第1章 静电场-1

1、Chapter 1 Electrostatics 第1章 静电场 Armed with the necessary tools of vector operations and vector calculus, we are now ready to explore electromagnetic field theory.Introduction In this chapter, we study static electric fields, due to charges at rest. 本章研究静止电荷产生的静电场本章研究静止电荷产生的静电场。 (Static electric fie

2、ld, electrostatics)Definition: 相对于观察者静止且量值不随时间变化的电荷相对于观察者静止且量值不随时间变化的电荷(the charge is assumed to be constant in time)所产所产生的电场，称为静电场。生的电场，称为静电场。Contents：Electric field intensity, Electric potential静电场中最主要的场量静电场中最主要的场量:电场强度矢量和标量电位电场强度矢量和标量电位The basic equation(静电场的基本方程静电场的基本方程)The boundary condition(不同

3、媒质分界面上的衔接条件不同媒质分界面上的衔接条件)The boundary value problem(边值问题边值问题)The method of image(镜像法镜像法)Capacitor(电容电容) The energy(静电能量静电能量)Electric field intensityElectric potential静电场中最主要的场量静电场中最主要的场量:电场强度矢量电场强度矢量(vector)电位标量电位标量(scalar)The Experimental Law Of CoulombElectrostatics is based upon the quantitative

4、and experimentally verifiable statement of Coulombs law pertaining to the electric force that one charged particle exerts on another.xyzo1r2rq1q221R12FQuantities of charge两带电体的电荷量两带电体的电荷量Position vectors位置矢量位置矢量distance vectors距离矢量距离矢量The electric force acting on q1 due to q2带电体带电体q2对带电体对带电体q1的作用力的作

5、用力111,zyx222,zyx2121rrRElectric forceFrom his experiments, Coulomb, a French physicist, postulated that the electric force between two charged particles is:32121021124RRqqF21rrmF1291085. 81036121221221221zzyyxxDistance between two chargesUnit vector22102112214ReqqFR212121RReR1r2rq1q221R12F222,zyx111

6、,zyxopermittivity of free space (vacuum)真空中的介电常数真空中的介电常数Coulombs lawo Directly proportional to the product of their charges,o Inversely proportional to the square of the distance between them,o Directed along the line joining them, ando Repulsive for like charges, and attractive for unlike charges.2

7、2102112214ReqqFR Coulombs law was specified calculating the force between the two charged particles. We know the magnitude and the direction. But we dont know how does this force produce. Action at a distance (超距超距)：The force that one charge to another charge is produced directly and instantaneously

8、.一个电荷对另一个电荷的作用超越时空，直接地、一个电荷对另一个电荷的作用超越时空，直接地、瞬时地发生瞬时地发生。点电荷之间的作用力靠什么来传递？思考Action by contact (近距近距)：That the information about the motion of one charge will take some time to reach the other charge because no signal can travel faster than the speed of light. Thus, the increase in the force acting on

9、the charges cannot be instantaneous. There must exist an extra entity, this extra entity is called the field. 一个电荷对另一个电荷的作用是通过一种中间物为媒介，以一一个电荷对另一个电荷的作用是通过一种中间物为媒介，以一定的、有限的速度传递过去定的、有限的速度传递过去。Electric field (电场电场)o We say that there exists an electric field everywhere in space surrounding the charge.o

10、When another charge is brought into this electric field, it experiences a force acting on it.电场的一个重要特性是对处在其中的任何其它电荷都产生作用力电场的一个重要特性是对处在其中的任何其它电荷都产生作用力Electric field intensity 电场强度矢量电场强度矢量引入引入 Electric field intensity ，表征电场的特性表征电场的特性The electric field intensity is defined as the force per unit charge.

11、So the electric field intensity due to a single point charges:(单个点电荷在空间任一点单个点电荷在空间任一点P P处处所产生的电场强度所产生的电场强度)tqFE 204ReqrER rrRtqtest charge204ReqqFRt FtqFErrRoThe electric field intensity due to n point charges,可以利用叠加原理得可以利用叠加原理得： Principle of superposition (叠加积分法计算电场强度叠加积分法计算电场强度) kRnkkkeRqrE120411E

12、2EE1210114ReRqE2220224ReRqE21220221012144RReRqeRqEEEA charge can be either concentrated at a point or distributed in some fashion. Such as the continuous distribution of charges in volumes, on surface, and on linear .Volume charge distribution 体电荷分布Volume charge density 体电荷密度体电荷密度the electric field

13、intensity at point P due to a volume charge distribution(此带电体在空间任一点产生的电场强度此带电体在空间任一点产生的电场强度) 3lim0mCVVddqVqr VRVdeRrrE2041Pxzy0Vd)(r 202020444ReVdRedqrEdReqrERRRVddqrrrrThe charge per unit volumeSurface charge distribution 面电荷分布Surface charge density 面电荷密度面电荷密度 SRRSd4120errE 2lim0mCSdSdqSqrSddqThe c

14、harge per unit areaLine charge distribution 线电荷分布Line charge density 线电荷密度线电荷密度 lRRld 4120errE mCll ddqlqrlim0l ddqThe charge per unit lengthThe electric potential 电位电位o In this section we define a scalar field, the electric potential, because it enables us to simplify a number of otherwise complica

15、ted calculations. o It is always easier to work with a scalar quantity than a vector quantity.o If we place a positive test charge q0 in an electric field produced by q, there will be a force on the charge given by .o Under this force, the charge moves a differential distance .o As the charge moves,

16、 work is being done by the electric field.EEqF0l dl dEql dFdWE0BABAEl dEql dFW0o If we move the charge around a closed path, the work done must be zero.llE0dBArrBArErrqqrdrqqrl deqqWBA1144400200200S)AlAlddS(0)(rE 204ReqrERBAEl dEqW0Circuital Law 静电场的环路定律静电场的环路定律o The field under static conditions is

17、 irrotational or conservative.o If the curl of a vector field is zero, the vector field can be represented in terms of the gradient of a scalar field. Thus, we can express the field in terms of a scalar field as llE0d0)(rE静电场是无旋场静电场是无旋场EEEo 例：真空中长度为例：真空中长度为2l的直线段，均匀带电，的直线段，均匀带电，电荷线密度为电荷线密度为 ，求线段外任一点

18、，求线段外任一点P的电场的电场强度。强度。 矢量恒等式FCFCFC) (1) (1333rrrrrrrrrrrr0) (3) (153rrrrrrrrrr故0)(rE静电场是无旋场1. 静电场的旋度旋度和环路定律 (Curl and Circuital Law )304)(rrrrqrE 点电荷电场304)(rrrrqrE 取旋度01. E 与 的微分关系 负号表示电场强度的方向从高电位指向低电位。在直角坐标系中,0E矢量恒等式0由zyxzyxEeee 根据E与 的微分关系，试问静电场中的某一点 ( ) ( )00E?00E? E所以图1.1.6 E 与 的积分关系线积分00ddPPPPllE

19、式中)ddd()(dzyxzyxzyxzyxeeeeeelddddzzyyxx设P0为电位参考点，即 ，则P点电位为00P0dPPPlE 000ddPPPPPPlE所以2. 与 E 的积分关系3. 电位参考点例如：点电荷产生的电位：Cqr04 00r C0r r04 q0C点电荷所在处不能作为参考点0Rr Rqq0044 rRqC04 场中任意两点之间的电位差与参考点无关。选择参考点尽可能使电位表达式比较简单。电位参考点可任意选择，但同一问题，一般只能选取一个参考点。电偶极子电偶极子 Electric dipole两点电荷两点电荷+q和和-q相距为相距为d。当当rd时，这一对等量异号时，这一对

20、等量异号的电荷称为电偶极子。的电荷称为电偶极子。(见书见书p10，eg.1-5)表示电偶极矩（dipole moment)，方向由-q 指向 +q。单位cmdpqConductors in an electric fieldo We have paid adequate attention to the fields produced by various charge distributions in free space (vacuum), and we are now at a stage when we can discuss materials in order to complet

21、e our study of electrostatic fields. We classify materials into three broad categories: conductors, semiconductors, and insulators.o Let us look first at electrostatic systems involving conductors.o静电场中的导体静电场中的导体Conductors in an electric field A conductor is a material, such as a metal, that possess

22、es a relatively large number of free electrons.导体是一种拥有大量自由电子导体是一种拥有大量自由电子(electrons)的物质。的物质。 In this subsection, our aim is to investigate the behavior of an isolated conductor when placed in a static electric field. We remind you that an isolated conductor is electrically neutral. In other words, t

23、he conductor has as many positive charges as it has electrons.o We place an isolated conductor in an electric field. The externally applied electric field exerts a force on the free electrons and causes them to move in a direction opposite the field. One side of the conductor becomes negatively char

24、ged, and the other side becomes positively charged.-+-+Positive chargenegative chargeinduced chargeE当将导体引入外电场中以后，当将导体引入外电场中以后，其自由电荷将会在导体中移动，其自由电荷将会在导体中移动，其运动的范围不会超过导体的外表面。其运动的范围不会超过导体的外表面。 o The effect of these induced charges is to produce an electric field within the conductor which is finally equ

25、al and opposite to the externally applied electric field.o In other words,the net electric fieldinside the conductoris zero when the steadyState is reached.-+-+在静电平衡条件下：在静电平衡条件下：1.导体内电场为零。导体内电场为零。2.静电场中，导体必为一等位体，导体表面必静电场中，导体必为一等位体，导体表面必为等位面。为等位面。3.导体表面上的电场强度必定垂直于表面。导体表面上的电场强度必定垂直于表面。4.导体如带电，则电荷只能分布于

26、其表面。导体如带电，则电荷只能分布于其表面。oNeither volume charge density nor electric field can be maintained within an isolated conductor under static conditions. Each conductor forms an equipotential region of space.Dielectrics in an electric field静电场中的电介质静电场中的电介质o Strictly speaking, an ideal dielectric is a material

27、 with no free electrons in its lattice structure.o 把自由电子非常少，也就是导电率非常低的媒把自由电子非常少，也就是导电率非常低的媒质近似地看成不导电的媒质，称为电介质或绝质近似地看成不导电的媒质，称为电介质或绝缘体缘体(insulator)电介质的分子分为两大类：电介质的分子分为两大类：非极性分子非极性分子non-polar molecule 极性分子极性分子polar molecule 极化 PolarizedEE非极性分子非极性分子non-polar molecule 极性分子极性分子polar molecule 电介质在外电场作用下发生

28、极化，形成有向排列；电介质内部和表面产生极化电荷 (polarized charge)； 极化电荷与自由电荷都是产生电场的源。 极化强度P ( polarization intensity )表示电介质的极化程度，即VVpPlim0C/m2电偶极矩体密度由eg1-5知，电偶极子元 所产生的电位为：整个极化电介质所产生的电位为：dVP204)(ReprR VdRePdR204 VdRerPRrVR220)(41)( 再由矢量恒等式：则VdRerPRrVR220)(41)( VdRrPrRRReVR)1()(41)()1()1(02 RPPRRP11VdPRVdRPrVV1)(41)(0 o 对上

29、式用散度定理：第一项体积可化为闭合面积分VdPRVdRPrVV1)(41)(0 SdRerPVdRrPVdRrPSdRerPrSnVVSn004141)( SdRerVdRrrSnPVP 041)(两式对比o 极化电荷面密度极化电荷面密度(bound surface charge density)：o 极化电荷体密度极化电荷体密度(bound volume charge density):o 极化强度与电场强度的关系：极化强度与电场强度的关系：nPeP PP EP0 相对极化率，无量纲纯数。相对极化率，无量纲纯数。Electric susceptibilityGausss Theoremo D

30、: electric flux density D高斯定律的微分形式SqSDd高斯定律的积分形式ED The constitutive relation between D and EThe Basic equations of electrostatic &The boundary conditions一、一、The equations of electrostatic 静电场的基本方程静电场的基本方程SqSDd电通量密度的闭合面积分等于面内所包围的总自由电荷l0dlE电场强度的环路积分恒等于零，电场强度的环路积分恒等于零，静电场是一个守恒场静电场是一个守恒场o This is the fi

31、rst of Maxwells four equations as they apply to electrostatics and steady magnetic fields.o Notice: the right-hand side of this equation is simply the free charge density. D高斯定律的微分形式o A field is conservative if its curl is zero.0)(rE静电场是无旋场ED构成方程构成方程静电场是有源无旋场，静止电荷是静电场的源。Boundary conditionso In this

32、section, we investigate the conditions that govern the behavior of electric fields at the boundary (interface) between two media.o The interface may be between a dielectric and a conductor or between two different dielectrics.o The equations governing the behavior of electric fields on either side o

33、f an interface are known as the boundary conditions.The normal component of n1n2DDD 的法向分量不连续当 时， D 的法向分量连续。0n2n1DDD 12DDen取分界面上P点为观察点，围绕P点领域作一小扁圆柱体，保持两个端面在分界面两侧Medium 1Medium 2InterfaceThe tangential component ofttEE12 E 的切向分量连续。E取P点为观察点，围绕P点做一狭小矩形环路ABCDA。012EEenMedium 1Medium 2InterfaceoThe normal components of the electric flux density are d