Presentation on theme: "1 《电磁波基础及应用》沈建其讲义 Chapter 2. Maxwell equations 1) Displacement current 2) Maxwell equations 3) Boundary conditions of time-dependent electromagnetic field."— Presentation transcript:
1 《电磁波基础及应用》沈建其讲义 Chapter 2. Maxwell equations 1) Displacement current 2) Maxwell equations 3) Boundary conditions of time-dependent electromagnetic field 4) Poynting’s Theorem and Poynting’s Vector 5) The generalized definition of conductors and insulators 6) The Lorentz potential
2 （ 1 ） Gauss’ law in electrostatic field （ 2 ） The loop theorem of the electrostatic field 1. Basic principles of time-dependent electric and magnetic fields Conservative field ( 保守场 ) Active field 1) Displacement current
3 H 是涡旋场, 因为 它的旋度不为零 Passive field 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. （ 3 ） Gauss’ law for magnetic field （ 4 ） The loop theorem of the magnetostatic field
4 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? （ 5 ） Faraday’s law of electromagnetic induction
5 2. Displacement current 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
6 Due to the current continuity equation On the polar plate S 2 surface where J D is a displacement current density
7 Now Ampere’s law can be rewritten as: The differential form of Ampere’s law can be expressed as andin the loop C obey the right-hand screw rule
8 3. 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.
9 Example: 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 is The amplitude is given by The density of the conduction current is with the amplitude So,
10 1. Maxwell equation set in integral and differential forms 2) Maxwell equations (1)The characteristic of electric field ( 电场特性 ) Gauss’ law The dielectric flux through a closed surface equals the total charges Q inside the closed surface. Integral form The source of a electric field is the free charge Differential form
11 (2) The characteristic of magnetic field Continuity of magnetic flux Magnetic 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 law The integral of magnetic field strength H along closed loop C equals the sum of conduction current and displacement current The 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 induction Time-dependent magnetic flux can produce electromotive force, the vorticity of an electric field is the time-dependent magnetic field
14 Maxwell equation set ( 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 form Differential form
15 2. Constitutive equation 材料的本构关系（方程）
16 3. The relationship between electric field and magnetic field chargecurrent Magnetic field Electric field motion varying Agitation （电流能激发 磁场） Agitation （电荷能激发 电场） varying
17 3) 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 H 1, 1 2, 2 h Rectangle loop in the interface J ST is the component of J vertical to l. When h 0, the second term in the right equation is 0. Then we have When J S ＝ 0 or tangential: 切向的 normal: 法向的
18 2. Boundary condition of electric field strength E When h 0, the right term in the above equation is 0 or 1, 1 2, 2 h Rectangle loop in the interface tangential: 切向的 normal: 法向的
19 3. Boundary condition of magnetic field strength B or The normal component of the magnetic flux density B in the interface is continuous 4. Boundary condition of dielectric flux density D When S ＝ 0 or
20 5. Summary of the boundary conditions or
21 or Interface of two passive media 无源，交界面上的边界条件
22 Ideal medium 1 and ideal conductor 2 理想介质１与理想导体２ or
23 4) 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。
24 1. Poynting’s theorem Combine the above equations From Maxwell equation set, we have If we assume that the medium is linear, we can obtain
25 For linear media E i is impressed electric field, J E i 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 equation we have
26 Let us multiply this equation by a volume element d and integrate over an arbitrary volume of the field, we have The power of all the sources inside v Transformed inside v into heat （焦耳热） change rate of the energy localized in the electric and magnetic field inside v Power transferred through S to a region outside S How the power is classified
27 2. Poynting’s vector W/m 2 Energy flow density （能流密度）
28 Example: 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)
29 (2) (3)
30 5) 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
31 Therefore 6) The Lorentz potential Ｍ agnetic vector potential and electric field strength in terms of retarded potentials ( 延迟势 ) magnetic vector potential （ Wb /m ） Scalar potential (V) Electric field strength in terms of retarded potential
32 If We get Helmholtz theorem
33 Lorentz condition: D’ Alembert’s equation ( 达兰伯方程 ) For sinusoidal electromagnetic field Lorentz condition:
35 Lorentz potential and gauge transformation ( 规范变换 ) can be written as This 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 equation 1) Vector and scalar potential
36 can 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.
37 2) Gauge Transformations, Lorentz Gauge, Coulomb Gauge We choose The transformations
38 * Lorentz Gauge Also,
39 These two equations are equivalent to 4 Maxwell equations. Under the Lorentz condition: In other words, as long as satisfies the above equation, the Lorentz condition preserves under the gauge transformation. ** Coulomb Gauge The solution is
40 The scalar potential is just the instantaneous Coulomb potential due to the charge density. This is the origin of the name “ Coulomb gauge ”. The vector potential satisfies The term involving the scalar potential V can, in principle, be calculated from the previous integral.
41 With the help of the continuity equation The Coulomb gauge is often used when no sources are present. Then V=0