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1、Field and Wave Electromagnetic电磁场与电磁波电磁场与电磁波2015. 11.101电磁场与电磁波第15讲边界条件电感磁能1. The Magnetic Dipole2. Magnetization and Equivalent Current DensitiesReview2电磁场与电磁波第15讲边界条件电感磁能3. Magnetic Field Intensity and Relative PermeabilityDifferential formIntegral formPostulates of Magnetostatics in magnetic mate
2、rial3电磁场与电磁波第15讲边界条件电感磁能Main topic 静磁场静磁场1. 静磁场的边界条件静磁场的边界条件2. 电感和电感器电感和电感器3. 静磁场能量静磁场能量4电磁场与电磁波第15讲边界条件电感磁能1. 静磁场的边界条件静磁场的边界条件 1 2B2H1B1H2an2Js5电磁场与电磁波第15讲边界条件电感磁能E2E1 2 1at w hacdban2hS 2 1an2D1D2 s6电磁场与电磁波第15讲边界条件电感磁能(a) 在分界面处在分界面处B的法向分量是连续的的法向分量是连续的.For linear isotropic media, we have(b) 磁场磁场H 的
3、切向分量在跨越存在面电流的分界面时的切向分量在跨越存在面电流的分界面时,是不连续的是不连续的.在在跨跨越越几几乎乎所所有有物物理理媒媒质质的的边边界界时时, H 的的切切向向分分量量都都是是连连续续的的; 只只有有当分界面为理想导体或超导体时当分界面为理想导体或超导体时,它才不连续它才不连续.7电磁场与电磁波第15讲边界条件电感磁能 Example 1. A loop magnetic core with a gap is closely wound by a coil with N turns, as shown in the figure. When the coil carries a
4、current I, and the leakage magnetic flux outside the coil is neglected, find the magnetic flux density and the magnetic field intensity in the core and the gap. Solution: Since the leakage magnetic flux is neglected, the direction of the magnetic flux density is around the circle, and it is perpendi
5、cular to the end faces of the gap. From the boundary condition, we know that the magnetic flux density Bg in the gap is equal to Bf in the core, i.e. 8电磁场与电磁波第15讲边界条件电感磁能 Since r0 a , the magnetic field in the core can be considered to be uniform. Using Amperes circuital law in media, and taking the
6、 circle of radius r0 as the integral path, then we have Considering , we have ThenIn the gapIn the core9电磁场与电磁波第15讲边界条件电感磁能2. 电感和电感器电感和电感器互磁通互磁通 12 (Mutual flux)Faradays law of electromagnetic inductionBiot-Savart law 其其中中,比比例例系系数数 L12 称称为为回回路路C1 和和 C2之之间间的的互互感感, with the unit henry (H). In case C2
7、has N2 turns, the flux linkage 12 (磁链数磁链数)due to 1210电磁场与电磁波第15讲边界条件电感磁能 由由上上式式,两两个个电电路路之之间间的的互互感感是是一一电电路路通通以以单单位位电电流流时时,另另一一电电路路所交链的磁通链所交链的磁通链. 这个关系只适用于这个关系只适用于线性媒质线性媒质.L12 更一般的定义为更一般的定义为:I1 所所产产生生的的某某些些磁磁通通量量只只与与回回路路 C1 自自身身交交链链,而而不不与与 C2交交链链.the total flux linkage (磁链数磁链数) with C1 caused by I1 is
8、Neumann formula11电磁场与电磁波第15讲边界条件电感磁能回路回路 C1 的自感定义为在回路本身通以单位电流时所产生的磁链的自感定义为在回路本身通以单位电流时所产生的磁链,即即:对线性媒质而言对线性媒质而言,更一般的表达式更一般的表达式 L11 为为: 一一个个回回路路或或电电路路的的自自感感,取取决决于于构构成成这这个个回回路路或或电电路路的的导导体体的的几几何何形形状状和和物物理理排排列列以以及及媒媒质质的的磁磁导导率率.在在线线性性媒媒质质中中,自自感感与与回回路路或或电电路路的的电电流流无无关关.事事实实上上,无无论论回回路路或或电电路路是是开开路路还还是是闭闭合合的的,
9、也也无论它是否靠近另一回路或电路无论它是否靠近另一回路或电路,其自感总是存在的其自感总是存在的.12电磁场与电磁波第15讲边界条件电感磁能 排排列列成成适适当当形形状状(例例如如由由导导线线缠缠绕绕而而成成的的线线圈圈)以以提提供供一一定定数数量量自自感感的的导导体体称称为为电电感感器器. 就就象象电电容容器器可可以以储储存存电电能能一一样样,电电感感器器能能够够储储存磁能存磁能.The procedure for determining the self-inductance of an inductor is as follows:1. 对于给定的几何形状选择适当的坐标系对于给定的几何形状
10、选择适当的坐标系. 2. 假设导线中的电流为假设导线中的电流为 I. 3. 如如果果存存在在对对称称性性,可可以以根根据据安安培培环环路路定定理理由由I 求求 B; 如如果果不不存存在在对对称称性性,则可以用则可以用 Biot-Savart law.4. 用积分的方法由用积分的方法由B求出每一圈所交链的磁通求出每一圈所交链的磁通 5. Find the flux linkage by multiplying by the number of turns. 6. Find L by taking the ratio L= /I.13电磁场与电磁波第15讲边界条件电感磁能EXAMPLE 6-14
11、P18114电磁场与电磁波第15讲边界条件电感磁能 Example. Calculate the mutual inductance (互互感感) between an infinitely long straight line and a rectangular coil. The line and the coil are at the same plane, and in vacuum.abdrrD0I1I2zS2 Solution: Select cylindrical coordinate system, and let the infinitely long straight li
12、ne to be at the z-axis. The magnetic flux density produced by current I1 is then The magnetic flux linkage 1212 with current I2 by current I1 is15电磁场与电磁波第15讲边界条件电感磁能We have If the flowing direction of the current I2 is counter clockwise, then the B1 and dS are opposite, and L12 0.abdrrD0I1I2zS2 If t
13、he flowing direction of the current I2 is clockwise, dS and B1 have the same direction. Then16电磁场与电磁波第15讲边界条件电感磁能例题例题: :求无限长导线与三角形回路之间的互感求无限长导线与三角形回路之间的互感I1rdrzx017电磁场与电磁波第15讲边界条件电感磁能3. 磁能磁能(Magnetic Energy) If an impressed source (外外源源)is applied to a circuit(回回路路), a current will be generated in t
14、he circuit. In the process of establishing the current, the reaction magnetic flux (感感应应磁磁通通)in the circuit will resist (阻止阻止)the increment of the current. Assume the current is increased very slowly(非非常常慢慢) so that radiation loss can be neglected, all energy provided by the impressed source will be
15、 stored(储存储存) in the magnetic field around the circuit. Based on the work done by the impressed source, the energy stored in the magnetic field can be calculated. In order to overcome the back electromotive force due to the reaction magnetic flux and to maintain the current, the impressed source has
16、 to do work(做功做功).18电磁场与电磁波第15讲边界条件电感磁能 考考虑虑一一个个自自感感为为 L1的的单单独独闭闭合合回回路路,其其初初始始电电流流为为0,将将电电流流源源与与回回路路连连接接,电电流流从从0增增加加到到 I1. 从从物物理理学学中中知知道道回回路路中中将将产产生生感感应应电电动动势势并并阻阻碍碍电电流流的的变变化化. 要要克克服服这这个个感感应应电电动动势势,必必须须有有外外源源作作功功. 假假设设电电感感两两端的的电压为端的的电压为 v1=L1di1/dt ,则所需做的功为则所需做的功为:对于线性媒质有对于线性媒质有L1= 1/I1 ,则上式变为则上式变为:
17、这个功看作是以磁能形式储存起来这个功看作是以磁能形式储存起来.Now consider two closed loops C1 and C2 carrying currents i1 and i2, respectively. The total amount of work done in rasing the currents in loops C1 and C2 from zero to I1 and I2 , respcetively, is 19电磁场与电磁波第15讲边界条件电感磁能which is stored as magnetic energy. Where W21 isGen
18、eralizing this result to a system of N loops (N N个个回回路路) carrying currents I1 , I2 , In , we obtain Which is the energy stored in the magnetic field. For a current I flowing in a single inductor(单单个个电电感感器器)with inductance L, the stored magnetic energy is 20电磁场与电磁波第15讲边界条件电感磁能It is often desirable to
19、 express the magnetic energy in terms of field quantities B and H. We haveNoting that H=B/ , we can rewriteandWhere wm is defined as magnetic energy density(磁能密度磁能密度)21电磁场与电磁波第15讲边界条件电感磁能静静 电电 场场恒定磁场恒定磁场物理量物理量媒质特性媒质特性场方程式场方程式边界条件边界条件能量密度能量密度力力无旋无旋无散无散有散有散有旋有旋电场强度电场强度 E磁通密度磁通密度 B电通密度电通密度 D磁场强度磁场强度 H介
20、电常数介电常数 磁导率磁导率 22电磁场与电磁波第15讲边界条件电感磁能1. Introduction2. Fundamental Postulates of Magnetostatics in free spaceLorentzs force equationDifferential formIntegral formPostulates of Magnetostatics in Free SpaceReview23电磁场与电磁波第15讲边界条件电感磁能3. Vector Magnetic Potential4. The Biot-Savart Law and Applications24电
21、磁场与电磁波第15讲边界条件电感磁能5. The Magnetic Dipole6. Magnetization and Equivalent Current Densities25电磁场与电磁波第15讲边界条件电感磁能7. Magnetic Field Intensity and Relative PermeabilityDifferential formIntegral formPostulates of Magnetostatics in magnetic material26电磁场与电磁波第15讲边界条件电感磁能8. Boundary Conditions for Magnetostaic Fields9. Inductances and Inductors磁通量；磁链数；自感；互感磁通量；磁链数；自感；互感27电磁场与电磁波第15讲边界条件电感磁能10. Magnetic Energy28电磁场与电磁波第15讲边界条件电感磁能homeworkThank you! Bye-bye!答疑安排答疑安排时间：时间：地点：地点：1401， 1403P. 6-3829电磁场与电磁波第15讲边界条件电感磁能
