Presentation on theme: "Chapter 2. Maxwell equations"— Presentation transcript:
1Chapter 2. Maxwell equations 《电磁波基础及应用》沈建其讲义Chapter Maxwell equations1) Displacement current2) Maxwell equations3) Boundary conditions of time-dependent electromagnetic field4) Poynting’s Theorem and Poynting’s Vector5) The generalized definition of conductors and insulators6) The Lorentz potential
21) Displacement current 1. Basic principles of time-dependent electric and magnetic fields（1）Gauss’ law in electrostatic fieldActive field（2）The loop theorem of the electrostatic fieldConservative field(保守场)
3（3）Gauss’ law for magnetic field Passive field（4）The loop theorem of the magnetostatic fieldH是涡旋场,因为它的旋度不为零In the above four equations, D, E, B, H are the fields produced by rest charges or steady current. q is the sum of charges enclosed by Gauss’ surface, and I is algebraic sum of conduction current through the closed loop.
4（5）Faraday’s law of electromagnetic induction The relationship between the circulation (环量) of rotational field and the varying magnetic field is:The equation indicates that the varying magnetic field can produce a rotional electric field. Then a new question could be asked: can the varying electric field produce a magnetic field?
52. Displacement currentAssume 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
6Due to the current continuity equation On the polar plateS2 surfacewhereJD is a displacement current density
7Now Ampere’s law can be rewritten as: The differential form of Ampere’s law can be expressed asandin the loop C obey the right-hand screw rule
83. The relationship between the displacement current and the conduction current (I.e., the connection and difference between… )(1) They can both produce the magnetic field with the same strength, provided that the displacement current and the conduction current have the same current density.位移电流与传导电流在产生磁效应上是等价的.(2)They are produced in different ways (They originate from different sources): specifically, the conduction current is caused by the motion of the free charges, while the physical essence of the displacement current is the varying electric field.(3)They will exhibit different effects when passing through the metal conductor: the conduction current can produce the Joule heat while the displacement current cannot.
9Example: the conductivity of sea water is 4S/m and its relative dielectric constant is 81, determine the ratio of the displacement current to the conduction current at 1MHz frequency.We assume that the electric field is of the sinusoidal form,The density of the displacement current isThe amplitude is given byThe density of the conduction current iswith the amplitudeSo,
102) Maxwell equations1. Maxwell equation set in integral and differential formsThe characteristic of electric field(电场特性)Gauss’ lawThe dielectric flux through a closed surface equals the total charges Q inside the closed surface.Integral formThe source of a electric field is the free chargeDifferential form
11(2) The characteristic of magnetic field Continuity of magnetic fluxMagnetic field is passive field. There is no free magnetic charge in nature.
12(3) The relationship between the varying electric field and the magnetic field General Ampere lawThe integral of magnetic field strength H along closed loop C equals the sum of conduction current and displacement currentThe vorticity source of a magnetic field is the conduction current and displacement current.
13(4) The relationship between the varying magnetic field and the electric field Faraday’s law of electromagnetic inductionTime-dependent magnetic flux can produce electromotive force, the vorticity of an electric field is the time-dependent magnetic field
14(需熟记Maxwell方程组，并明确各个方程的物理含义) Maxwell equation set(需熟记Maxwell方程组，并明确各个方程的物理含义)Integral formDifferential form
16Magnetic field Electric field 3. The relationship between electric field and magnetic fieldchargecurrentMagnetic fieldElectric fieldmotionvaryingAgitation（电流能激发磁场）Agitation（电荷能激发电场）
173) Boundary conditions of time-dependent electromagnetic field (电磁场边界条件,具体讨论可见谢处方、饶克谨《电磁场与电磁波》pp )1. Boundary condition of magnetic field strength H1, 12, 2hRectangle loop in the interfaceJST is the component of J vertical to l. When h0, the second term in the right equation is 0. Then we haveWhen JS＝0ortangential:切向的normal:法向的
182. Boundary condition of electric field strength E When h0, the right term in the above equation is 0or1, 12, 2hRectangle loop in the interfacetangential:切向的normal:法向的
193. Boundary condition of magnetic field strength B orThe normal component of the magnetic flux density B in the interface is continuous4. Boundary condition of dielectric flux density DWhen S＝0or
22Ideal medium 1 and ideal conductor 2 理想介质１与理想导体２or
234) Poynting’s Theorem and Poynting’s Vector Poynting’s theorem is the mathematical expression for the law of conservation of energy of the electromagnetic fields. Poynting’s vector describes the flow of electromagnetic energy.Poynting定律是电磁能量的守恒定律，其中Poynting矢量的物理意义是：电磁能流密度。
241. Poynting’s theoremFrom Maxwell equation set, we haveCombine the above equationsIf we assume that the medium is linear, we can obtain
25For linear mediaEi is impressed electric field, JEi is the power of impressing (external) sources per unit volume. 外电源也产生了一个电场If we substitute the above expression into the equationwe have
26Let us multiply this equation by a volume element d and integrate over an arbitrary volume of the field, we haveThe power of all the sources inside vTransformed inside v into heat （焦耳热）change rate of the energy localized in the electric and magnetic field inside vPower transferred through S to a region outside SHow the power is classified
272. Poynting’s vectorW/m2Energy flow density （能流密度）
28Example: in passive free space, the time-dependent electromagnetic field is Determine: (1) magnetic field strength;（2）instantaneous Poynting’s vector；（3） average Poynting’s vector(1)
305) The generalized definition of conductors and insulators (导体与绝缘体的推广定义)For linear media and time-harmonic （时谐）fields
316) The Lorentz potential Ｍagnetic vector potential and electric field strength in terms of retarded potentials (延迟势)magnetic vector potential （Wb /m）ThereforeScalar potential (V)Electric field strength in terms of retarded potential
35Lorentz potential and gauge transformation (规范变换) 1) Vector and scalar potentialcan be written asThis implies that the term in ( ) can be written as the gradient of a scalar potential V, i.e.,At this stage it is convenient to consider only the vacuum case. Then the Maxwell equation35
36can be expressed in terms of the potentials as We have now reduced the set of four Maxwell equations to two equations. But they are still coupled. The uncoupling can be achieved by using the arbitrariness in the definition of the potentials.36
372) Gauge Transformations, Lorentz Gauge, Coulomb Gauge We chooseThe transformations37
39These two equations are equivalent to 4 Maxwell equations. Under the Lorentz condition:In other words, as long as l satisfies the above equation,the Lorentz condition preserves under the gauge transformation.** Coulomb GaugeThe solution is39
40density . This is the origin of the name “Coulomb gauge”. The scalar potential is just the instantaneous Coulomb potential due to the chargedensity This is the origin of the name “Coulomb gauge”.The vector potential satisfiesThe term involving the scalar potential V can, in principle, be calculated from the previous integral.40
41With the help of the continuity equation The Coulomb gauge is often used when no sources are present. Then V=041