工程電磁場第一章

1、Engineering Electromagnetics工程電磁場主講 彭潤玲 講師The aim of having this lesson1. Electromagnetics is both the oldest and most basic branch of electrical engineering2. It deals with the following questions: vWhats electricity? How does it behave?vWhat can it do? How can we control it? 3. So it is essential

2、to understand electromagnetics for us to fully understand the operation of many electrical devices and effects.vTransformer Computer wireless communicationChapter 1 Vector AnalysisChapter 1 Vector Analysis1. Vector analysis is the branch of mathematics that was developed to describe quantities that

3、are both directional in nature and distributed over regions of space.2. The reason for starting our study of electromagnetics with vector analysis3. The main contents in the chapter1.1 Scalars and Vectors1.1 Scalars and Vectors1. 1. A scalar: a quantity that can be specified by a single numberA scal

4、ar: a quantity that can be specified by a single numbervWhats a scalar?vIn mathematics-any real quantityvIn physics-given a physical unit, with physical significance.2. 2. A vector:A vector: vWhats a vector?3. 3. Scalar and Vector fieldsScalar and Vector fields4. 4. Boldface type-Vector and Italic t

5、ype-ScalarBoldface type-Vector and Italic type-ScalarvA field is defined as some function of that vector which connects an arbitrary origin to a general point in space.5. 5. When writing, please draw a line or an arrow over a vectorWhen writing, please draw a line or an arrow over a vector1.2 Vector

6、 Algebra1.2 Vector Algebra 1. 1. The main contents of this subsectionThe main contents of this subsectionvThe rules of vector algebra: addition, subtraction, multiplication by a scalar2. 2. VectorialVectorial addition additionvFollows the parallelogram lawvFollows the commutative lawvFollows the ass

7、ociative lawABBACBACBA)()(1.2 Vector Algebra1.2 Vector Algebra 3. 3. Vector-two or three componentsVector-two or three components Vector addition-adding the corresponding components4. Vector subtraction4. Vector subtraction)( BABA5. Scalar times Vector5. Scalar times VectorvIf the scalar0, the vecto

8、r is changed only with magnitudevIf the scalar0, both magnitude and direction are changed6. 6. Division of a Vector by a ScalarDivision of a Vector by a Scalar AssA18.8. BABA07. 7. BsAsBrArBAsBArBAsr)()()(Obeys associative and distributive laws1.3 The Cartesian Coordinate System1.3 The Cartesian Coo

9、rdinate System 1. 1. Three coordinate systemsThree coordinate systems2. The Cartesian coordinate system has 2. The Cartesian coordinate system has three axesthree axesvMutually normal to each othervA right-handed systemvThe intersection of three surfacesx=constant;y=constant;z=constantvCartesian sys

10、tem or rectangular system vCircular cylindrical systemvSpherical coordinate systemQ1.3 The Cartesian Coordinate System1.3 The Cartesian Coordinate System 3. 3. In other coordinate systems, points are also located at the In other coordinate systems, points are also located at the common intersection

11、of three surfaces, not necessarily planes, common intersection of three surfaces, not necessarily planes, but still normal at the point of intersection.but still normal at the point of intersection.4. Shown in Fig.1.2c, the coordinate of P is x,y,z respectively and 4. Shown in Fig.1.2c, the coordina

12、te of P is x,y,z respectively and that of Q is that of Q is x+dxx+dx, , y+dyy+dy, z+dz., z+dz. vThe differential volume is vThe differential area of three surfacesvThe distance dL from P to Q is dxdydzd dxdzdydzdxdy;222)()()(dzdydxdL1.4 Vector Components and Unit Vectors1.4 Vector Components and Uni

13、t Vectors 1. 1. A way to identify a vector - giving the three component A way to identify a vector - giving the three component vectors, lying along the three coordinate axes.vectors, lying along the three coordinate axes.zyxrzyxazayaxr2. Provided that , , are the unit vectors2. Provided that , , ar

14、e the unit vectors xayazaNote that the unit vectors are directed toward increasing Note that the unit vectors are directed toward increasing coordinate value and perpendicular to the surface on which coordinate value and perpendicular to the surface on which that coordinate value is constant.that co

15、ordinate value is constant.1.4 Vector Components and Unit Vectors1.4 Vector Components and Unit Vectors 3. 3. Shown in Fig. 1.3c, a vector pointing from the origin to point Shown in Fig. 1.3c, a vector pointing from the origin to point P(1,2,3) is writtenP(1,2,3) is writtenzyxPaaar32The vector from

16、P to Q(2,-2,1) can be obtained by the rule of The vector from P to Q(2,-2,1) can be obtained by the rule of vector addition.vector addition.zyxPQPQaaarrR24B1.4 Vector Components and Unit Vectors1.4 Vector Components and Unit Vectors 4. 4. If we discuss a force vector , or other vectors, we should If

17、 we discuss a force vector , or other vectors, we should use suitable letters for the three component vectors.use suitable letters for the three component vectors.FzzyyxxaFaFaFF5. Any vector ,not only the displacement type, may be 5. Any vector ,not only the displacement type, may be describeddescri

18、bed BzzyyxxaBaBaBB222zyxBBBBBThe magnitude of is B222zyxBBBBBaA unit vector in the direction of the vector is B6. Example: Specify the unit vector extending from the origin 6. Example: Specify the unit vector extending from the origin toward the point G(2,-2,-1)toward the point G(2,-2,-1)7. Problem:

19、 D1.1(P9)7. Problem: D1.1(P9)1.6 The Dot Product1.6 The Dot Product 1. 1. The dot product of two vectors: -a scalarThe dot product of two vectors: -a scalarABBABAcos2. Dot product obeys the commutative law2. Dot product obeys the commutative lawABBA3. For example, the work of applied over a straight

20、 3. For example, the work of applied over a straight displacement is :displacement is : FLLFworkLdFworkF( has a constant magnitude )F( varies along the path )4. 4. zzyyxxzzyyxxzzyyxxBABABAaBaBaBaAaAaABA)()(0yzzyxzzxxyyxaaaaaaaaaaaa1.6 The Dot Product1.6 The Dot Product 5.5. ;22AAAA1AAaa6. One of the

21、 important application of the dot product:6. One of the important application of the dot product:Finding the component of a vector in a given directionFor example: The component of in the direction of unit vector is :BaBaBaBcosaaB is the projection of in the direction BThe component vector of in the

22、 direction of is BaaB)(a1.6 The Dot Product1.6 The Dot Product 7. Example 1.2: the vector field and the point Q(4,5,2), find: (1) at Q; (2) the scalar component of at Q in the direction of ; (3) the vector component of at Q in the direction of ; (4) the angle between and 8. Drill problem: D1.3(P13)z

23、yxaaxayG35 . 2GG)22(31zyxNaaaaGNaGa)(QrGNa1.7 The Cross Product1.7 The Cross Product 1. The cross product of two vectors:ABNBAaBAsinThe magnitude: ; The direction: ABBAsinNa2. The cross product is not commutative: )(ABBA3. For the unit vectors, , ,zyxaaaxzyaaayxzaaaTo find the area of a parallelogra

24、m4. An application of the cross product: 5. The evaluation of the cross product() ()()()()xxyyzzxxyyzzyzzyxzxxzyxyyxzA BA aA aA aB aB aB aA BA B aA BA B aA BA B a1.7 The Cross Product1.7 The Cross Product 6. zyxzyxzyxBBBAAAaaaBA7. For example:23;425xyzxyzAaaa Baaa 231131416425xyzxyzaaaA Baaa 8. Dril

25、l problem:D1.4(P15)1.8 Circular Cylindrical Coordinate 1. The polar coordinate of analytic geometry: 、2. The location of a point in a cartesian coordinate system :the intersection of three planes:x=constant;y=constant;z=constantbut in cylindrical coordinates: =constant;=constant;z=constant3. Three u

26、nit vectors in cylindrical coordinate:aaza1.8 Circular Cylindrical Coordinate 4. In cartesian coordinates, unit vectors are not changed; 6. A differential volume element in cylindrical coordinates:dzddaabut in cylindrical coordinates, and are changed with their directions. So in integration or diffe

27、rentiation with respect to , then and must not be treated as constants.aa5. A righted-handed cylindrical coordinate system:zaaaThe differential surfaces:d d d dz ddz1.8 Circular Cylindrical Coordinate 7. Express the cylindrical variables in terms of x, y and z:zzxyyxzzyx122tansincosPlease note the sign of , and the proper value of the angle1.8 Circular Cylindrical Coordinate 8. Know , how to getsincos)(yxzzyyxxAAaaAaAaAaAAzzyyxxaAaAaAAzzaAaAaAAcossin)(yxzzyyxxAAaaAaAaAaAAzzzzyyxxzzAaaAaAaAaAA)(aazaxayazacoscossinsin000011.8 Circular Cylindrica