7.2 Faraday’s Law of Electromagnetic Induction 7.3 Maxwell’s Equations

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1 7.2 Faraday’s Law of Electromagnetic Induction 7.3 Maxwell’s Equations
D. K. Cheng Field and Wave Electromagnetics Chapter 7 Time-Varying Fields and Maxwell’s Equations 7.1 Introduction 7.2 Faraday’s Law of Electromagnetic Induction 7.3 Maxwell’s Equations 7.4 Potential Functions 7.5 Electromagnetic Boundary Equations 7.6 Wave Equations and Their Solutions 7.7 Time Harmonics Fields

2 Fundamental relations for electrostatic and magnetostatic models
7.1 Introduction Fundamental relations for electrostatic and magnetostatic models - In the static case (non-time-varying), electric vectors E and D are independent of magnetic vectors B and H. - In a conductive medium, electric and magnetic fields may both exist and form an electromagnetostatic field.

3 7.2 Faraday’s Law of Electromagnetic Induction
Faraday’s Law (1831): the quantitative relationship between the induced emf (electromotive force, 电动势) and the rate of change of magnetic flux linking a conducting loop. Fundamental postulate for Electromagnetic Induction: Michael Faraday (1791—1867) - It applies to every point in the space. E is nonconservative in a region of time-varying magnetic flux density. 现在，E的旋度不为零，故而E不再是保守的了。 The integral form over a surface: （with the help of the Stokes theorem）

4 7.2.1 A Stationary Circuit in a Time-varying Magnetic Field
For a stationary circuit with a contour C and surface S, we have (7-3) If we define,

5 Faraday’s Law of Electromagnetic Induction
The electromotive force induced in a stationary closed circuit is equal to the negative rate of increase of the magnetic flux linking the circuit. Lenz’s Law: the induced current in the closed loop in such a direction as to oppose the change in the linking magnetic flux. Transformer emf: the emf induced in a stationary loop caused by a time-varying magnetic field. 楞次定律：感应电流具有这样的方向，即感应电流的磁场总要阻碍引起感应电流的磁通量的变化。 楞次定律还可表述为: 感应电流的效果总是反抗引起感应电流的原因。简单地说，就是“来拒去留”。 如果感应电流是由组成回路的导体作切割磁感线运动而产生的，那么楞次定律可具体表述为：“运动导体上的感应电流受的磁场力（安培力）总是反抗（或阻碍）导体的运动。 楞次定律(Lenz's law)是一条电磁学的定律,可以用来判断由电磁感应而产生的电动势的方向.它是由俄国物理学家海因里希·楞次（Heinrich Friedrich Lenz）在1834年发现的。 海因里希·楞次 (H.F.E.Lenz, )

6 7.2.2 Transformers Transformer: an alternating-current (ac) device that transforming voltages, currents and impedances. For closed magnetic circuit, we have R: reluctance of the magnetic circuit magnetic flux Φ: Magnetic flux. - The induced mmf in the secondary circuit, N2i2, opposes the flow of the magnetic flux created by the mmf in the primary circuit, N1i1.

7 a) Ideal transformer (assume μ ∞)
(l: length of the magnetic core, S: cross section, μ: permeability) We have: a) Ideal transformer (assume μ ∞) Faraday’s law ` The ratio of current is inversely proportional to the ratio of the number of turns. The ratio of voltage is proportional the ratio of the number of turns. Ideal transformer（理想变压器）满足

8 Or, Effective load (R1)eff （有效负载）seen by the source:
RL: load of the secondary windings Impedance transformation for a sinusoidal source v(t) and a load impedance Zl: 变压器阻抗是指变压器里的线圈的绕组的阻抗，这包括：电阻，感抗，容抗。

9 We can write the magnetic flux linkage as
b) Real transformer. We can write the magnetic flux linkage as D. K. Cheng, p. 313 According to Faraday’s law, we have (Self-inductance of primary windings) Where, (Self-inductance of secondary windings) (mutual-inductance) 对ideal transformer，左右两部分线圈的互感是它们自感乘积的平方根，但是对于real transformer, 由于有磁漏（leakage flux）, 互感系数会小一些。 For real transformers, (k: coefficient of coupling)

10 Real-life conditions for real transformers:
D. K. Cheng, p 各种复杂要素（一般了解） Real-life conditions for real transformers: Flux leakage, nonlinear inductance, nonzero winding resistance, hysteresis and eddy-current losses Equivalent circuit Dot convention X1,X2: leakage inductive reactances, Rc: power loss due to hysteresis and eddy-current effects, Xc: nonlinear inductive reactance

11 7.2.3 A Moving Conductor in A Static Magnetic Field
The forces applied to the free charges in a moving conductor: Magnetic force Colombian force F=qE u: velocity of the conductor in magnetic field B At equilibrium, which is rapidly reached, the net force on the free charges in the moving conductor is zero.

12 Flux cutting emf or motional emf （动生电动势）
For an observe moving with the inductor, the magnetic force per unit charge can be regarded as an induced electric field acting along the conductor and producing a voltage 也可以这样理解：在一个参考系内看到的是磁力，在另一个参考系中看到的是电力（在实验室坐标系看，运动电子受到磁力，但在电子自身参考系看来，它受到的是电力，E来自B的感应，因为电子看到磁场B在后退，“运动”的磁场感应出一个电场E）。 For a closed circuit C, then the emf generated around the circuit is Flux cutting emf or motional emf Note: only the part of the circuit that moves in a direction not parallel to B will contribute to V’. （只有导体切割或者部分切割磁力线才有电动势）

13 7.2.4 A Moving Circuit in A Time-varying Magnetic Field
Lorentz’s force (for a charge q moves in an electric field E and a magnetic field B): For an observer moving with q, there is no apparent motion, and the force on q can be interpreted by an electric field E’ Or E来自B的感应，因为电子看到磁场B在后退，“运动”的磁场感应出一个电场E。 晃动一个磁铁，在空中感应出一个电场。 在一个参考系内看到的是磁现象，在另一个参考系中看到的是电现象。电与磁在涉及运动状态时，就不再独立。电与磁是同一事物（电磁场）的两种表现。

14 Faraday’s law in a time varying magnetic field
When a conducting circuit with contour C and surface S moves with a velocity u, we have the following form (7-34) emf induced in the moving frame of reference Transformer emf Motional emf The division of the induced emf between transformer and motional parts depends on the chosen frame of reference. 感应电动势包含两种：动生电动势（ Motional emf ）和感生电动势（ Transformer emf ）。

15 ◆Loop cuts the magnetic lines of force, while the magnetic field does not change
感应电动势包含两种：动生电动势和感生电动势。前者是磁场B不变，但导线运动切割磁力线；后者是B本身随时间变化。本质均是磁通量发生变化。下图是动生电动势示意图。 Motional electromotive force 动生电动势

16 A moving circuit in a time-varying magnetic field
From time t to t+Δt, consider a contour C moves from C1 to C2 in a changing magnetic field in an arbitrary manner. Time-varying change of magnetic flux through the contour (7-35) D. K. Cheng, p. 318

17 (7-36) Express B(t+Δt) by Taylor’s expansion,
D. K. Cheng, p. 318 Express B(t+Δt) by Taylor’s expansion, (7-36) High-order terms Substituting Eq. (7-36) in Eq. (7-35) yields (identical as the negative of the right of Eq. (7-34))

18 If we designate D. K. Cheng, p. 319 请注意关于时间的全导数与偏导数的区别
（偏导数仅仅包含B的时间变化，全导数还进一步包含了线路面积的扩张与收缩） V’ reduces to V for a stationary circuit; Faraday’s law applies to both stationary and moving circuit, thus for a general case;

19 Example 7-3: find the voltage in Faraday disk generator
Example 7-2: find the voltage of a sliding metal bar over conducting rails D. K. Cheng, p. 315 D. K. Cheng, p Example 7-3: find the voltage in Faraday disk generator

20 D. K. Cheng, p. 321 7.3 Maxwell’s Equations A time-varying magnetic field induces an electric field and from Table 7-1: （7-47a） Question； Satisfy these equations at time-varying conditions? （7-47b） （7-47c） （7-47d） In addition we know the principle of conservation of charge must be satisfied. We have （7-48） 电荷守恒定律(conservation law of charge)的数学表达式

21 Taking the divergence of Eq. (7-47b), we have
(7-49) In time varying case, In general, not true For the consistence of Eqs. (7-47a,b,c, and d) to Eq. (7-48), a term must be added at the right side of Eq. (7-49) (7-50) (7-52)

22 Time-varying electric field will generate magnetic field
is necessary for the consistence with the principle of conservation of charge : Displacement current density - One of major contributions of James Clerk Maxwell ( )

23 Displacement current概念引入的另一种思路：安培环路定理在电路导线上成立，但在电容器内不成立（因为电容器两板之间没有传导电流经过），传导电流在电容器内“断”掉了。但是，两板之间的电场一直在含时变化。于是Maxwell认为，为了保持安培环路定律在电容器内也成立，需要假设含时变化的电场也代表一种“电流密度”。这就是位移电流概念的由来。 Assume that the capacitor is charged. The charges on plates A and B is +q and –q, respectively. The charge densities are ＋ and －  , respectively. So, one can readily obtain 为电容器上面电荷密度，它在数值上等于D（电位移矢量）

24 Due to the current continuity equation
On the polar plate S2 surface Maxwell方程组与电荷守恒定律的相容性，得益于Maxwell引入的位移电流密度概念。可以说，在Maxwell方程组中，除了位移电流密度外，其他任何项都是实验总结的产物。只有位移电流密度概念是思维的产物。当然，这个思维并不是凭空的，而是通过分析已有矛盾开始的（譬如，没有位移电流贡献，那么安培环路定律并不是处处成立的，电荷也不再守恒）。所以，Maxwell引入位移电流概念，实在是画龙点睛之笔。因此，我们把电磁学整套方程组称为Maxwell方程组（尽管方程里面大部分贡献属于Maxwell之前的科学家，如奥斯忒、安培、库仑、法拉第、高斯等）。 画龙点睛，神来之笔，四两拨千斤，令人感慨万分。Maxwell遂成为一代宗师，是牛顿和爱因斯坦之间的过渡性人物。 where JD is a displacement current density

25 谈谈电荷守恒和能量守恒 (A)电荷守恒（charge conservation） 什么是电荷守恒？ 电荷守恒的数学表达式是什么？
答：古典的“电荷守恒”定律，是指电荷不能凭空创生，也不能凭空消失。自从量子电动力学(quantum electrodynamics)诞生（1940年代）以来，电荷可以创生，也可以湮灭，如一个高能光子（gamma射线光子）可以变成正负电子对，正电子（带正电）与普通电子（带负电）相遇可以变成光子。 正电子质量与普通电子一样，只是带正电。反质子（带负电）与正电子可以构成反氢原子，即反物质。反物质世界好比正物质世界的“镜像世界”（如反物质世界的左、右定义与我们正物质世界的左、右定义相反）。来自反物质世界的友好人士与你握手，你俩瞬间变为光子和各种射线, 真正的“灰飞烟灭”。所以要谨慎交友。

26

27 (B)能量守恒(energy conservation)

28 Maxwell’s equations 令人神魂颠倒的方程
J: density of free current (= convection current + conduction current) ρ: volume density of free charge 千万要记住：这里的电流密度是自由电流密度（不含极化电流密度），电荷密度是自由电荷密度，不含极化电荷密度。

29 Maxwell方程礼赞 麦克斯韦电磁理论不但为人类改造自然与文明进步做出了巨大贡献，具有很大实用
意义，而且该理论本身就是一件艺术珍品，其形式非常优美，我们可以从多角度(如代数 、几何、拓扑等)来欣赏其数学艺术价值。作为人类历史上第一个相对论性理论，亦可 以说是第一个量子理论(一次量子化理论)的麦克斯韦电磁理论，其张量结构理论形式在 相对论和量子论中完全被保留了下来，却比相对论与量子力学的建立几乎早了半个世 纪。正是因为其理论中所包含的博大精深的广延性，电磁理论在历史上曾引导物理学 家创建了相对论和量子力学。 以1865年Maxwell提出系统的电磁理论为标志而创立了经典电动力学，尽管其理论比相对论 的诞生早了40年，比量子力学完整框架的提出早了60年，但可以好不夸张地说，Maxwell电动力 学是第一个相对论，也是第一个量子力学理论。Maxwell方程组天生具有Lorentz协变性，对它的 考察使得Poncaré, Lorentz和Einstein等人在1900年代初期革新了传统的牛顿力学，创建了相对论 ；Maxwell电动力学作为光的波动理论，其方程组其实属一次量子化理论范畴，就场论角度看， 其与Schrödinger方程、Klein-Gordon方程和Dirac方程地位等价（它们都是单粒子的量子场论 ）。作为经典场方程，Maxwell方程在进入量子场论之后，其基本形式仍旧保持，这不 免令人惊叹。 事实也证明，麦克斯韦电磁理论是建立新理论的典范。 模仿Maxwell理论的规范理论（把E、 B看作矩阵）是Maxwell理论的“亲表哥”，它是杨-米尔斯(Yang-Mills)非阿贝尔规范场理论，已经 成为了描述强、弱相互作用的基本理论，如 年代格拉肖、温伯格、萨拉姆等很多人一 起建立了包括强、弱相互作用和电磁相互作用的统一理论。

30 电磁理论与Maxwell 1865年，英国物理学家麦克斯韦在电磁学的三大实验定律（库仑定律、毕奥-沙伐定律和法拉第电磁感应定律）基础上，提出了位移电流的基本假设，归纳总结出麦克斯韦方程，奠定了宏观电磁理论的基础。麦克斯韦方程组给出了电磁场的空间分布和随时间变化的全部规律，预言了电磁波的存在, 统一了光、电、磁。 James Clerk Maxwell ( )

31 Maxwell equations（要能熟练写出）
Differential form Integral form Ampère law 磁场的涡源是传导电流和位移电流 Faraday electromagnetic induction law 电场的涡源是随时间变化的磁场 Principle of continuity of magnetic flux 自然界中没有自由的磁荷或者说磁场没有磁源 高斯定律 一个电场的源就是自由电荷 介质本构方程（constitutive relations)：

32 位移电流（displacement current） 与传导电流（conduction current ）的关系
位移电流与传导电流在产生磁效应上是等效的。 (2) 产生的原因不同：传导电流是由自由电荷运动引起的，而位移电流本质上是变化的电场。 (3)通过导体时的效果不同：传导电流通过导体时产生焦耳热，而位移电流不产生焦耳热。

33 Charge conservation equation
Maxwell’s equations Charge conservation equation Lorentz equation All macroscopic electromagnetic phenomena could be explained by the above equations. 注意：无论材料是线性的还是非线性的、有损耗的还是无损耗的、有色散的还是无色散的、 均匀的还是非均匀的、各向同性的还是各向异性的，本页方程均不变。 总之，海枯石烂、山无棱、夏飘雪、冬雷阵阵，我心依旧。除非质子衰变（半衰期10^32年，是宇宙年龄的10^22倍），此时需要依靠大统一理论，电磁相互作用已归位于大统一理论之中。 （“海枯石烂、山无棱、夏飘雪、冬雷阵阵”均属于电磁相互作用范畴）

34 Not all independent 12 unknowns: E, D, B, H 14 equations (but 12 independent equations) Constitutive relations （本构关系）:

35

36 在金属中，电子平均速度与电场强度成正比（思考：为什么？ ） 此为Ohm law
四个电流密度 自由传导电流(free conduction current)、位移电流(displacement current) 、(电)极化电流(polarized current) 、磁化电流(magnetized current) 称为自由电荷密度(free charge density)与自由传导电流密度(free conduction current density). 有时也用 表示。 在金属中，电子平均速度与电场强度成正比（思考：为什么？ ） 此为Ohm law

37 此为流导数（flow derivative）
满足电荷守恒定律(conservation law of charge) 什么是“电荷守恒定律”？ 。 此为流导数（flow derivative） 除了自由电流密度（主要是传导电流密度）外，还有三种电流密度： 位移电流(displacement current) 电极化电流(electrically polarized current) 磁化电流(magnetized current) 代 入 此式右边就含四个电流密度

38 那么这三种电流密度有无对应的电荷密度呢？ 有的有，有的没有。只要看看是否满足电荷守恒定律即可（ ）。
那么这三种电流密度有无对应的电荷密度呢？ 有的有，有的没有。只要看看是否满足电荷守恒定律即可（ ）。 电极化电流 有对应的（极化）电荷密度(polarized charge density) 满足电荷守恒定律： 磁化电流 是一个“鳏夫”，它没有对应的“电荷密度”，因为它的电荷守恒定律是这么简单： 位移电流 有对应的“电荷密度” 它也满足电荷守恒定律

39 7.3.1 Integral form of Maxwell’s equations
In a physical environment we must deal with finite objects of specified shape and boundaries. Take the surface integral of both sides of the curl equations over an open surface S with a contour C and apply Stokes’s theorem.

40 Taking the volume integral of both sides of the divergence equations over a volume V with a closed surface S and using divergence theorem (no isolated magnetic polars) 孤立的磁极是不存在的（也即磁荷不存在）

41 Maxwell’s equations Maxwell方程的积分形式特征：方程左边的积分都是带圈的（即闭合曲面积分、闭合曲线积分），右边积分不带圈。

42 7.4 Potential Functions （用场E、B可以表示电磁场，同时，我们也可以采用“势”的观点，利用electric scalar potential V与magnetic vector potential A来表示场E、B 。用E、B ，采用的是“场与力”的观点，用A、V，采用的是“势与能”的观点。两者体系在经典电动力学中是等价的，但在量子力学中略有区别（Aharonov-Bohm效应）。在经典电动力学中， E、B是基本的， A、V被看作辅助量。但在量子力学中， A、V被认为是基本的、不可或缺的） Vector magnetic potential A: Substitute the above equation into Faradays’ law or

43 From the above, we obtain
(7.57) In the static case, and Eq. (7.57) reduces to For time-varying fields, E depends on both V and A. E and B are coupled. In the static or quasi-static state we have the solutions of Poisson’s equations of Eqs. (4-6) and (6-21)

44 Substitute Eqs. (7-55) and (7-57) into Eq
Substitute Eqs. (7-55) and (7-57) into Eq. (7-53b) and make use of the constitutive relations or (7.61)

45 We take the liberty to choose the divergence of A
(7.62) Lorentz conditions or Lorentz gauge for the potentials Use the above relations, Eq. (7.61) could be rewritten as (7.63) Inhomogeneous （非齐次）wave equations for vector potential A 以上方程称为d’Alembert equation (second order differential equation) 所谓非齐次方程，即方程右边含有源项（不含A的项）。 说明：为什么要引入一个Lorentz条件？ 在经典电动力学中，我们可以用Ｅ，Ｂ，Ｈ，Ｄ来描述电磁场．与之等价的方案是，可以用上面提到的四维电磁势来代替Ｅ，Ｂ，Ｈ，Ｄ．但是我们发现，同一个电场Ｅ与磁场Ｂ可以对应无穷多套电磁势．也就是说，对于描述电磁场，电磁势是“欠定的”，不是“超定的”．为了把电磁势定下来，我们需要额外的约束．这个约束条件就是Lorentz条件．当然，约束条件是可以随意选的．我们不一定必须选Lorentz条件．有时我们可以选择库仑规范． 无论选择什么约束条件，都是等价的．选择什么约束条件，主要看问题方便而定．如为了照顾到狭义相对论不变性，我们就使用Lorentz条件．

46 For constant ε and use Eq. (7-62), we have
A corresponding equation for V can be obtained by substitute Eq. (7-57) into Eq. (7-53c) For constant ε and use Eq. (7-62), we have Inhomogeneous wave equations for scalar potential V 以上方程称为d’Alembert equation (second order differential equation) 所谓非齐次方程，即方程右边含有源项（不含V的项）。 Lorentz condition in Eq. (7.62) uncouples wave equations for A and V. 历史补遗：通常我们在教材上看到的“洛仑兹（Lorentz）规范”名称其实是张冠李戴的。这一名称实际上应该称作“洛伦茨(Lorenz)规范” （发表在：L.V. Lorenz, Phil. Mag. Ser. 3, Vol. 34, p.287 (1867)）. 洛伦茨（Ludvig V. Lorenz）是丹麦人(Danish)，洛仑兹（Hendrik A. Lorentz, ）是荷兰人(Dutchman; Hollander)。后者名气大（经典电子论创始人，获得1902年Nobel奖）。当前者提出“洛伦茨(Lorenz)规范” ，荷兰的洛仑兹只有14岁。

47 7.5 Electromagnetic Boundary Conditions
D. K. Cheng, p. 329 It is necessary to know the boundary conditions for E, D, H, and B in order to solve the electromagnetic problems in contiguous regions. 问题求解的两个要素：场方程+边界条件 Methodologies to acquire the boundary equations For curl equations, apply the integral form to a flat closed path at a boundary with top and bottom sides in the two touching media yielding the boundary conditions for the tangential components For divergence equations, apply the integral form to a shallow pillbox at an interface with top and bottom surface yielding the boundary conditions for the normal components.

48 Boundary condition for tangential E
Boundary condition for tangential H

49 Boundary condition for normal D
Boundary condition for normal B Note: The boundary equations are not completely independent.

50 Interface between two lossless linear media

51 Interface between a dielectric and a perfect conductor
说明：理想导体（如类似超导体）的电导率σ为无穷大，根据欧姆定律J=σE，理想导体内部不可有非零的E，否则电流密度J就为无穷大（无意义，或者破坏介质）。理想导体具有抗磁性，磁场无法渗透入其内，故B为零（因为一旦理想导体被放入磁场内，它立即感应出较大的电流密度，产生新的B抵消外界提供的B）。理想导体内E、B、D、H都为零。

52 7.6 Wave Equations and their Solutions
For given charge and current distribution,

53 7.6.1 Solutions of Wave Equations for Potentials
Assume an point charge at time t, ρ(t)Δv’, at the origin of a spherical coordinate and out of the source region we have Introduce a new variable 这是一个重要的技巧，将三维波动方程化为了一维波动方程。 (7-73) One-dimensional homogeneous （齐次） wave equation

54 Wave travelling in the positive R direction with a velocity
Any twice differential function of or of is a solution of Eq. (7-73). But here we only select a function of for causality. Hence we have Wave travelling in the positive R direction with a velocity (7-75)

55 Electric potential due to a charge distribution over a volume V’:
Retarded scalar potential （因为含有-R/u，这是“推迟标量势函数”，t时刻的势实际上是t -R/u时刻的电荷所产生的势）。太阳如果突然消失，那么要在八分钟后，我们才感觉到，从而地球甩出轨道。

56 Retarded vector potential
Magnetic vector potential due to a current distribution over a volume V’ (could be processed in the same way): Retarded vector potential The electric and magnetic fields derived from A and V will be functions of and therefore retarded in time. It takes for electromagnetic waves to travel and to be felt at a distance.

57 7.6.2 Source-free Wave Equations （无源波动方程，source-less wave equations）
For source-free regions (ρ=0 and J=0) in simple (linear, isotropic and homogenous) non-conducting (σ=0) medium, Maxwell’s equations are reduced to

58 Take the curl of Eq. (7-79a) and use Eq. (7-79b)
d’ Alembert’s equation Homogeneous vector wave equations (could be decomposed into three one-dimensional homogeneous scalar wave equations.) Similarly,

59 7.7 Time-Harmonic Fields Arbitrary periodic time functions can be expanded into Fourier series of harmonic sinusoidal components Transient nonperiodic functions can be expressed as Fourier integrals Sinusoidal time variations of source functions will produce sinusoidal variations of E and H with the same frequency Electrodynamic fields can be determined in terms of those caused by the various frequency components of an arbitrary time-varying source function. The principle of superposition (Maxwell方程是线性的，故而 满足叠加原理)

60 7.7.1 The Use of Phasors- A Review
A sinusoidal quantity is defined by three parameters: amplitude, frequency and phase. For example, (7-83) It is not convenient to work directly with an instantaneous expressions such as the cosine function when differentiation or integration of i(t) are involved. Example: a series RLC circuit with an applied voltage Complicated mathematical manipulation is required.

61 Phasors （相量）: contains amplitude and phase but independent of it

62 Is can be solved very easily
感抗 容抗 Is can be solved very easily

63 7.7.2 Time-Harmonic Electromagnetics
For a time-harmonic E field referring to cosωt can be written as (7-93)

64 Time-harmonic Maxwell’s equations in a simple (linear, isotropic and homogeneous) medium （用了工程师习惯）
Almost exclusively deal with time-harmonic fields (and therefore with phasors) Phase quantities are not functions of t; Any quantity containing j must necessarily be a phasor.

65 Time-harmonic wave equations for scalar potential V and vector potential A
non-homogenous Helmholtz equations Where wavenumber: and Lorentz condition

66 The phasor solutions (retarded) for the potential equations:
（用了工程师习惯） (7-99) (7-100) If kR << 1, equations (7-99) and (7-100) then simplify t be the static expressions in Eqs. (7-58) and (7-59). Formal procedure to determine E and H due harmonic charges and currents:

67 7.7.3 Source-free Fields in Simple Media
Homogenous vector Helmholtz’s equations Homogenous vector wave equations

68 (Principle of duality)

69 In general case, Similarly,
In a conducting medium (σ≠0), equation (7-104b) should be changed into Complex permittivity: In general case, -Damping loss due to the inertial property of charged particles; -Ohmic loss in metal or semiconductor (good conductor: σ >> ωε) Similarly, Complex wavenumber: Loss tangent (δc-loss angle):

70 （用了物理学家习惯）

71 （用了物理学家习惯）

72 7.7.4 The Electromagnetic Spectrum
一些典型频段的波长或频率要作为常识记住： 可见光频率大约10^14—10^15Hz， 微波波长大约1mm到1m。