電磁場(chǎng)及電磁波第10講電流密度電動(dòng)勢(shì)電壓電流定理-wb

1、1Field and Wave Electromagnetics電磁場(chǎng)與電磁波電磁場(chǎng)與電磁波2012. 3. 2321. Poissons and Laplaces Equations2V 20V222222222aaaaaaxyzxyzVVVVxyVVVVVVxyzxyzz In Cartesian coordinates:Review 4. Solution of Electrostatic Problems33. Method of Images 4. Boundary-Value Problems in Cartesian Coordinates 2. Uniqueness of El

2、ectrostatic Solutionsuniqueness theorem: means that a solution of Poissons equation (of which Laplaces equation is a special case) that satisfies the given boundary conditions is a unique solution. Essence: The effect of the boundary is replaced by one or several equivalent charges, and the original

3、 inhomogeneous region with a boundary becomes an infinite homogeneous space.( , , )( ) ( ) ( )V xyzX x Y y Z z0dd222XkxXx 0dd222YkyYy0dd222ZkzZz4Main topic Steady Electric Currents3. Equation of Continuity and Kirchhoffs Current Law1. Current Density and Ohms Law2. Electromotive Force and Kirchhoffs

4、 Voltage Law51. Current Density and Ohms LawElectrolytic current(電解電流電解電流): are the result of migration of positive and negative ions; （正離子和負(fù)離子徙動(dòng)的結(jié)果正離子和負(fù)離子徙動(dòng)的結(jié)果）Convection current(對(duì)流電流對(duì)流電流): result from motion of electrons and/or ions in a vacuum or rarefied gas;（電子和（或）離子在電子和（或）離子在真空中的運(yùn)動(dòng)真空中的運(yùn)動(dòng)）Condu

5、ction current(傳導(dǎo)電流傳導(dǎo)電流): in conductors and semiconductors are caused by drift motion of conduction electrons and/or holes;（導(dǎo)體和半導(dǎo)體中的電子和（或）空穴的移動(dòng)引起的導(dǎo)體和半導(dǎo)體中的電子和（或）空穴的移動(dòng)引起的）6unasThe amount of charge Q passing through the element of surface S with the time t, is (C)nQNqu as tNqus t Current is the time rat

6、e of change of charge, so QINqusJst Consider the steady motion of one kind of charge carriers, each of charge q (which is negative for electrons), across an element of surface s with a velocity u. If N is the number of charge carriers per unit volume, then in time t each charge carrier moves a dista

7、nce u t 7Define vector current density(電流密度矢量電流密度矢量):;IJs SIJ dsA2 ( /)JuA mIt is convenient to define a vector point function, volume current density, or simply current density J, in amperes per square meter,2 (A/m )JNquThe total current I flowing through an arbitrary surface S is then the flux of

8、the J vector through S:JNquNqWe may rewrite:which is the relation between the convection current density and the velocity of the charge carrier.8電流場(chǎng)：電流場(chǎng)：大塊導(dǎo)體中各點(diǎn)的電流密度矢量大塊導(dǎo)體中各點(diǎn)的電流密度矢量 J J，在不同的點(diǎn)，在不同的點(diǎn)有不同的大小和方向，給定空間每一點(diǎn)處的電流密度矢量有不同的大小和方向，給定空間每一點(diǎn)處的電流密度矢量J J的大小和方向，構(gòu)成一個(gè)矢量場(chǎng)，的大小和方向，構(gòu)成一個(gè)矢量場(chǎng)，即：電流場(chǎng)。即：電流場(chǎng)。電流線：電流線：

9、電流場(chǎng)由電流場(chǎng)由電流線電流線來描述，電流線上每點(diǎn)的切來描述，電流線上每點(diǎn)的切線方向都和該點(diǎn)的電流密度矢量方向一致。線方向都和該點(diǎn)的電流密度矢量方向一致。電流管：電流管：由一束電流線圍成的管狀區(qū)域叫由一束電流線圍成的管狀區(qū)域叫電流管，電流管，通通過同一個(gè)電流管的各個(gè)截面的電流強(qiáng)度都相等過同一個(gè)電流管的各個(gè)截面的電流強(qiáng)度都相等 電流線：電流場(chǎng)J1S2S9In the case of conduction currents there may be more than one kind one charge carriers drifting with different velocities 2

10、(A/m )iiiiJN quIt can be justified analytically that for most conducting materials the average drift velocity is directly proportional to the electric field intensity. For metallic conductors we write ( / )euEm sWhere e is the electron mobility in (m/Vs). We have2 (A/m )eeJNquEE Where e=-Ne is the c

11、harge density of the drifting electrons and is a negative quantity. =- e e, is a macroscopic constitutive parameter of the medium called conductivity.1012lIVIRIsGllRss-resistance電阻電阻1GR-conductance電導(dǎo)電導(dǎo)1-resistivity電阻率電阻率-conductivity 電導(dǎo)率電導(dǎo)率parallel： series：12sRRR1212111 ppRRRGGG11The conductivities

12、of several mediaunit in S/mMediaConductivitiesMediaConductivitiesSilverSea waterCopperPure waterGoldDry soilAluminumTransformer oilBrassGlassIronRubber71017. 671080. 531071010. 451071054. 3111071057. 11210710151041219111911年荷蘭物理學(xué)家發(fā)現(xiàn)電阻完全消失的現(xiàn)象僅發(fā)生在物質(zhì)處于極端年荷蘭物理學(xué)家發(fā)現(xiàn)電阻完全消失的現(xiàn)象僅發(fā)生在物質(zhì)處于極端物理狀態(tài)的溫度物理狀態(tài)的溫度- -臨界溫度

13、。臨界溫度。19331933年，德國(guó)科學(xué)家發(fā)現(xiàn)超導(dǎo)體的抗磁效應(yīng)。年，德國(guó)科學(xué)家發(fā)現(xiàn)超導(dǎo)體的抗磁效應(yīng)。19621962年，英國(guó)劍橋大學(xué)博士研究生證明兩個(gè)以薄的絕緣層相隔的超導(dǎo)年，英國(guó)劍橋大學(xué)博士研究生證明兩個(gè)以薄的絕緣層相隔的超導(dǎo)體之間會(huì)產(chǎn)生一種特殊現(xiàn)象體之間會(huì)產(chǎn)生一種特殊現(xiàn)象- -隧道效應(yīng)。隧道效應(yīng)。19621962年，美國(guó)年，美國(guó)WestinghoseWestinghose公司用鈦化鈮合金拉制成超導(dǎo)電線。（公司用鈦化鈮合金拉制成超導(dǎo)電線。（15-25K)15-25K)19861986年，在蘇黎世附近一家年，在蘇黎世附近一家IBMIBM實(shí)驗(yàn)室工作的兩位瑞士物理學(xué)家宣布，實(shí)驗(yàn)室工作的兩位瑞士物理

14、學(xué)家宣布，他們研制成含鑭和鋇的銅酸鹽（氧化銅）化合物，其臨界溫度為他們研制成含鑭和鋇的銅酸鹽（氧化銅）化合物，其臨界溫度為30K30K。更令人驚奇的是此物質(zhì)為一種陶瓷材料，在常溫下具有絕緣體的所用更令人驚奇的是此物質(zhì)為一種陶瓷材料，在常溫下具有絕緣體的所用特征。特征。在隨之而來的高溫超導(dǎo)熱中又發(fā)現(xiàn)了一批具有較高臨界溫度的化合物，在隨之而來的高溫超導(dǎo)熱中又發(fā)現(xiàn)了一批具有較高臨界溫度的化合物，目前臨界溫度的最高紀(jì)錄約是目前臨界溫度的最高紀(jì)錄約是150K150K（-123-1230 0C C）13超導(dǎo)材料的基本物理特征超導(dǎo)材料的基本物理特征：v零電阻現(xiàn)象零電阻現(xiàn)象v完全抗磁性完全抗磁性（邁斯納效應(yīng)）

15、（邁斯納效應(yīng)）v超導(dǎo)態(tài)并非僅取決于溫度超導(dǎo)態(tài)并非僅取決于溫度( (臨界電流和臨界磁場(chǎng)臨界電流和臨界磁場(chǎng)) )普通普通導(dǎo)體導(dǎo)體超導(dǎo)體超導(dǎo)體14高溫超導(dǎo)材料制備所面臨的問題：高溫超導(dǎo)材料制備所面臨的問題： 材料制造成本高材料制造成本高, 價(jià)格昂貴價(jià)格昂貴。 在長(zhǎng)距離超導(dǎo)線材的制造上面仍然在長(zhǎng)距離超導(dǎo)線材的制造上面仍然有很大的難度。（氧化物高溫超導(dǎo)陶有很大的難度。（氧化物高溫超導(dǎo)陶瓷材料各向異性和短的電子相干長(zhǎng)度瓷材料各向異性和短的電子相干長(zhǎng)度以及大量晶界的存在嚴(yán)重影響線材的以及大量晶界的存在嚴(yán)重影響線材的超導(dǎo)電性。）超導(dǎo)電性。） 高溫超導(dǎo)材料臨界電流和臨界磁場(chǎng)高溫超導(dǎo)材料臨界電流和臨界磁場(chǎng)的提高仍

16、是科學(xué)家研究的難題。的提高仍是科學(xué)家研究的難題。-Robert F. Serverce.-Robert F. Serverce.ScienceScience,295,786(2002).,295,786(2002).15節(jié)省大量資金節(jié)省大量資金緩解環(huán)境污染緩解環(huán)境污染超導(dǎo)電纜、超導(dǎo)發(fā)電超導(dǎo)電纜、超導(dǎo)發(fā)電機(jī)、超導(dǎo)電纜機(jī)、超導(dǎo)電纜預(yù)預(yù) 測(cè)測(cè)低電力低電力低能耗低能耗靈敏度度高靈敏度度高釔鋇銅超導(dǎo)薄膜釔鋇銅超導(dǎo)薄膜-應(yīng)用于諧振器、濾波應(yīng)用于諧振器、濾波器、天線等有源器件器、天線等有源器件商品化商品化低電力低電力低能耗低能耗釔鋇銅超導(dǎo)超導(dǎo)塊材釔鋇銅超導(dǎo)超導(dǎo)塊材-用于磁懸浮、儲(chǔ)能飛用于磁懸浮、儲(chǔ)能飛輪等方

17、面輪等方面即即 將將實(shí)業(yè)化實(shí)業(yè)化預(yù)計(jì)在預(yù)計(jì)在20202020年年左右會(huì)形成左右會(huì)形成1500-20001500-2000億美億美元的超導(dǎo)市場(chǎng)，元的超導(dǎo)市場(chǎng)，其中高溫超導(dǎo)其中高溫超導(dǎo)占一半占一半162 (A/m )JEThis equation is a constitutive relation of the conducting medium. Isotropic materials for which the linear relation holds are called ohmic media. It is generally referred to as the point form

18、 of Ohms law. It holds at all points in space. The unit for is ampere per volt-meter (A/Vm), or siemens per meter (S/m). The reciprocal of conductivity is called resistivity, in ohm-meter ( m). 172. Electromotive Force and Kirchhoffs Voltage LawThis equation tells us that a steady current cannot be

19、maintained in the same direction in a closed circuit by an electrostatic field. A steady current in a circuit is the result of the motion of charge carriers, which, in their paths, collide with atoms and dissipate energy in the circuit. This energy must come from a non-conservative field, These elec

20、trical energy sources, when connected in an electric circuit, provide a driving force for the charge carriers. This force manifests itself as an equivalent impressed electric field intensity Ei.100CCE dJ dll18iEE Chemical action creates a cumulation of positive and negative charges at electrodes 1 a

21、nd 2, respectively. These charges give rise to an electrostatic field intensity E both outside and inside the battery. Inside the battery, E must be equal in magnitude and opposite in direction to the nonconservative Ei produced by chemical action, since no current flows in the open-circuited batter

22、y and the net force acting on the charge carriers must vanish.EConducting mediumPNEImpressed sourceEi19The electromotive force is a measure of the strength of the nonconservative source, denoted by , we have1122dd (Inside the source)iEl =El 211200CE dE dE dlllOutside the sourceInside the source11212

23、12221dd =d-VV ViEl =El El Outside the sourceInside the sourceWe have expressed the emf of the source as a line integral of the conservative E and interpreted it as a voltage rise. In spite of the nonconservative nature of Ei, the emf can be expressed as a potential difference between the positive an

24、d negative terminals. EConducting mediumPNEImpressed sourceEi20EConducting mediumPNEImpressed sourceEi()1()iiCCJEEEEdJ dRIllIf there are more than one source of electromotive force and more than one resistor (including the internal resistances of the sources) in the closed path, we generalize (V)jkk

25、jkR IEquation is an expression of Kirchhoffs voltage law. It states that around a closed path in an electric circuit the algebraic sum of the emfs (voltage rises) is equal to the algebraic sum of the voltage drops across the resistances.213. Equation of Continuity and Kirchhoffs Current LawThe princ

26、iple (law) of conservation of charge: Electric charges may not be created nor destroyed, but merely transported; all charges either at rest or in motion must be conserved for at all time；any change of charge in a region must be accompanied by a flow of charge across the surface bounding the region.S

27、naVJISVVVVdQdIJ dsdVdVdtdttJdVdVt 22Since the equation must hold regardless of the choice of V, the integrands must be equal. Thus we have3 (A/m )Jt This point relationship derived from the principle of conservation of charge is called the equation of continuity.For the steady currents, / t=0, the e

28、quation of continuity is :0J0SJ ds0jjI Kirchhoffs current law :the algebraic sum of all the currents out of a junction in an electric circuit is zero.a1I2I3I23We are now in a position to prove this statement and to calculate the time it takes to reach an equilibrium.tEJEJt E0twhere 0 is the initial

29、charge density at t = 0.The time constant is called the relaxation time(馳豫時(shí)間）馳豫時(shí)間）.銅，銅， 1.521.5210-19SAn initial charge density 0 will decay to 1/e or 36.8% of its value in a time equal to0ttCee24summary1. Current Density and Ohms Law SIJ dsA2 ( /)JuA m2 (A/m )JE12lIVIRIsG253. Equation of Continuity and Kirchhoffs Current Law2. Electromotive Force and Kirchhoffs Voltage Law1121212221dd =d-VV ViEl =El El Outside the sourceInside the source