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Field and Wave Electromagnetic电磁场与电磁波电磁场与电磁波2015.10.311.Current Density and Ohms LawReview23.Equation of Continuity and Kirchhoffs Current Law2.Electromotive Force and Kirchhoffs Voltage LawOutside the sourceInside the sourceEConducting mediumPNEImpressed sourceEi3We are now in a position to prove this statement and to calculate the time it takes to reach an equilibrium.where 0 is the initial charge density at t=0.The time constant is called the relaxation time(驰豫时间）驰豫时间）.铜，=1.521.5210-19SAn initial charge density 0 will decay to 1/e or 36.8%of its value in a time equal to4For a homogeneous conducting mediumWe know that a curl-free vector field can be expressed as the gradient of a scalar potential field.Let us writeSubstitution of this equation into yields a Laplaces equation in ;that is5Two fields are found to be very similar in source-free region.Steady Electric Current Field Electrostatic Field The electric current density J corresponds to the electric field intensity E,and the electric current lines to the electric field lines.In some cases,since the steady electric current field is easy to be constructed and measured,the electrostatic field can be investigated based on the steady electric current field with the same boundary conditions,and this method is called electrostatic simulation.6Capacitance JEResistanceBased on the equations for two fields,we can find the resistance and conductance between two electrodes as7In certain situations,electrostatic and steady-current problems are not exactly analogous,even when the geometrical configurations are the same.This is because current flow can be confined strictly within a conductor(which has a very large in comparison to that of the surrounding medium),whereas electric flux usually cannot be contained within a dielectric slab of finite dimensions.The range of the dielectric constant of available materials is very limited,and the fluxfringing around conductor edges makes the computation of capacitance less accurate.8Main topic 恒定电流恒定电流3.电阻的计算电阻的计算1.功率耗散和焦耳定律功率耗散和焦耳定律2.电流密度的边界条件电流密度的边界条件91.功率耗散和焦耳定律功率耗散和焦耳定律 宏宏观观上上，导导体体中中的的电电子子受受电电场场的的影影响响，发发生生漂漂移移运运动动；在在微微观观上上，这这些些电电子子与与晶晶格格上上的的原原子子发发生生碰碰撞撞。因因此此，能能量量从从电电场场传传到到作作热热振振动动的的原原子子上上。将将电电荷荷移移动动了了一一段段距距离离 ，电电场场E 作作的的功功为为q E ，则则其其所所对对应应的功率为：的功率为：其中其中 u 为漂移速度。传递到体积为漂移速度。传递到体积 dv 内所有电荷载体的总功率为内所有电荷载体的总功率为:10The total electric power converted into heat in volume V:This is known as Joules law.The point function EJ is a power density under steady-current conditions.In a conductor of a constant cross section,we can written as112.电流密度的边界条件电流密度的边界条件When current obliquely crosses an interface between two media with different conductivities(1 2),the current density vector changes both in direction and in magnitude.A set of boundary conditions can be derived for J in a way similar to that used in Section 3-9 for obtaining the boundary conditions for D and E.The governing equations for steady current density J in the absence of non-conservative energy sources areDifferential formIntegral formGoverning Equations for Steady Current Density12E2E1 2 1at w hacdban2hS 2 1an2D1D2 s13J2J1 2 1at w hacdban2hS2 1an2J1J2 sthe normal component of current density vector J being continuous.the ratio of the tangential components of current density vector J at two sides of an interface is equal to the ratio of the conductivities.14 1,1an2E2,D2,J2 s 2,2E1,D1,J1When a steady-current flows across the boundary between two different lossy dielectrics:15For a homogeneous conducting mediumWe know that a curl-free vector field can be expressed as the gradient of a scalar potential field.Let us writeSubstitution of this equation into yields a Laplaces equation in ;that is16Example 5-4 P214:An emf is applied across a parallel-plate capacitor of area S.The space between the conductive plates is filled with two different lossy dielectrics of thickness d1 and d2,permittivities 1 and 2,and conductivities 1 and 2 respectively.Determine(a)the current density between the plates,(b)the electric field intensities in both dielectrics,and(c)the surface charge densities on the plates and at the interface.x12+-yo171.x12+-yo2.Method oneOr method two183.191.Choose an appropriate coordinate system for the given geometry.2.Assume a potential difference V0 between conductor terminals.3.Find E from E=-V(2V=0),or other relations.4.Find the total current where S is the cross-sectional area over which I flows.5.Find resistance R by taking the ratio V0/I.The procedure for computing the resistance of a piece of conducting material between specified equipotential surfaces(or terminals)is as follows:3.电阻计算电阻计算(Resistance Calculations)20 Example 5-6.一一导导电电材材料料形形状状是是四四分分之之一一的的扁扁平平圆圆垫垫圈圈,其其均均匀匀厚厚度度为为 h,电电导导率率为为 ,垫垫圈圈内内半半径径为为 a,外外半半径径b,如如下下图图所所示示,求求端端面面之间的电阻之间的电阻.思路思路V0yxhabr0(r,)0 21 Solution:The cylindrical coordinate system should be selected.Assume the electric potential difference between two end faces is V0,and let Since the electric potential V is related to the angle ,it should satisfy the following equationThe general solution isThe electric potential at The electric potential atV0yxhabr0(r,)0 22 Based on the given boundary conditions,we findThe current density J in the conducting medium is Then the current I flowing into the conducting medium across the end face at isConsequently,the resistance R between two end faces is23summary1.Power Dissipation and Joules Law3.Resistance Calculationselectrostatic simulation242.Boundary Conditions for Current DensityDifferential formIntegral formGoverning Equations for Steady Current DensityWhen a steady-current flows across the boundary between two different lossy dielectrics:25homeworkThank you!Bye-bye!答疑安排答疑安排时间：时间：地点：地点：1401，1403P.5-15261.Current Density and Ohms LawReview Steady Electric Currents273.Equation of Continuity and Kirchhoffs Current Law2.Electromotive Force and Kirchhoffs Voltage LawOutside the sourceInside the sourceEConducting mediumPNEImpressed sourceE284.Power Dissipation and Joules Law6.Resistance Calculationselectrostatic simulation295.Boundary Conditions for Current DensityDifferential formIntegral formGoverning Equations for Steady Current DensityWhen a steady-current flows across the boundary between two different lossy dielectrics:30abc 1,1 2,2V0I31abc 1,1 2,2V0I32abc 1,1 2,2V0I33
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