電磁場(chǎng)與電磁波第一課

1、Electromagnetic field theory fundamentals (bilingual course)電磁場(chǎng)與電磁波(雙語(yǔ)課程）(1) Grasp the basic attributes and movement rules about macro electromagnetic fields. (2) Grasp the basic solving methods of macro electromagnetic field problems. (3) Know the main application of macro electromagnetic fields. (

2、4) Develop ability to solve practical problems using mathematics.Northeast petroleum university學(xué)習(xí)方法和要求Learning approaches and requirements(1) Preview before class.(2) Pay attention in class.(3) Take the notes in class.(4) Do some homework after class. (5) Read intensively one or two reference booksN

3、ortheast petroleum university1 Vector Analysis1.1 Scalar and Vector QuantitiesNortheast petroleum universityScalar A scalar quantity has magnitude only. Some examples：time, temperature, mass, work, current and electric charge.t, T, m, W, I or q. VectorA vector quantity has both magnitude and directi

4、on. Some examples: force, velocity, electric field intensity and magnetic flux density. F, , E or BNortheast petroleum universityUnit vector: A vector of unit magnitude (length) .Vector A can be written as What is the meaning of “A = B ” ?The magnitude of vector A isWhat is the zero vector (null vec

5、tor) ?Northeast petroleum universityField Classificationa. 按物理量的性質(zhì)分： properties of physical quantity scalar field 標(biāo)量場(chǎng) vector field 矢量場(chǎng)按場(chǎng)量與時(shí)間的關(guān)系分 relationship between field quantity and time static field 靜態(tài)場(chǎng) (穩(wěn)態(tài)場(chǎng) steady field) dynamic field 動(dòng)態(tài)場(chǎng)(時(shí)變場(chǎng)time-varying field)Field What is a field?A field is a

6、 function that describes a physical quantity at all points in space (or in a region). Northeast petroleum universityScalar FieldA scalar field is specified by a scalar quantity at all points in space (or in a region). The distribution of a scalar field u can be described by means of equiscalar surfa

7、ces(equiscalar lines).The equation of an equiscalar surface isNortheast petroleum universityVector FieldA vector field is specified by a vector quantity at all points in space (or in a region). orwhereare three components of vector A. These are three scalar fields.static field ANortheast petroleum u

8、niversityThe distribution of a vector field can be described by means of lines of force (or streamlines) in the region.Note: The tangent line direction of any point on the streamlines is the same as the direction of the vector at the point.+ The equation of the streamlines satisfiesNortheast petrole

9、um university1.2 Vector Operations Northeast petroleum university Vector Addition Parallelogram law Commutative law Associative law 多個(gè)矢量首尾相連組成閉合多邊形，其矢量和必為零。Northeast petroleum university Position vector r: from origin o to point PNote: Each point can determine a position vector, and vice versa. A re

10、ctangular coordinate system is a system formed by three mutually orthogonal straight lines. (x, y, and z axes) Unit vectors:The position vector is written asRectangular coordinates: point P(x,y,z)Northeast petroleum university The unit vector er is The position vector r can be written asThe magnitud

11、e r is Northeast petroleum universityIf vectors A and B are the sum of the two vectors isNortheast petroleum university Scalar Product of Two Vectors (dot product) Scalar projection of B on A Distributive lawCommutative lawMagnitude of vector ANortheast petroleum university In the rectangular coordi

12、nate system,and The scalar product of two vectors isNortheast petroleum university Cross Product of Two Vectors (vector product) where is a unit vector normal to the plane of A and B. It points in the direction of motion of a right-handed screw rotating from A to B. Distributive lawNortheast petrole

13、um university含義：(1)兩矢量叉積，結(jié)果得一新矢量，(2)其大小為這兩個(gè)矢量組成的平行四邊形的面積 (area of parallelogram)，(3)方向?yàn)樵撁娴姆ň€方向 (normal direction)， 符合右手螺旋法則(right-handed screw law).Northeast petroleum universityIn the rectangular coordinate system, andVector product of two vectors isNortheast petroleum university Scalar triple prod

14、uctNote: The scalar triple product yields its volume of a parallelepiped. 三個(gè)非零矢量共面的條件Northeast petroleum university 三重積 triple product：矢量，標(biāo)量與矢量相乘。標(biāo)量，標(biāo)量三重積。矢量，矢量三重積。Northeast petroleum university Example 1.2.1 Calculate the vector R that is directed from a point to a point Distance vector: A vector f

15、rom one point to another pointNortheast petroleum university Solution Let r and r be the position vectors of points Q and P, respectively, as shown in Figure. We have and From Figure, the vector R from point Q to P is The magnitude of the distance vector R is Northeast petroleum university例： 已知求：確定垂

16、直于 、 所在平面的單位矢量。解：已知所得矢量垂直于 、 所在平面。Northeast petroleum university1.3 Coordinate Systems Northeast petroleum university Cylindrical Coordinate System 1 Cylindrical coordinates: pointNortheast petroleum university 2 Relations of coordinates 3 Unit vectors: and Note: The unit vectors and vary with the c

17、oordinate .Northeast petroleum university 4 A vector in cylindrical coordinates where each component is given as a function of and z. 5 Transformation of the unit vectors Northeast petroleum university Example 1.3.1 Write an expression for a position vector at any point in space in the cylindrical c

18、oordinate system. Solution The position vector of any point P(x, y, z ) in space in the rectangular coordinate is In the cylindrical coordinate systemNortheast petroleum university Substituting and we have and Thus, the position vector P in the cylindrical coordinate system isFrom Table, we obtainNo

19、rtheast petroleum university Spherical Coordinate System 1 Spherical coordinates: pointNortheast petroleum university 2 Relations of the coordinates cylindrical and spherical coordinates spherical and rectangular coordinatesNortheast petroleum university 3 unit vectors 6 Transformation of the unit v

20、ectors4 A vector A can be expressed as5 Position vector at any point in space isNortheast petroleum university Northeast petroleum university1.4 Divergence of a Vector Field Northeast petroleum university In a vector field A, the flux of a vector across a surface S is If the surface S is closed, the

21、 net outward flux (flow) passing through the closed surface S is 若 S 為閉合曲面,根據(jù)通量的大小判斷閉合面中源的性質(zhì)通量 ( Flux ) 矢量A沿有向曲面 S 的面積分Northeast petroleum university Consider a velocity field of an incompressible fluid. If there must exist the source of the fluid in the volume . If there must exist the sink of the

22、fluid in the volume. If the amount of fluid entering and leaving the volume must be equal. = 0 (無(wú)源) 0 (有正源) 0 (有負(fù)源)Northeast petroleum university Divergence of a vector field A at a point where is the volume bounded by the closed surface S. Note: (1) The divergence is a scalar quantity. (2) The dive

23、rgence is positive at a source point and negative at a sink point. (3) If divA=0, A is said to be a continuous or solenoidal vector field. Northeast petroleum universityThe divergence represents the distribution of flux source of a vector field. (no source） (positive source) (negative source)圖 散度的物理

24、意義 Northeast petroleum university Divergence of a vector field A is Hamilton operator del Northeast petroleum university Divergence Theorem Divergence theorem states that the volume integral of the divergence of A over a volume is equal to the surface integral of A over the boundary of the volume. w

25、here volume V is bounded by the closed surface S. Note: Convert a volume integral into an equivalent surface integral.Northeast petroleum universityExample Verify divergence theorem for a vector field in the region bounded by the planes and Northeast petroleum university 1.5 Curl of a Vector Field N

26、ortheast petroleum university The circulation of a vector field A is where C is a closed path.環(huán)量 ( Circulation )矢量 A 沿空間有向閉合曲線 C 的線積分環(huán)量的大小與閉合路徑有關(guān)，它表示繞環(huán)線旋轉(zhuǎn)趨勢(shì)的大小。Northeast petroleum university水流沿平行于水管軸線方向流動(dòng)無(wú)旋轉(zhuǎn)（渦旋）運(yùn)動(dòng) Irrotational motion流體做旋轉(zhuǎn)（渦旋）運(yùn)動(dòng)有產(chǎn)生旋轉(zhuǎn)（渦旋）的源Rotational motionConsider a velocity field of

27、 an incompressible fluid.圖 流速場(chǎng)Northeast petroleum university We take the limit as shrinks to zero. where is bounded by the closed path C. （right-handed screw） en is the unit normal of . 環(huán)量密度circulation density:the circulation per unit areaNortheast petroleum universityNote： is the scalar projection

28、of curl A along the unit vector.Physical significance : Magnitude of a curl is the maximum of the circulation density. It is in the normal direction of the area element when the circulation density is maximum.Curl of a vector field ANortheast petroleum university Northeast petroleum university Physi

29、cal significance: (1) The curl of a vector field yields a vector field. (2) It represents the maximum “circulation” density at each point and orients perpendicularly to the plane of circulation. (3) If the line integral of a vector field about a closed elementary path is nonzero, the curl of the vec

30、tor field is also nonzero and we say that the vector field is rotational. (4) If the curl of a vector field is zero everywhere, the vector field is said to be irrotational or conservative.Northeast petroleum university旋度的物理意義矢量的旋度仍為矢量，是空間坐標(biāo)點(diǎn)的函數(shù)。某點(diǎn)旋度的大小是該點(diǎn)環(huán)量密度的最大值，其方向是最大環(huán)量密度的方向。在矢量場(chǎng)中，若 A=J 0 稱之為旋度場(chǎng)（或

31、渦旋場(chǎng)），J 稱為旋度源rotation source（或渦旋源）。若矢量場(chǎng)處處 A= 0 ，稱之為無(wú)旋場(chǎng)。Northeast petroleum university Stokes Theorem Stokes theorem states that the area integral of the curl of a vector field is equal to the line integral of the field around the boundary of the area. where S is bounded by the closed path C.Note:Conv

32、ert a line integral into an equivalent surface integral.Northeast petroleum university Example 1.5.1 If the three components Ax, Ay and Az of a vector A are continuously differentiable scalar functions, show that Solution The curl of a vector is The divergence of isNortheast petroleum university Hen

33、ce Conclusion: If the divergence of a vector field F is zero everywhere, the vector field F can be expressed in terms of a curl of another vector A. That is, Northeast petroleum university1.6 The Gradient of a Scalar Field Northeast petroleum university Let u(x,y,z) be a real-valued differentiable f

34、unction of x, y, and z that describes a scalar field. The differential change du from P to Q is (1) Northeast petroleum universitySince the differential length element is we can rewrite (1) aswhere Definition : Gradient of the scalar field u is What is the meaning of the gradient of a scalar field?

35、Northeast petroleum university where is a unit vector from P to Q in the direction of dl . The maximum of rate of change is dl 最小時(shí)，du/dl最大。 此時(shí)el在P點(diǎn)垂直于等值面u。enDirectional derivative gives the rate of change of u in some direction (l or el) .Northeast petroleum university圖 等溫線分布 Note： (1) The gradient

36、of a scalar function yields a vector. (2) Its magnitude is equal to the maximum rate of the function with distance. That is, the maximum directional derivative. (3) Its direction is in the direction of that maximum rate of change. (4) is also normal to the equiscalar surface (line ) . Northeast petr

37、oleum university圖0.1.3 等溫線分布梯度的方向?yàn)樵擖c(diǎn)最大方向?qū)?shù)的方向。梯度的大小為該點(diǎn)標(biāo)量函數(shù) 的最大變化率，即最大方向?qū)?shù)。標(biāo)量場(chǎng)的梯度是一個(gè)矢量，是空間坐標(biāo)點(diǎn)的函數(shù)。梯度的意義Northeast petroleum university Example 1.6.1 If u(x,y,z) is a continuously differentiable scalar function, show that Solution The gradient of a scalar function u(x,y,z) is The curl of isNortheast pet

38、roleum university Since u(x, y, z) is continuously differentiable, we have , and . Hence Conclusion: If the curl of a vector field is zero the vector field A can be expressed in terms of a gradient of a scalar function u. That is, where the choice of plus (+) or minus (-) sign depends upon the physi

39、cal interpretation of u.Northeast petroleum university Example 1.6.2 Let r be the magnitude of the position vector in the rectangular coordinate system. Show (a) (b) Solution (a) The magnitude of the position vector r is The partial derivative of r with respect to x isNortheast petroleum universityS

40、imilarly, we have andThus, the gradient of r isNortheast petroleum university(b) The derivatives of with respect to x is,Similarly, we haveand Thus, the gradient of isNortheast petroleum universityIf R is the magnitude of the distance vector R from a point to a point, show thatwhere Northeast petroleum university Laplacian Definition:The Laplacian for a scalar function is the divergence of the gradient of the scalar function. That is, where is called the Laplacian operator. Note: