyperboloid双曲面radii半径asymptote渐近线apex顶点rigorous精确地manifestation体现表明wave diffraction波衍射aperture孔径complex beam radius复光束半径lenslike medium类透镜介质be adjacent to与之相邻confocal beam共焦光束a unity determinant单位行列式waveguide波导illustration说明induction归纳symmetric 对称的steady-state稳态be consistent with与之一致solid curves实线dashed curves虚线be identical to相同eigenvalue本征值noteworthy关注的counteract抵消reinforce加强the modal dispersion模式色散the group velocity dispersion群速度色散channel波段repetition rate重复率overlap重叠intuition直觉material dispersion材料色散information capacity信息量feed into 注入derive from由之产生semi-intuitive半直觉intermode mixing模式混合pulse duration脉宽mechanism原理dissipate损耗designate by命名为to a large extent在很大程度上etalon 标准具archetype圆形interferometer干涉计be attributed to归因于roundtrip一个往返infinite geometric progression无穷几何级数conservation of energy能量守恒free spectral range自由光谱区reflection coefficient（fraction of the intensity reflected）反射系数transmission coefficient（fraction of the intensity transmitted）透射系数optical resonator光学谐振腔unity 归一optical spectrum analyzer光谱分析grequency separations频率间隔scanning interferometer扫描干涉仪sweep移动replica复制品ambiguity不确定simultaneous同步的longitudinal laser mode纵模denominator分母finesse精细度the limiting resolution极限分辨率the width of a transmission bandpass透射带宽collimated beam线性光束noncollimated beam非线性光束transient condition瞬态情况spherical mirror 球面镜locus（loci）轨迹exponential factor指数因子radian弧度configuration不举intercept截断back and forth反复spatical mode空间模式algebra代数in practice在实际中symmetrical对称的a symmetrical conforal resonator对称共焦谐振腔criteria准则concentric同心的biperiodic lens sequence双周期透镜组序列stable solution稳态解equivalent lens等效透镜verge 边缘self-consistent自洽reference plane参考平面off-axis离轴shaded area阴影区clear area空白区perturbation扰动evolution渐变decay减弱unimodual matrix单位矩阵discrepancy相位差longitudinal mode index纵模指数resonance共振quantum electronics量子电子学phenomenon现象exploit利用spontaneous emission自发辐射initial初始的thermodynamic热力学inphase同相位的population inversion粒子数反转transparent透明的threshold阈值predominate over占主导地位的monochromaticity单色性spatical and temporal coherence时空相干性by virtue of利用directionality方向性superposition叠加pump rate泵浦速率shunt分流corona breakdown电晕击穿audacity畅通无阻versatile用途广泛的photoelectric effect光电效应quantum detector 量子探测器quantum efficiency量子效率vacuum photodiode真空光电二极管photoelectric work function光电功函数cathode阴极anode阳极formidable苛刻的恶光的irrespective无关的impinge撞击in turn依次capacitance电容photomultiplier光电信增管photoconductor光敏电阻junction photodiode结型光电二极管avalanche photodiode雪崩二极管shot noise 散粒噪声thermal noise热噪声1.In this chapter we consider Maxwell’s equations and what they reveal about the propagation of light in vacuum and in matter. We introduce the concept of photons and present their density of states.Since the density of states is a rather important property，not only for photons，we approach this quantity in a rather general way. We will use the density of states later also for other(quasi-) particles including systems of reduced dimensionality.In addition,we introduce the occupation probability of these states for various groups of particles.在本章中，我们讨论麦克斯韦方程和他们显示的有关光在真空中传播的问题。
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• Suppose that the displacement current were uniformly distributed over the entire area of the capacitor, and we can simply use •For CI the entire displacement current is enclosed by an imaginary closed path; for CII only 1/4 of the total displacement current is enclosed. Therefore, BII(r0) = BI(r0)/4.
I
Compare the result with the distribution of magnetic field due to a long cylinder carrying current
R
I
I
ACT Suppose that at time t the currents flowing into capacitors CI and CII are identical, and that CII has twice the radius (and 4 times the area) of CI , as shown.
Path l
The Electric Field E between the plates of the capacitor is determined by the charge Q on the plate of area A:
Because there is current flowing through the wire, there must be a change in the charge on the plates:
x Ex z1 By y Ex Dx z
DZ
z2
Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane
x Ex DZ z1 z2
z Dy
y
By
By
Step 4 Combine results from steps 2 and 3 to eliminate By the wave equation One solution is the harmonic plane wave propagating in the +z direction
IV. Faraday’s law: changing magnetic flux creates electric field
28-3 Electromagnetic Waves
Note the symmetry of Maxwell’s Equations in free space, meaning when no charges or currents are present
Current Sheet
Magnetic Field of Closely Spaced Parallel Currents
Current oscillates in the x-direction Magnetic field oscillates in the y-direction Electric field oscillates in the x-direction Plane waves will propagate in the z-direction
a closed path
In the figure we have two (imaginary) surfaces that are bounded by a closed path, (i.e., surface 1 a flat disk, surface 2 like a little pouch). A wire carrying current I passes through each of them. The line integral of the magnetic field is the same. (the current enclosed by the path is the same)
We could derive for By:
→ Energy
→ Energy flow
→ Momentum
• Dipole Radiation
• Electromagnetic Radiation as Particles
28-1 Maxwell’s Displacement Current Applying Ampere’s law
1
2 I
Consider applying Ampere’s Law to the current Path l shown in the diagram.
If the surface is chosen as S1,the enclosed current = I If the surface is chosen as S2, the enclosed current = 0 (i.e., there is no current between the plates of the capacitor) If currents are not constant in time, the concept of the current enclosed by a path becomes ambiguous.
(1)Compare the displacement current for the two cases.
(a) ID,I < ID,II (b) ID,I = ID,II
(c) ID,I > ID,II
ro
I
2ro
CI
ro
I
• Although there is no actual current flowing between the capacitor plates, the displacement is always equal to the real current. • Therefore, since the two capacitors have the same real current, they must have the same total displacement current.
Chapter 28 Maxwell’s Equations And Electromagnetic Waves
Review
?
Main Points of Chapter 28
• Maxwell’s Displacement Current
• Maxwell’s Equations
• Electromagnetic Waves
•Only accelerating charges will create electromagnetiபைடு நூலகம் waves • Once created they can continue to propagate without further input
The Propagation of Electromagnetic Waves
Electric and magnetic fields are coupled through Ampère’s and Faraday’s laws
A changing magnetic field induces a changing electric field, and vice versa
CII
(2) Compare the magnetic fields at a radial distance r0 from the axes of CI and CII.
ro
I
2ro
CI
ro
(a) BI(r0) < BII(r0)
(b) BI(r0)= BII(r0)
I CII
(c) BI(r0) > BII(r0)
I. Gauss’ law for electric fields: equivalent to Coulomb’s law
II. Gauss’ law for magnetic fields: no magnetic monopole
III. Generalized Ampè re’s law: changing electric flux creates magnetic field
But where does the “displacement current” come from?
Although there is no actual charge moving between the plates, nevertheless, something is changing – the electric field between them!
In order to have
Maxwell proposed there was an extra “displacement current” in the region between the plates, equal to the current in the wire Modified Ampere’s law
Maxwell’s Displacement Current
Modified Ampere’s Law:
Discussion (1)The surface through which the sum of I and Id passes is any surface whose boundary is the closed surface.