# Chapter 2. Maxwell equations

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Chapter 2. Maxwell equations
《电磁波基础及应用》沈建其讲义 Chapter 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

1) Displacement current
1. Basic principles of time-dependent electric and magnetic fields （1）Gauss’ law in electrostatic field Active field （2）The loop theorem of the electrostatic field Conservative field (保守场)

（3）Gauss’ law for magnetic field
Passive field （4）The loop theorem of the magnetostatic field H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.

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?

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

Due to the current continuity equation
On the polar plate S2 surface where JD is a displacement current density

Now Ampere’s law can be rewritten as:
The differential form of Ampere’s law can be expressed as and in the loop C obey the right-hand screw rule

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.

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,

2) Maxwell equations 1. Maxwell equation set in integral and differential forms 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

(2) The characteristic of magnetic field
Continuity of magnetic flux Magnetic field is passive field. There is no free magnetic charge in nature.

(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.

(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

(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

2. Constitutive equation

Magnetic field Electric field
3. The relationship between electric field and magnetic field charge current Magnetic field Electric field motion varying Agitation（电流能激发磁场） Agitation （电荷能激发电场）

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 JST is the component of J vertical to l. When h0, the second term in the right equation is 0. Then we have When JS＝0 or tangential: 切向的 normal: 法向的

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

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

5. Summary of the boundary conditions
or

Interface of two passive media

Ideal medium 1 and ideal conductor 2

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。

1. Poynting’s theorem From Maxwell equation set, we have Combine the above equations If we assume that the medium is linear, we can obtain

For linear media Ei is impressed electric field, JEi 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

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

2. Poynting’s vector W/m2 Energy flow density （能流密度）

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)

(2) (3)

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

6) The Lorentz potential
Ｍagnetic vector potential and electric field strength in terms of retarded potentials (延迟势) magnetic vector potential （Wb /m） Therefore Scalar potential (V) Electric field strength in terms of retarded potential

If Helmholtz theorem If We get

D’ Alembert’s equation
(达兰伯方程) Lorentz condition: For sinusoidal electromagnetic field Lorentz condition:

Lorentz potential and gauge transformation (规范变换)
1) Vector and scalar potential 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 35

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. 36

2) Gauge Transformations, Lorentz Gauge, Coulomb Gauge
We choose The transformations 37

* Lorentz Gauge Also, 38

These 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 Gauge The solution is 39

density . This is the origin of the name “Coulomb gauge”.
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. 40

With the help of the continuity equation
The Coulomb gauge is often used when no sources are present. Then V=0 41

(B)能量守恒(energy conservation)