高斯单位制(CGS)的意义： This MKS system is convenient in engineering. For a treatment of the fundamental physics of fields and matter, it has one basic defect. Thus the MKS system, as it has been constructed, tends to obscure either the fundamental electromagnetic symmetry of the vacuum, or the essential asymmetry of the sources. That was one of our reasons for preferring the Gaussian CGS system in this book. ------------------------------------------ Ch 1 We try to treat carefully a question that is some times avoided and sometimes beclouded in introductory texts, the meaning o f the macroscopic fields E and B inside a material. 。 ”电荷不变“源自经验，目前为止无实验违反之。 the total electric charge o f an isolated system is a relativistically invariant number. 。 On present ideas, the electron and the proton are about as unlike as two elementary particles can be. N o one yet understands why their charges should have to be equal to such a fantastically precise degree. Evidently the quantization o f charge is a deep and universal law o f nature. The fact of charge quantization lies outside the scope o f classical electromagnetism , o f course. Still, it is worth remembering that classical theory cannot be expected to explain the structure o f the elementary particles. (It is not certain that present quantum theory can either!) W hat holds the electron together is as mysterious as what fixes the precise value o f its charge. Something m ore than electrical forces must be involved, for the electrostatic forces between different parts o f the electron would be repulsive. 。 Coulomb’s law：10^-13 cm to many kilometers 。 保守力做功与路径无关，因为保守力是central force 。 electrical potential energy of this paticular system： U=∑_{i!=j} qiqj/rij = 1/2∑_{i=1-->N} (∑_{j!=i} qiqj/rij ) 。 E=lim_{q0->0} F/q0, 批评E的这种定义，因为现实中q0不可能小于e。 。 Gauss’s law and Coulomb’s law are not two independent physical laws, but the same law expressed in different ways 。 Gauss law is applicable to any inverse-square field in physics, 。 P33 problem 1.20 很有趣 。 ---------------------------------- CH 2 potential function φ=-∫_{P1->P2}|E>*ds potential energy U field function is in the derivative of the potential function φ: |E> = - ∇φ single point charge: |E> = q/r 。 φ(x, y,z) = ∫_{all sources} ρ(x', y', z') dx'dy'dz' / r, r=sqrt((x-x')^2+(y-y')^2+(z-z')^2) 。 As we should expect, at a considerable distance from the disk (relative to its diameter), it doesn’t matter much how the charge is shaped;only the total charge matters, in first approximation. 。 圆盘边沿的电势： we see that, as we should expect, the potential, falls off from the center to the edge of the disk. The electric field, therefore, must have an outward component in the plane of the disk. T hat is why we re marked earlier that the charge, if free to move, would redistribute itself toward the rim. To put it another way, our uniformly charged disk is not a surface o f constant potential, which any conducting surface must be unless charge is moving. As y approaches zero from the positive side, Ey approaches 2πσ. On the negative y side o f the disk, which we shall call the back, E points in the other direction and its y component Ey is -2πσ. This is the same as the field o f an infinite sheet o f charge o f density a, derived in Sec. 1.10. It ought to be, for at points close to the center of the disk, the presence or absence of charge out beyond the rim can’t make much difference. 【In other words, any sheet looks infinite if viewed from close up】. Indeed, Ey has the value 2πσ not only at the center but all over the disk. 。 2.8 Energy Associated w ith an Electric Field U = 1/(8π)∫_{entire space} E^2 dv Our accounting comes out right if we think o f it as stored in space with a density o f E^2/(8π) in ergs/cm 3. There is no harm in this, but in fact we have no way of identifying, quite independently o f anything else, the energy stored in a particular cubic centimeter o f space. 。 potential energy U v.s. electric potential φ: - The potential energy U of a stationary system of charges is the work required to assemble it out of its parts, energy which we may think of as stored in the assembled system. It is a single scalar quantity and a property of the system as a whole. - The electric potential φ is a function of position in space, for a given distribution of electric charges. It is expressed in units o f ergs per esu, or statvolts. The difference between the values o f φ at two points in space is the work per unit charge required to transport charge from one place to the other. - U = 1/(8π)∫_{entire space} |∇φ|^2 dv - 也可以这样解释U： U=1/2∑_{i=1-->N} qj(∑_{j!=i} qi/rij )。其中(∑_{j!=i} qi/rij )≡ φ j表示 the potential at qj due to all other charges。所以，U = ½ ∫ρ(x,y,z)φ(x,y,z)dv, 其中φ(x,y,z) is the electric potential for the whole system. 。 div|F> ≡ lim_{Vi->0}1/Vi ∫_{Si}|F>*d|ai> div F is the flux out of Vi, per unit of volume, in the limit of infinitesimal Vi. 。 Gauss’s theorem (or the Divergence Theorem): ∫_S |F> d|a> = ∫_V div|F> dv (is meaningful on a macroscopic scale only(P63)) 。 div E = 4πρ (Gauss’s law in differential form) （(P63)前提是the inverse-square law成立, 而the inverse-square law是Coulumb做实验总结出来的“经验定律”） 。 |F>≡(Fx, Fy, Fz), div|F>= ∂Fx/∂x +∂Fy/∂y +∂Fz/∂z If div|F> has a positive value at some point, we find—thinking of F as a velocity field—a net “outflow” in th at neighborhood. 。 The divergence is a quantity that expresses only one aspect o f the spatial variation o f a vector field. 。 div|E> = - div grad φ = 。 Poisson’s equation: Δφ = -4πρ (is meaningful on a macroscopic scale only(P63)) Laplace’s Equation: Δφ =0, (解φ被称为harmonic function) 。 harmonic function的性质之一： If φ(x,y,z) satisfies Laplace’s equation, then the average value o f cp over the surface o f any sphere (not necessarily a small sphere) is equal to the value o f cp at the center o f the sphere. 。 Stokes’ Theorem: ∫_C |F>*d|s> = ∫_S curl|F>*d|a> 。 in the electrostatic field, curl |E> around any closed path is zero(因为是保守场) 。 P76， Problem 2.15, 对 ∇*(∇xF) = 0的直观理解。 。 ∇*(f∇f) = (∇f)^2 + f Δf ------------------------------- CH 3 Δφ =0的解的唯一性源自Δ算子的特性----Δ表示的是周围点的均值。 导体内部电场为零，源自导体外边界是等势面。 。 the conservation of charge: div J = 0 (time-independent charge distribution) steady current div J = - ∂ρ/∂t (time-dependent charge distribution) 。 |J> = σ|E> (经验公式) ---------------------------------------------- CH 5 这一章非常赞！大开眼界 But how do we know that Gauss’s law holds when charges are moving? Fortunately it does. We can take that as an experimental fact. 。 There is conclusive experimental evidence that the total charge in a system is not changed by the motion o f the charge carriers.(粗糙的反证见P153 5.4) 。 The experiments we have described, and many others, show that the value o f our Gauss’s law surface integral ∫_S |E> d|a> depends only on the number and variety o f charged particles inside S, and not on how they are moving. According to the postulate o f relativity, such a statement must be true for any inertial frame o f reference if it is true for one. 。 charge conservation：div |J> = - ∂ρ/∂t the relativistic invariance of charge: ∫_S(t) |E> d|a> = ∫_S'(t') |E'> d|a'> 。 P156 Energy is conserved, but energy is not a relativistic invariant. Charge is conserved and charge is a relativistic invariant. In the language o f relativity theory, energy is one component o f a four-vector, while charge is a scalar, an invariant number, with respect to the Lorentz transformation. This is an observed fact with far-reaching implications. It completely determines the nature of the field of moving charges. 。 relativity postulates + the relativistic invariance of charge ==>运动的电场产生波，该波的传播速度为光速。(P167） 。 If charge is to be invariant under a Lorentz transformation, the electric field E has to transform in a particular way. 。 If the electric field E at a point in space-time is to have a unique meaning, then the way E appears in other frames of reference cannot depend on the nature of the sources. In other words, the observer in F, having measured the field in his neighborhood at some time, ought to be able to predict from these measurements alone what observers in other frames of reference would measure at the same space-time point. Were this not true, field would be a useless concept. The evidence that it is true is the eventual agreement of our field theory with experiment. 。 the force acting on a charged particle in motion through F is q times the electric field E in that frame, strictly independent o f the velocity o f the particle. F=qE，在任意惯性系下 电荷在电场中的受力F与电荷的运动速度无关(这是电荷不变的结果) 推导过程在Section 5.8 。 It is a remarkable fact that the force on the moving test charge does not depend separately on the velocity or density o f the charge carriers in the wire, but only on the combination that determines the net charge transport. 。 If we had to analyze every system of moving charges by transforming back and forth among various coordinate systems, our task would grow both tedious and confusing. There is a better way. The overall effect of one current on another, or o f a current on a moving charge, can be described completely and concisely by introducing a new field, the magnetic field. 。 ------------------------------ CH 6 In other words, if two quite different systems o f moving charges happen to produce the same E and B at a particular point, the behavior o f any test particle at the point w ould be exactly the same in the two systems. It is for this reason only that the concept o f field, as an in term ediary in the interaction o f particles, is useful. And it is for this reason that we think o f the field as an independent entity Is the field more, or less, real than the particles whose interaction, as seen from our present point of view, it was invented to describe? T hat is a deep question which we would do well to set aside for a long time. People to whom the electric and magnetic fields were vividly real—Faraday and Maxwell, to name two—were led thereby to new insights and great discoveries. Let’s view the magnetic field as concretely as they did and learn some of its properties. electromagnetic field: (Ex, Ey, Ez, Bx, By, Bz)-- tensor ------------------------------------------- CH 7 It is a remarkable fact that for any two circuits mutual inductance, M12=M21 (证明见P249，本质上是由双重积分导致的) electromotive force: Ƹ = -1/c * dΦ/dt Ƹ21 = -M21 * dI1/dt (M21: mutual inductance) Ƹ11 = -L1 * dI1/dt (L1: self-inductance) Maxwell's Equations: div |B> = 0 div |E> = 4πρ curl |B> = 4π/c |J> + 1/c * ∂|E>/∂t (div |J> = - ∂ρ/∂t) curl |E> = - 1/c * ∂|B>/∂t Maxwell’s Equations in empty space(ρ=0 and |J>=0): div |B> = 0 div |E> = 0 curl |B> = 1/c * ∂|E>/∂t curl |E> = - 1/c * ∂|B>/∂t Displacement Current |J>_d ≡ 1/(4π) * ∂|E>/∂t (比如圆盘电容充放电时两极的E，会形成div|E>!=0) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 给定函数f(x)=0之后，利用求导法则可以得出新的关系式f'(x) = 0。 同样，施加div，curl算子后能得到更多的关系式：div(f) = 0; curl(f)=0; div(curl(f))≡0; 这应该算是向量分析里的一个技巧吧。 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< In em pty space, the term s with ρ and J are zero, and Maxwell’s equations become： E ^2-B^2 and |E>*|B> remain invariant in a transformation to another inertial frame In this case, since E = B at any point, the invariant quantity E^2-B^2 has the value zero. Also, because E is perpendicular to B, the other invariant |E>*|B> is zero. It follows that in any other frame the transform ed fields E' and B' must be equal to each other in magnitude and perpendicular in direction. A light wave looks like a light wave in any frame of reference. --------------------------- Ch 8 In fact, we have now reduced the ac network problem to the dc network problem , with only this difference: the numbers we deal with are complex numbers. --------------------------- CH 9 |dipole moment> = charge * |displacement> the moments o f the charge distribution： K0: the net charge, the monopole moment, or monopole strength K1:|dipole moment>.z K2: quadrupole moment of the distribution The advantage to us o f describing a charge distribution by this hierarchy o f m om ents is that it singles out just those features o f the charge distribution which determ ine the field at a great distance. If we were concerned only w ith the field in the im m ediate neighborhood o f the distribution, it w ould be a fruitless exercise. F o r our m ain task, understanding w hat goes on in a dielectric, it turns out th at only the m onopole strength (the net charge) and the dipole strength o f the m olecular building blocks m atter. We can ignore all other m om ents. --------------------------- CH 10 The world around us appears totally asymmetrical in the sense that we find no magnetic charges at all. There has been serious speculation, however, that pairs o f poles, like pairs of elementary particles, might be created and fly apart in very energetic nuclear events. Several recent searches for such particles, termed magnetic monopoles, have detected none. Whether they cannot exist, and if so why not, remains an open question. We are forced to conclude that the only sources of the magnetic field are electric currents. This takes us back to the hypothesis of Ampere, his idea that magnetism in matter is to be accounted for by a multitude o f tiny rings o f electric current distributed through the substance. the magnetic dipole moment: |m> = I |a>/c the field o f a magnetic dipole: |A> = |m> cross |r>/r^2 It turns out that the lines o f H inside the m agnet look ju st like the lines o f E inside the polarized cylinder o f Fig. 10.21a. T hat is as it should be, for if magnetic poles really were the source o f the m agnetization, rather than electric currents, the m acroscopic m agnetic field inside the m aterial would be H , not B, and the sim ilarity o f m agnetic polarization to electric polarization would be complete. Auxiliary field H v.s. Magnetic field B ------------------------- ¹²³⁴⁵⁶⁷⁸⁹ ᵃᵇᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ ʳˢᵗᵘᵛʷˣʸᶻ ᵅᵝᵞᵟᵠ ₀₁₂₃₄₅₆₇₈₉ ₐ ₑ ₕᵢⱼₖₗₘₙₒₚ ᵣₛₜᵤᵥ ₓ ᵦ º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθ ικ λ μ νξοπρστυφχψω ∽ ⊥ ∠ ⊙ ⊕ ⊗∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜ ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ ∈⊂∂Δ∇∀∃e̅Ζ͏͏͏͏͏͏ Z̅ ▹◃ ∧† ┘˩⌋⎦┙┚┛
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