第1章 靜電場(chǎng)-1

1、Chapter 1 Electrostatics 第1章 靜電場(chǎng) 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. 本章研究靜止電荷產(chǎn)生的靜電場(chǎng)本章研究靜止電荷產(chǎn)生的靜電場(chǎng)。 (Static electric fie

2、ld, electrostatics)Definition: 相對(duì)于觀察者靜止且量值不隨時(shí)間變化的電荷相對(duì)于觀察者靜止且量值不隨時(shí)間變化的電荷(the charge is assumed to be constant in time)所產(chǎn)所產(chǎn)生的電場(chǎng)，稱為靜電場(chǎng)。生的電場(chǎng)，稱為靜電場(chǎng)。Contents：Electric field intensity, Electric potential靜電場(chǎng)中最主要的場(chǎng)量靜電場(chǎng)中最主要的場(chǎng)量:電場(chǎng)強(qiáng)度矢量和標(biāo)量電位電場(chǎng)強(qiáng)度矢量和標(biāo)量電位The basic equation(靜電場(chǎng)的基本方程靜電場(chǎng)的基本方程)The boundary condition(不同

3、媒質(zhì)分界面上的銜接條件不同媒質(zhì)分界面上的銜接條件)The boundary value problem(邊值問(wèn)題邊值問(wèn)題)The method of image(鏡像法鏡像法)Capacitor(電容電容) The energy(靜電能量靜電能量)Electric field intensityElectric potential靜電場(chǎng)中最主要的場(chǎng)量靜電場(chǎng)中最主要的場(chǎng)量:電場(chǎng)強(qiáng)度矢量電場(chǎng)強(qiáng)度矢量(vector)電位標(biāo)量電位標(biāo)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兩帶電體的電荷量?jī)蓭щ婓w的電荷量Position vectors位置矢量位置矢量distance vectors距離矢量距離矢量The electric force acting on q1 due to q2帶電體帶電體q2對(duì)帶電體對(duì)帶電體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)真空中的介電常數(shù)真空中的介電常數(shù)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、.一個(gè)電荷對(duì)另一個(gè)電荷的作用超越時(shí)空，直接地、一個(gè)電荷對(duì)另一個(gè)電荷的作用超越時(shí)空，直接地、瞬時(shí)地發(fā)生瞬時(shí)地發(fā)生。點(diǎn)電荷之間的作用力靠什么來(lái)傳遞？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. 一個(gè)電荷對(duì)另一個(gè)電荷的作用是通過(guò)一種中間物為媒介，以一一個(gè)電荷對(duì)另一個(gè)電荷的作用是通過(guò)一種中間物為媒介，以一定的、有限的速度傳遞過(guò)去定的、有限的速度傳遞過(guò)去。Electric field (電場(chǎng)電場(chǎng))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.電場(chǎng)的一個(gè)重要特性是對(duì)處在其中的任何其它電荷都產(chǎn)生作用力電場(chǎng)的一個(gè)重要特性是對(duì)處在其中的任何其它電荷都產(chǎn)生作用力Electric field intensity 電場(chǎng)強(qiáng)度矢量電場(chǎng)強(qiáng)度矢量引入引入 Electric field intensity ，表征電場(chǎng)的特性表征電場(chǎng)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:(單個(gè)點(diǎn)電荷在空間任一點(diǎn)單個(gè)點(diǎn)電荷在空間任一點(diǎn)P P處處所產(chǎn)生的電場(chǎng)強(qiáng)度所產(chǎn)生的電場(chǎng)強(qiáng)度)tqFE 204ReqrER rrRtqtest charge204ReqqFRt FtqFErrRoThe electric field intensity due to n point charges,可以利用疊加原理得可以利用疊加原理得： Principle of superposition (疊加積分法計(jì)算電場(chǎng)強(qiáng)度疊加積分法計(jì)算電場(chǎng)強(qiáng)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 體電荷分布Volume charge density 體電荷密度體電荷密度the electric field

13、intensity at point P due to a volume charge distribution(此帶電體在空間任一點(diǎn)產(chǎn)生的電場(chǎng)強(qiáng)度此帶電體在空間任一點(diǎn)產(chǎn)生的電場(chǎng)強(qiáng)度) 3lim0mCVVddqVqr VRVdeRrrE2041Pxzy0Vd)(r 202020444ReVdRedqrEdReqrERRRVddqrrrrThe charge per unit volumeSurface charge distribution 面電荷分布Surface charge density 面電荷密度面電荷密度 SRRSd4120errE 2lim0mCSdSdqSqrSddqThe c

14、harge per unit areaLine charge distribution 線電荷分布Line charge density 線電荷密度線電荷密度 lRRld 4120errE mCll ddqlqrlim0l ddqThe charge per unit lengthThe electric potential 電位電位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 靜電場(chǎng)的環(huán)路定律靜電場(chǎng)的環(huán)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靜電場(chǎng)是無(wú)旋場(chǎng)靜電場(chǎng)是無(wú)旋場(chǎng)EEEo 例：真空中長(zhǎng)度為例：真空中長(zhǎng)度為2l的直線段，均勻帶電，的直線段，均勻帶電，電荷線密度為電荷線密度為 ，求線段外任一點(diǎn)

18、，求線段外任一點(diǎn)P的電場(chǎng)的電場(chǎng)強(qiáng)度。強(qiáng)度。 矢量恒等式FCFCFC) (1) (1333rrrrrrrrrrrr0) (3) (153rrrrrrrrrr故0)(rE靜電場(chǎng)是無(wú)旋場(chǎng)1. 靜電場(chǎng)的旋度旋度和環(huán)路定律 (Curl and Circuital Law )304)(rrrrqrE 點(diǎn)電荷電場(chǎng)304)(rrrrqrE 取旋度01. E 與 的微分關(guān)系 負(fù)號(hào)表示電場(chǎng)強(qiáng)度的方向從高電位指向低電位。在直角坐標(biāo)系中,0E矢量恒等式0由zyxzyxEeee 根據(jù)E與 的微分關(guān)系，試問(wèn)靜電場(chǎng)中的某一點(diǎn) ( ) ( )00E?00E? E所以圖1.1.6 E 與 的積分關(guān)系線積分00ddPPPPllE

19、式中)ddd()(dzyxzyxzyxzyxeeeeeelddddzzyyxx設(shè)P0為電位參考點(diǎn)，即 ，則P點(diǎn)電位為00P0dPPPlE 000ddPPPPPPlE所以2. 與 E 的積分關(guān)系3. 電位參考點(diǎn)例如：點(diǎn)電荷產(chǎn)生的電位：Cqr04 00r C0r r04 q0C點(diǎn)電荷所在處不能作為參考點(diǎn)0Rr Rqq0044 rRqC04 場(chǎng)中任意兩點(diǎn)之間的電位差與參考點(diǎn)無(wú)關(guān)。選擇參考點(diǎn)盡可能使電位表達(dá)式比較簡(jiǎn)單。電位參考點(diǎn)可任意選擇，但同一問(wèn)題，一般只能選取一個(gè)參考點(diǎn)。電偶極子電偶極子 Electric dipole兩點(diǎn)電荷兩點(diǎn)電荷+q和和-q相距為相距為d。當(dāng)當(dāng)rd時(shí)，這一對(duì)等量異號(hào)時(shí)，這一對(duì)

20、等量異號(hào)的電荷稱為電偶極子。的電荷稱為電偶極子。(見書見書p10，eg.1-5)表示電偶極矩（dipole moment)，方向由-q 指向 +q。單位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靜電場(chǎng)中的導(dǎo)體靜電場(chǎng)中的導(dǎo)體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.導(dǎo)體是一種擁有大量自由電子導(dǎo)體是一種擁有大量自由電子(electrons)的物質(zhì)。的物質(zhì)。 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當(dāng)將導(dǎo)體引入外電場(chǎng)中以后，當(dāng)將導(dǎo)體引入外電場(chǎng)中以后，其自由電荷將會(huì)在導(dǎo)體中移動(dòng)，其自由電荷將會(huì)在導(dǎo)體中移動(dòng)，其運(yùn)動(dòng)的范圍不會(huì)超過(guò)導(dǎo)體的外表面。其運(yùn)動(dòng)的范圍不會(huì)超過(guò)導(dǎo)體的外表面。 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.-+-+在靜電平衡條件下：在靜電平衡條件下：1.導(dǎo)體內(nèi)電場(chǎng)為零。導(dǎo)體內(nèi)電場(chǎng)為零。2.靜電場(chǎng)中，導(dǎo)體必為一等位體，導(dǎo)體表面必靜電場(chǎng)中，導(dǎo)體必為一等位體，導(dǎo)體表面必為等位面。為等位面。3.導(dǎo)體表面上的電場(chǎng)強(qiáng)度必定垂直于表面。導(dǎo)體表面上的電場(chǎng)強(qiáng)度必定垂直于表面。4.導(dǎo)體如帶電，則電荷只能分布于

26、其表面。導(dǎo)體如帶電，則電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靜電場(chǎng)中的電介質(zhì)靜電場(chǎng)中的電介質(zhì)o Strictly speaking, an ideal dielectric is a material

27、 with no free electrons in its lattice structure.o 把自由電子非常少，也就是導(dǎo)電率非常低的媒把自由電子非常少，也就是導(dǎo)電率非常低的媒質(zhì)近似地看成不導(dǎo)電的媒質(zhì)，稱為電介質(zhì)或絕質(zhì)近似地看成不導(dǎo)電的媒質(zhì)，稱為電介質(zhì)或絕緣體緣體(insulator)電介質(zhì)的分子分為兩大類：電介質(zhì)的分子分為兩大類：非極性分子非極性分子non-polar molecule 極性分子極性分子polar molecule 極化 PolarizedEE非極性分子非極性分子non-polar molecule 極性分子極性分子polar molecule 電介質(zhì)在外電場(chǎng)作用下發(fā)生

28、極化，形成有向排列；電介質(zhì)內(nèi)部和表面產(chǎn)生極化電荷 (polarized charge)； 極化電荷與自由電荷都是產(chǎn)生電場(chǎng)的源。 極化強(qiáng)度P ( polarization intensity )表示電介質(zhì)的極化程度，即VVpPlim0C/m2電偶極矩體密度由eg1-5知，電偶極子元 所產(chǎn)生的電位為：整個(gè)極化電介質(zhì)所產(chǎn)生的電位為：dVP204)(ReprR VdRePdR204 VdRerPRrVR220)(41)( 再由矢量恒等式：則VdRerPRrVR220)(41)( VdRrPrRRReVR)1()(41)()1()1(02 RPPRRP11VdPRVdRPrVV1)(41)(0 o 對(duì)上

29、式用散度定理：第一項(xiàng)體積可化為閉合面積分VdPRVdRPrVV1)(41)(0 SdRerPVdRrPVdRrPSdRerPrSnVVSn004141)( SdRerVdRrrSnPVP 041)(兩式對(duì)比o 極化電荷面密度極化電荷面密度(bound surface charge density)：o 極化電荷體密度極化電荷體密度(bound volume charge density):o 極化強(qiáng)度與電場(chǎng)強(qiáng)度的關(guān)系：極化強(qiáng)度與電場(chǎng)強(qiáng)度的關(guān)系：nPeP PP EP0 相對(duì)極化率，無(wú)量綱純數(shù)。相對(duì)極化率，無(wú)量綱純數(shù)。Electric susceptibilityGausss Theoremo D

30、: electric flux density D高斯定律的微分形式SqSDd高斯定律的積分形式ED The constitutive relation between D and EThe Basic equations of electrostatic &The boundary conditions一、一、The equations of electrostatic 靜電場(chǎng)的基本方程靜電場(chǎng)的基本方程SqSDd電通量密度的閉合面積分等于面內(nèi)所包圍的總自由電荷l0dlE電場(chǎng)強(qiáng)度的環(huán)路積分恒等于零，電場(chǎng)強(qiáng)度的環(huán)路積分恒等于零，靜電場(chǎng)是一個(gè)守恒場(chǎng)靜電場(chǎng)是一個(gè)守恒場(chǎng)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靜電場(chǎng)是無(wú)旋場(chǎng)ED構(gòu)成方程構(gòu)成方程靜電場(chǎng)是有源無(wú)旋場(chǎng)，靜止電荷是靜電場(chǎng)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 的法向分量不連續(xù)當(dāng) 時(shí)， D 的法向分量連續(xù)。0n2n1DDD 12DDen取分界面上P點(diǎn)為觀察點(diǎn)，圍繞P點(diǎn)領(lǐng)域作一小扁圓柱體，保持兩個(gè)端面在分界面兩側(cè)Medium 1Medium 2InterfaceThe tangential component ofttEE12 E 的切向分量連續(xù)。E取P點(diǎn)為觀察點(diǎn)，圍繞P點(diǎn)做一狹小矩形環(huán)路ABCDA。012EEenMedium 1Medium 2InterfaceoThe normal components of the electric flux density are d