1、Chapter 3,Conductors in electrostatic field & capacitors,物理学中册，马文蔚改编,P56-99,Contents in this chapter:,* 1)Conductors in electrostatic equilibrium 导体静电平衡,2) Electric properties of conductors and dielectrics in E filed. 电场中导体和电介质的电学性质,* 3) Gausss law with dielectrics 有介质时的高斯定理,* 4) Properties and calc
2、ulation of capacitors. 电容器的性质和计算,5) Energy of electrostatic field 静电场的能量,3.1 conductors in electrostatic field,recall,Induction 感应,Induction process of a conductor in E field 导体在静电场中的感应过程,(a) Conductor without external E field,(b) Conductor in an external E field (at the beginning),Induced charge,In
3、duced charge,(c) Conductor in an external E field (at equilibrium),Properties of an isolated conductor in electrostatic equilibrium 孤立导体静电平衡时的性质,The E field is zero everywhere inside the conductor. 导体内部电场处处为零。,Reasoning:,The conductor is in electrostatic equilibrium.,2) If the isolated conductor car
4、ries a net charge, the net charge resides entirely on its surface. 如果导体本身带有净电荷，那么这些净电荷将全部分布在 导体表面。,Reasoning:,Gausss law,Both the net charge and the induced charge reside on the Surface.净电荷和感应电荷都分布在表面上。,S is an arbitrary Gaussian surface inside the conductor.,3) The E field on the surface of the cha
5、rged conductoris perpendicular to the conductor surface and has a magnitude , where is the surface charge density at that point. 导体表面的电场与表面垂直，大小为 。,Reasoning:,Gausss law,S is a small Gaussian surface, whose end faces parallel to the surface of the conductor.,The E field is perpendicular to the surfa
6、ce because the conductor is in electrostatic equilibrium.,The magnitude of the E field on the conductor surface is:,Reasoning:,Gausss law,Example,one uniform E field,the E field after a metal ball was put into the field (in equilibrium),4) On an irregularly shaped conductor, the surface charge densi
7、ty is highest at locations where the radius of curvature of the surface is smallest. 不规则形状导体曲率半径最小的地方表面电荷密度最大。,越尖的地方感应电场越强。,尖端放电,5) The surface of the conductor in electrostatic equilibriumis an equi-potential surface and the conductor is an equi-potential body. 处于静电平衡的导体的表面是一等势面，导体是一等势体。,Reasoning:
8、,Take one arbitrary path from point A to point B inside the conductor, the E potential difference between the two points is:,Applications of properties of conductors in electrostatic Equilibrium导体静电平衡性质的应用,1) Discharge 放电,Losing charges of charged particle/object. 带电物体失去电荷。,Link: the atmospheric E f
9、ield & lightening,了解：大气电场和闪电,There is a huge E field in the atmosphere of the Earth.,The Earth is a good conductor.,The average magnitude of the E field at the surface of the Earth is:,Very strong E fields in the thundercloud cause lightening discharge between the cloud and the ground.,Typical E fie
10、lds during a thunderstorm are as high as 25000N/C.,2) Electrostatic shielding 静电屏蔽,Cavity conductor空腔导体,No net charges inside the cavity conductor空腔导体无净电荷,The external E field has no effect on the space inside the cavity. 外电场对空腔内部无影响。可将仪器等放入空腔内部起到屏蔽作用。,b) There is net charges inside the cavity condu
11、ctor空腔内有净电荷,* The external E field has no effect on E field inside the cavity. 腔内电场不受外电场影响。,Gausss law,由于导体内表面上电量与腔内电荷等量异号，在发出的电场线全部终止在内表面上，则及在腔外产生的合场强为。,* The external E field does not depend on the E field inside the cavity. 腔外电场也不受腔内电场的影响。It does not relate to the charge distribution inside the c
12、avity, but the charge quantity. 外电场与腔内电荷分布无关，但与腔内带电体电量有关。,The external E field is the superposition of the E fields due to Q+q.导体外的电场是产生的电场的叠加。,外表面电荷全部流入地下，导体外部由产生的电场随之消失。,* If connect the external surface of the cavity to the ground, the external E field does not relate to E field inside the cavity
13、. 如果腔外表面接地，则外场不受腔内电场影响。,Summary: the charge distribution of a conductor in electrostatic equilibrium,小结： 静电平衡导体的电荷分布,1) No charges inside the conductor 导体内部无电荷,Net charges entirely on the external surface.静电荷全部位于外表面,2) cavity conductors with charges空腔导体带电荷Q,a) There is no charges inside the cavity c
14、onductor空腔导体内部无电荷,b) There is charges inside the cavity conductor空腔导体内部有电荷,Example 3-1,As shown in the figure, there are one ball A and its coaxial spherical shell B. The radius of the ball is R1. The inner and outer radii of the spherical shell are R2 and R3 respectively. The charge on the ball is
15、q and Q on external surface of the shell.,Find: 1) the charge in the space. 2) the E field in the space. 3) the potential in the space. 4)if connect A and B with a wire, calculate 1), 2) and 3).,Solution:,1) since the ball and the shell will arrive electrostatic equilibrium, we can get the charge di
16、stribution in the space as following:,q: on the surface of the core,-q: on the inner surface of the shell,Q+q: on the outer surface of the shell,2) according to Gausss law, the E field in the space is:,0,3) the potential of the core:,3) the potential of the shell:,4) if connect A and B with a wire,t
17、he charge on the surface of the core and the charge on the inner surface of the shell are neutralized. There is no charge inside the shell.,1) The charge distribution is:,0: inside the shell,Q+q: on the outer surface of the shell,2) The E field distribution is:,3) the potential distribution:,3.2 cap
18、acitance &capacitor 电容 电容器,Capacitor is one kind of electrical components.,In general, a capacitor consists of two conductors of any shape, which have charges of equal magnitude and opposite sign. 两个含有等值异号的任意形状的导体组成的系统叫做电容器。,The capacitor stores charges.,A capacitor stores energy as well as charge.,
19、Capacitance C of a capacitor is defined as:,Q: the charge on the capacitor.,: the potential difference across the capacitor.,The unit of capacitance： coulomb per volt (farad F),Capacitance C is always a positive quantity.,1) The parallel-plate capacitor平行板电容器,The E field between the plates:,The pote
20、ntial difference between the plates:,The capacitance of parallel-plate capacitor is:,A: the area of the plate.,d: the distance between the plates.,2) The cylindrical capacitor 圆柱形电容器,The potential difference between the two cylinders:,Gaussian surface,From the previous result, we know:,The capacitan
21、ce of cylindrical capacitor is:,3) The spherical capacitor 球形电容器,Gaussian surface,The potential difference between the two spherical shells:,The capacitance of spherical capacitor is:,Combinations of capacitors 电容器的并联和串联,1) Parallel combination 并联,The maximum charges on the two capacitors Q1 and Q2,
22、The total charge Q stored by the two capacitors is:,The equivalent capacitance of the two capacitors is:,The equivalent capacitance of a parallel combination ofcapacitors is the algebraic sum of the individual capacitances and is larger than any of the individual capacitances.并联电容器的等效电容等于这些电容器的电容的代数
23、和，并且比单个电容器的电容都大。,2) series combination 串联,both capacitors must have the same charge Q.,The equivalent capacitance of the two capacitors in series combination is:,？,The inverse of the equivalent capacitance is the algebraic sum of the inversed of the individual capacitances and the equivalent capacit
24、ance of a series combination is always less than any individual capacitance in the combination.串联电容器的等效电容的倒数等于这些电容器电容的倒数的代数和。等效电容总是小于任意单个电容。,3.3 capacitors with dielectrics 有电介质的电容器,One parallel-plate capacitor with distance d and charge Q. The potential between the plates is V0,If fill some uniform
25、 dielectric into the two plates, keep the distance and the charge unchanged. The potential is found to be,The capacitance of the capacitor with the dielectric is:,C0 is the capacitance of the capacitor without dielectric.,will change with different dielectrics.,is a physical quantity to describe the
26、 electrical properties of dielectric.,is called relative permittivity of dielectric.电介质的相对电容率,dielectric constant,is permittivity of dielectric.电介质的电容率,In this case,That means, for the identical charge distribution, the E field in a certain dielectric is only that in free space. 对于同一电荷分布，在电介质中产生的场强仅
27、仅是真空中的,Parallel-plate capacitor in free space.,The charge on the plate：,The E field between plates：,Filling a dielectric into the plates,The polarized charge is,The E field due to polarized charge is .,极化电荷面密度 与自由电荷面密度 的关系,Polarized charge density & free charge density,The free charge density：,The p
28、olarized charge density：,The total E field of the capacitor with dielectric:,The capacitance of parallel-plate capacitor is:,The capacitance when the capacitor is filled with a dielectric is:,Therefore, we can increase the capacitance by:,1) decreasing the distance d. But d is limited.,2)increasing
29、the area of the capacitor.,3) filling a dielectric,The advantage of dielectrics:,1)It increases the capacitance of a capacitor.,2) It increases the maximum operating voltage of a capacitor.,3) It may provide mechanical support between the conducting plates.,3.4 Gausss law at presence of dielectrics
30、有电介质时的高斯定理,Recall: Gausss law in free space 真空中的高斯定理,But actually, in many cases, there are dielectrics in E field.,What will Gausss law be when there is dielectric in E field?,Get the E field between the plates when filling with a dielectric using Gausss law:,A cylindrical Gaussian surface,From Gau
31、sss law:,We define as electrical displacement.,We already have:,Then we get Gausss law with dielectric as:,where,电介质中的高斯定理,The net electric displacement flux through any closed surface is equal to the net free charge inside the surface.,在静电场中通过任意闭合曲面的电位移通量等于该闭合曲面包围自由电荷的代数和。,The polarized charge does
32、 not appear in this equation, but it is included in or,1) It is true only in the uniform isotropic dielectrics.,仅在均匀各向同性电介质中成立,2) The electric displacement has no obvious physical meaning.,电位移矢量没有明显的物理意义。,3) The electric displacement D does relate to the free charge and the polarized charge, but the
33、 electric displacement flux only relates to the free charge.,由极化电荷和自由电荷共同决定，不能认为 仅由自由电荷决定，但 的通量仅由自由电荷决定。,E 线,D 线,E field lines & electric displacement lines,Start from free positive charges, and end at free negative charges.,Start from positive charges, and end at negative charges.,Example 3-2,As sh
34、own in the figure, there are one metal ball with charge Q ，radius R1 and one uniform isotropic dielectric spherical shell with outer radius R2 relative permittivity in the space.,Find： at point A, B and C.,Solution:,the direction of is along the direction of radium.,the magnitude of can be obtained
35、from Gausss law with dielectrics:,since there is no free chargeinside the conductor,because the free charges inside the Gaussian surfaces surrounding B or C are both Q.,because,therefore,The E potential at point A,The E potential at point B,The E potential at point C,Exercise 3-1,As shown in the fig
36、ure, one parallel plate capacitor formed by conducting plate A and B is filled with two dielectrics with distance d1 and d2, relative permittivity,Find： in dielectric 1 and 2.,Exercise 3-2,One metal ball with net charge Q and radius R is placed into an infinity dielectric with relative permittivity,2) the E potential of the conducting ball.,Find: 1) at any point outside the metal ball.,
