大学物理高斯定理课件-英文版.ppt

1、第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawChapter 20 Gausss LawA statement of the relationship between electric charge and electric field.1 Electric Flux2 Gausss Law3 Application of Gausss Law第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawNew Words and Expressionsflux 通量通量Gausss L

2、aw 高斯定理高斯定理gaussion surface 高斯面高斯面spherical 球面的球面的cylindrical 柱面的柱面的planar 平面的平面的第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law1.Electric Flux(通量通量)p487-489p487-489ESelectric flux The product of electric field and area of surface.ESE1)The cases that planes are in the uniform and the electric

3、 field direction is perpendicular to the surfaceE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law2)The cases that planes are in the uniform but the area S in not perpendicular to EThe number of electric field line through both area of S and are equal,soSEScoseES 均匀电场均匀电场，与平面夹角与平面夹角EneSEeE第八章静电

4、场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law3)For nonuniform fields and curved surfaces we rewrite this definition of electric flux in differential form:E 非均匀电场强度电通量非均匀电场强度电通量 sSEdcosdeesSEdeSEddenddeSSSdEne(electric flux through a Gaussian surface)第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Ga

5、usss Law规定：规定：取闭合面外法线方取闭合面外法线方向为正向。向为正向。We define of S or of dS,to point outward from the enclosed volume.E0d,2e220d,2e11 为封闭曲面为封闭曲面S1dS2dS22E11E3)For the surfaces which is closedFlux leaving the volume is positive.Whereas flux entering the volume is negative.第八章静电场第八章静电场Chapter 20 Gausss LawChapter

6、 20 Gausss LawSSSESEdcosde 闭合曲面的电场强度通量闭合曲面的电场强度通量SEddeESdESxyzEo例例:如图所示如图所示，有一个，有一个三棱柱体放置在电场强三棱柱体放置在电场强度度 的的1CN200iE匀强电场中匀强电场中.求通过此三求通过此三棱柱体的电场强度通量棱柱体的电场强度通量.第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawxyzEoPQRNM解解下右左后前eeeeee 下后前eee 0dsSE左左左左ESESsSEcosd enenene左右右右ESESsSEcosd e0 eeeeee下右左后

7、前第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawKarl Friedrich Gauss(1777-1835),German mathematician and physicist.He made a lot of contributions in the fields of experimental physics,theoretical physics and mathematics.He made major contributions to the theory of electromagnetism.3.Gauss Law

8、(P489-491)第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law1)Gausslaw for vacuum.The flux of the electric field through a closed surface of any shape equals to 1/0 times of the algebraic

9、sum of charges enclosed within the surface.niiSqSE10e1d 定理：在真空中定理：在真空中,通过任一通过任一闭合闭合曲面的电场强度通曲面的电场强度通量量,等于等于该曲面所包围的该曲面所包围的所有电荷的代数和除以所有电荷的代数和除以 .0（与（与面外面外电荷无关，闭合曲面称为高斯面）电荷无关，闭合曲面称为高斯面）0:permittivity of free space第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law1)The relation of the source of elect

10、ric filed and the field(反映反映场场和和源源 的关系的关系).niiSqSE10e1dDiscussion2)Where is the algebraic sum of all interior charges enclosed in the Gaussian surface.is not related to the way of distribution and outside charges.iq电通量只与面内电荷有关，与面外电荷无关。电通量只与面内电荷有关，与面外电荷无关。第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 G

11、ausss Law3)The quantity on the left side of above equation is the electric field resulting from all charges,both those inside and those outside the Gaussian surface.E 为高斯面上某点的场强，是由空间所有电荷产生为高斯面上某点的场强，是由空间所有电荷产生的，与面内面外电荷都有关。的，与面内面外电荷都有关。E5)It give a simple way to calculate the distribution of electric

12、 field for a given charge distribution with sufficient symmetry.4)Gausss Law and Coulombs law are equivalent(see P496).However,Gausss Law is hold for the produced by moving charges;Coulombs law is only true for electrostatic field.E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law2)The derivati

13、on of Gausss Law(高斯定理的导出高斯定理的导出)高斯高斯定理定理库仑定律库仑定律电场强度叠加原理电场强度叠加原理+Sd 点电荷位于球面中心点电荷位于球面中心(a spherical surface enclosing a point charge at its center)20 4rqESSSrqSEd 4d20e0eq r第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law+点电荷在任意封闭曲面内点电荷在任意封闭曲面内(with irregular surface)1S2SThe same number of fiel

14、d lines pass through surface,as pass through the spherical surface .1S2S012ddqSESESS第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Lawq 点电荷在封闭曲面之外点电荷在封闭曲面之外2dS2E0dd111SE0dd222SE0dd21 0dSSE1dS1E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law 由多个点电荷产生的电场由多个点电荷产生的电场21EEE SiiSSESEdde （外）内）iSi

15、iSiSESEdd(内）（内）(0e1diiiSiqSE0d（外）iSiSE1qiq2qsSdE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawniiSqSE10e1d高斯定理高斯定理1 1）高斯面上的电场强度为高斯面上的电场强度为所有所有内外电荷的总电场强度内外电荷的总电场强度.4 4）仅高斯面仅高斯面内内的电荷对高斯面的电场强度的电荷对高斯面的电场强度通量通量有贡献有贡献.2 2）高斯面为封闭曲面高斯面为封闭曲面.5 5）静电场是静电场是有源场有源场.3 3）穿进高斯面的电场强度通量为正，穿出为负穿进高斯面的电场强度通量为正，穿出

16、为负.总总 结结第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law3)Applying Gauss Law(p491-495)a.场对称性分析。场对称性分析。b.选取高斯面。选取高斯面。d.应用定理列方程求解。应用定理列方程求解。0cosqEdSSc.确定面内电荷代数和确定面内电荷代数和q。计算步骤计算步骤:第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Lawb.高斯面要高斯面要经过所研究的场点经过所研究的场点。a.要求电场具有要求电场具有高度对称性高度对称性。c.高斯面应选取规则

17、形状。高斯面应选取规则形状。d.面上各点的面上各点的场强大小相等，方向与高斯面法线方向一致场强大小相等，方向与高斯面法线方向一致。0qdSES,/SdE1cosSdSqE0写成写成高斯面选取的原则高斯面选取的原则e.高斯面上某一部分各点的高斯面上某一部分各点的场强方向与高斯面法线方向场强方向与高斯面法线方向垂直，该部分的通量为垂直，该部分的通量为0。,SdE0cos第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Lawspherical symmetry1.Spherical shell of uniform charge 一半径为一半径为

18、 ,均匀带电均匀带电 的薄的薄球壳球壳.求球壳内外任意点的电场强求球壳内外任意点的电场强 度度.RQ+ORA thin spherical shell of radius R possesses a total net charge Q that is uniformly distributed on it.Determine the electric field at point(a)outside the shell,and(b)inside the shell.第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawSolution:+

19、ORr1Sr2sBecause the charge is distributed symmetrically,the electric field must also be symmetric.Thus the field must be directed radially outward or inward.We choose our imaginary gaussian surface to be a sphere of radius r concentric with the shell as a dashed circle the shell.第八章静电场第八章静电场Chapter

20、20 Gausss LawChapter 20 Gausss Law+OR+OR0d1SSE0E02dQSESr1S20 4rQE02 4QEr（a）Rr 0Rr（b）r2s第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law1、均匀带电球面、均匀带电球面内部的场强处处为零内部的场强处处为零。2、均匀带电球面外面的场强分布正象球面上的电荷、均匀带电球面外面的场强分布正象球面上的电荷都集中在球心时所形成的一个都集中在球心时所形成的一个点电荷在该区的场强点电荷在该区的场强分布分布一样。一样。3、场强随距离的变化曲线、场强随距离的变化曲线-E-

21、r曲线如图。曲线如图。20 4RQrRoE讨论讨论0E20 4rQE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law2.Solid sphere of charge.An electric q is distributed uniformly throughout a nonconducting sphere of radius R.Determine the electric field(a)outside the sphere and inside the sphere.半径半径 R、带电量为、带电量为 q 的均的均匀带电球体，计

22、算球体内、匀带电球体，计算球体内、外的电场强度。外的电场强度。+OR+第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law+OR+(a)(a).球体外部球体外部 r R作半径为作半径为 r 的球面；的球面；面内电荷代数和为面内电荷代数和为Qq球面上各点的场强球面上各点的场强 E E 大小相等，大小相等，方向与法线同向。方向与法线同向。1cos,/SdE,cos0qEdSS0qdSES024qrE2041rqE与点电荷的与点电荷的场相同。场相同。21rr2s第八章静电场第八章静电场Chapter 20 Gausss LawChapter 2

23、0 Gausss Law(b).球体内部球体内部 r R作半径为作半径为 r 的球面；的球面；面内电荷代数和为面内电荷代数和为333434rRqq1cosqRr33球面上各点的场强球面上各点的场强 E E 大小相等，方向与法线相同。大小相等，方向与法线相同。,/SdE+O+Rr1S第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law0cosqEdSS0qdSES30324RqrrErRqE3041oRqREor2041Rqr第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawCalc

24、ulate the electric field at a distance r from the axis of the rod.cylindrical symmetry（柱对称）（柱对称）无限长均匀带电直线，无限长均匀带电直线，单位长度上的电荷，即电荷单位长度上的电荷，即电荷线密度为线密度为 ，求距直线为，求距直线为 处的电场强度处的电场强度.r+oxyzLong uniform line of charge.A very long straight wire possesses a uniform positive charge per unit length .第八章静电场第八章静电场

25、Chapter 20 Gausss LawChapter 20 Gausss LawOur Gaussian surface s h o u l d m a t c h t h e s y m m e t r y o f t h e problemaxis symmetry(轴对称性轴对称性)or cylindrical symmetry.场强沿垂直轴线的方向；场强沿垂直轴线的方向；距中心同远处场强相同距中心同远处场强相同.+oxyzhneneneE+rFrom symmetry,the direction of is radially outward from the line of cha

26、rge if the charge is positive;the magnitudes of E at the same r are same.E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law下底）上底）柱面）(dd dsssSESESESSEd柱面）(dsSE+oxyzhneneneE+r0hrE0 20 2hrhE 柱面）(ddsSSESE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law 无限大均匀带电平面，单位无限大均匀带电平面，单位面积上的电荷，即电荷面密度面积上

27、的电荷，即电荷面密度为为 ，求距平面为，求距平面为 处的电场处的电场强度强度.Infinite plane of charge.Charge is distributed uniformly,with a surface charge density s s,over a very large but very thin nonconducting flat plane surface.Determine the electric field at points near the plane.+Planar symmetry（面面对称）对称）sr第八章静电场第八章静电场Chapter 20 G

28、ausss LawChapter 20 Gausss LawPlanar Symmetry 垂直板面向外垂直板面向外,距板同远处距板同远处E大大小相同小相同.+SEESS选取闭合的柱形高斯面选取闭合的柱形高斯面02sE0dsSSES底面积底面积S20sSE Choose the gaussian surface a small,closed cylinder whose axis is perpendicular to the plane.第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law02sEEsEEsExEO)0(s第八章静电场第

29、八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law0022 s s s s EEEC0,0 BAEEDiscussion:If we arrange two infinite plates,with uniform opposite s s,to be close to each other and parallel as the figure shown.What are the in the region of A,B and C?EThe electric field between parallel-plate capacitor in ci

30、rcuit(直流电路中直流电路中的平行板电容器间的场强的平行板电容器间的场强)is such a case.ABC第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Lawss00s0sss0s00讨讨 论论无无限限大大带带电电平平面面的的电电场场叠叠加加问问题题第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawSummary:For certain symmetry arrangements of charge(as illustrated cylindrical,planar and

31、 Spherical symmetry),G.L.is very much easier to use than integration of field components.The key factors of choosing G.surface:The magnitude of E at the surface is constant;and the Gaussian surface should pass through the question-point.第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawMain Step

32、s by using Gauss(G.)Law:Step1:Analysis of symmetry;Step2:Choose a suitable G.surface;Step3:Calculating by G.law;第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawConductor p494-495Conductor:There exist a lot of free moved charges.Insulator(绝缘体绝缘体):Almost no free moved charges.Semiconductor(半导体半导

33、体):Between them.1.Interaction between Electrostatic Field and Conductor:A conductor in uniformly electric field :E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawElectrostatic Equilibrium(静电平衡状态静电平衡状态):):导体导体内部和表面都没有电荷的宏观移动内部和表面都没有电荷的宏观移动.导体导体静电感应静电感应Electrostatic FieldFree charges Redistribut

34、ing(electrostatic induction)E+-+EEE=0E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law2.Distribution of Charges on Conductor under The Electrostatic Equilibrium(P495)0EPS0/dqSES00SSdEEChoose an arbitrary G.S.in the internal conductor.From Gauss Law,Charges placed on an isolated conductor will

35、move entirely to the surface of the conductor.None of the excess charge will be found within the conductor(导体内净电荷为零导体内净电荷为零).1第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law面内电荷是否会等量异号？面内电荷是否会等量异号？缩小高斯面。缩小高斯面。与静电平衡条件矛盾。与静电平衡条件矛盾。,0q0 0E E所以静电平衡时导体内无净电荷。所以静电平衡时导体内无净电荷。高高斯斯面面 0q there is no net

36、 charge in any small volume of internal conductor静电平衡时导体内无净电荷，所有电荷分布于外表面。静电平衡时导体内无净电荷，所有电荷分布于外表面。第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawThe magnitude of the at a location just outside a conductor(导体外表面导体外表面)is proportional to surface charge density at that location on the conductor.E2

37、证明：证明：垂直导体表面作一垂直导体表面作一小高斯柱面，外底面上的小高斯柱面，外底面上的场强近似不变。场强近似不变。SEdScos,0qSqs+E作钱币形高斯面作钱币形高斯面 S S0E第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law内底侧外底0 0E,0内底,0侧SdE外底外底cosEdS外底面上电场大小相等，外底面上电场大小相等，,/SdE1cos外底dSE0qES00sSq0sE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawE:produced by all char

38、ged bodies in the space.s s:related to all charge distribution in the space.0sE第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law3An Isolated Conductor with a Cavity(P495):(a)There is no net charge on the inside the cavity walls if no charges inside the cavity空腔内表面无电荷全部电荷分布于外表面。空腔内表面无电荷全部电荷分布于外表

39、面。证明：证明：在导体内作高斯面，在导体内作高斯面，0qSdES,0 0E E导体内导体内 0q第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law面内电荷是否会等量异号？面内电荷是否会等量异号？如在内表面存在等量异号如在内表面存在等量异号电荷，则腔内有电力线，移动电荷，则腔内有电力线，移动电荷作功。所以导体不是等势电荷作功。所以导体不是等势体，与静电平衡条件矛盾。体，与静电平衡条件矛盾。所以内表面无电荷，所有电荷分布于外表面。所以内表面无电荷，所有电荷分布于外表面。第八章静电场第八章静电场Chapter 20 Gausss LawCha

40、pter 20 Gausss Law(b)If there is charge+q in the cavity,then it induces q at inner surface and all excess charge remains on the outer surface of the conductor.空腔原带有电荷空腔原带有电荷 Q，将，将 q 电电荷放入空腔内。荷放入空腔内。Qqqq内表面带有内表面带有 q 电电荷。电电荷。外表面带有外表面带有 Q+q 电荷。电荷。证明：在导体面内表面作高斯面，证明：在导体面内表面作高斯面，0qSdES第八章静电场第八章静电场Chapter

41、20 Gausss LawChapter 20 Gausss Law,0 0E E导体内导体内 0q由于腔内有由于腔内有 q 电荷，电荷，0)(qqQqqq腔内表面有腔内表面有 q 电荷电荷,由电荷守恒定律，在外表由电荷守恒定律，在外表面上产生等量的正电荷，外表面上产生等量的正电荷，外表面上的电荷为：面上的电荷为：qQ第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law腔内电荷变化会引起腔外电场的变化。腔内电荷变化会引起腔外电场的变化。Q0q 接地可屏蔽内部电场变化接地可屏蔽内部电场变化对外部的电场影响。对外部的电场影响。0q0q 例如：

42、如家电的接地保护；例如：如家电的接地保护；半导体中的中周外壳是金属的。半导体中的中周外壳是金属的。第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law221124rrrqs ss s s s 4The surface charge density outside the surface of a conductor is related to the radius at that location on the conductor(导导体外表面的电荷密度与该处曲率半径有关体外表面的电荷密度与该处曲率半径有关).).1221rrs ss s

43、 2021012144rqrqVV r1r2Far away from each otherR1s第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law导体表面尖锐处导体表面尖锐处 R小，小，s 大，表面大，表面E E也大；也大；RE1s导体表面平滑处导体表面平滑处 R大，大，s 小，表面小，表面E E也小；也小；带来一个现象带来一个现象-尖端放电尖端放电 在带电尖端附近，电离的分子与周围分子碰撞，在带电尖端附近，电离的分子与周围分子碰撞，使周围的分子处于激发态发光而产生电晕现象。使周围的分子处于激发态发光而产生电晕现象。第八章静电场第八章

44、静电场Chapter 20 Gausss LawChapter 20 Gausss Law 尖端放电会损耗电能，还会干扰精密测量和对尖端放电会损耗电能，还会干扰精密测量和对通讯产生危害。通讯产生危害。尖端放电现象的利与弊尖端放电现象的利与弊 例如避雷针。避雷针就是例如避雷针。避雷针就是利用其尖端的电场强度大，空利用其尖端的电场强度大，空气被电离，形成放电通道，使气被电离，形成放电通道，使云地间电流通过导线流入地下云地间电流通过导线流入地下而避免而避免“雷击雷击”的。的。第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law+第八章静电场第八

45、章静电场Chapter 20 Gausss LawChapter 20 Gausss LawExample:Two parallel-conducting-plates with same area(S d2).They charged QA and QB respectively.Find the surface density at each face of the plates under electrostatic equilibrium.AQBQ第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawSolution:Assume

46、surface density is s s From conservation of charges law,Considering the point PB in plate B,由静电平衡条件：由静电平衡条件：3s1s2s4sAQBQABSSQA21ssSSQB43ss040302012222ssssAE0040302012222ssssBE0第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law41ssSQQBA232ssSQQBA21.两外表面电荷等量同号。两外表面电荷等量同号。2.两内表面电荷等量异号。两内表面电荷等量异号。第八

47、章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawSQ/SQ/有有041ssSQ32ss讨论：讨论：QQQBA41ss32ss0BQSQA2SQA2第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss Law 0SdE32s ss s AQBQ1s s2s s3s s4s sSdPBSPlot a Gausss surface S,we haveAQSS 21s ss sBQSS 43s ss s0222204030201 s s s s s s s sBE32s ss s SQQBA241 s ss sSQQBA232 s ss ss s1=s s4=0.Think that if the plate B is connected to the ground,thenEAnother solution:第八章静电场第八章静电场Chapter 20 Gausss LawChapter 20 Gausss LawHomework:p498:3;4;6;8;13;15;16;18;19;21;22;23;24;27;28;29;36;47.