Electromagnetic Field

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Electromagnetic Field. Textbook and Reference Books. Textbook ： Electromagnetic Field Theory Fundamentals ， Bhag Singh Guru ， Huseyin R. Hiziroglu ， 机械工业出版社 Reference Books ： (1)《 电磁场与电磁波 》 ，谢处方，高等教育出版社 (2)《 电磁场理论 》 ，毕德显，电子工业出版社. 总评成绩的组成：. 考核成绩占 50% ，平时成绩占 50% 。. 有下列情况之一者，取消其考试资格：.

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Textbook and Reference Books

Textbook：

Electromagnetic Field Theory Fundamentals，Bhag Singh Guru，Huseyin R. Hiziroglu，机械工业出版社

Reference Books：

(1)《电磁场与电磁波》，谢处方，高等教育出版社

(2)《电磁场理论》，毕德显，电子工业出版社

1、全学期缺交作业三分之一以上；

2、旷课达10学时以上（课堂点名6次缺席）。

introduction

Electric Field and Magnetic Field

Electromagnetic Wave

Propertiesof Medium

Relationship between Electro-magnetic Field and Medium

Sources of Electromagnetic Field

Review of Events in History

19世纪以前，电、磁现象作为两个独立的物理现象，人们没有发现电与磁的联系。

Important events

1785: Coulomb’s law

1820: magnetic effect of current (Oersted), Ampere’s force law

1831: Faraday’s law of induction

1863: displacement current, Maxwell’s equations

1888: Hertz proved the existence of electromagnetic wave by experiment.

Aims, Methods and Requirements

Difficulty

Methods to analyze and deal with problems

—— Process of mathematical treatment

Vector Analysis

Chapter 1 Vector Analysis

• 矢量的基本概念和运算
• 常用坐标系
• 场论基础（标量场的梯度，矢量场的散度和旋度）

Vector analysis is the language used in the study of electromagnetic fields. It’s useful to simplify and unify field equations. For example, the cross product of two vectors is

In the rectangular coordinate system,

Three scalar equations are

1.1 Introduction of Vector Analysis

When expressed in scalar form, this equation yields a set of three scalar equations. The appearance of these scalar equations depends upon the coordinate system.

velocity

electric field intensity

force

1.2 Scalar and Vector Quantities

1.2.1 Scalar

a physical quantity that can be completely described by its magnitude

mass ( m), time ( t ), work ( W), electric charge ( q)

1.2.2 Vector

a physical quantity having a magnitude as well as a direction

1. Graphical representation of a vector

A vector quantity is depicted by a line segment. The magnitude of the vector is represented by the length of the line segment. The direction of the vector is indicated by an arrow.

Parallel arrows of equal length in the same direction represent the same vector.

is the unit vector in the same direction of

2.

means having the same magnitude and direction

The direction of zero vector is arbitrary.

3. Zero vector a vector of magnitude zero

4. Unit vectora vector of unit magnitude

Ais the magnitude

1.3 Vector Operations

1. Parallelogram Method

2. Triangle Method

3. Commutative Law of Addition

4. Associative Law of Addition

1.

or

2. k > 0,

is in the same direction as .

k < 0, is in the opposite direction from .

3.

is a dependent vector.

4.

1.3.3 Multiplication of a Vector by a Scalar

(2) 若两轴 不相交，则可自空间中的任一点 S 引两轴l1和l2，使之分别与 平行，且有相同指向，l1和l2的夹角即为 间的夹角。

1.3.4 Product of Two Vectors

Angle between two vectors?

• Angle between two axes( axis : a straight line having a direction )

(1) 若两轴 l1 和 l2 相交于点 S ，在两轴决定的平面上，把其中一轴绕点 S 旋转，使它的正向与另一轴的正向重合时所需要旋转的角度，称为两轴间的夹角。一般规定两轴间的夹角限定在0与p之间，且不区分轴的顺序。

1. Dot Product

(1) Dot product is a scalar. ( scalar product )

(2) The dot product is maximum when the two vectors are parallel. (q =0, p )

(3) If the dot product of two nonzero vectors is zero, the two vectors are orthogonal. (q = p/2 )

∵ Zero vector is thought to be orthogonal to any vector.

(4) Basic Properties of the Dot Product

Commutative:

Distributive:

Scaling:

(5) the scalar projection of on

(6) the vector projection of on

Scalar projection may be positive or negative.

q：

angle between and

：垂直于 和 决定

2. Cross Product

q

q

(2)

(3)

two nonzero vectors are parallel

(4) If and are the two sides of a parallelogram, then

(1) The cross product of two vectors is a vector. ( vector product )

two cases:

Distributive:

Scaling:

(5) Basic Properties of the Cross Product

Commutative law doesn’t exist.

(2)

3. Scalar Triple Product

(1) If the three vectors represent the sides of a parallelepiped, then the scalar triple product yields its volume.

variables:

unit vectors:

position vector (directed from the origin O to point P )

1.4 The Coordinate Systems

1.4.1 Rectangular coordinate system

constant vectors

X, Y, and Z are the scalar projections of the position vector on the x, y, and z axes.

1.

2. Representation of a vector

3.

, then

4.

Angles makes with the x, y, and z axes are .

5.

∵ 和 的夹角q 由下式计算：

1. variables

1.4.2 Cylindrical Coordinate System

r：位矢OP 在 xy 平面的投影

f ：+x轴至平面OTPM的夹角

z ：位矢OP在 z 轴上的标投影

2. unit vectors

is a constant vector, and change directions asf varies.

For example,

3. position vector

5. Representation of a vector

6.If two vectors and are defined either at a commonpoint or in aplane, we can add, subtract, and multiply these vectors as we did in the rectangular coordinate system.

For example, if the two vectors at point

are and , then，

7. 若 定义在点 上， 定义在点 上，且 ，则必须首先把 和 转换成矩坐标系中的矢量，然后进行运算。

8. Transformation of Unit Vectors

9. Transformation of a Vector

1. variables

1.4.3 Spherical Coordinate System

r ：位矢OP 的大小

q ：位矢OP与+ z 轴的夹角

f ：+ x 轴至平面OMPN的夹角

2. unit vectors

3. position vector

5. 若矢量 和 定义在同一点 或同一径向线 的不同点上，则矢量加法、减法和乘积运算规则与矩坐标系中的相同。

6. Transformation of Unit Vectors

7. Transformation of a Vector

1.5 Scalar and Vector Fields

1. Field Concept

2. Classifications

• according to the properties of the physical quantity
• Scalar Fields物理量为标量（温度场、电位场）
• VectorFields物理量为矢量（电场、磁场）
• according to the variability of the physical quantity
• Static Fields物理量不随时间的变化而变化

(1) 对于矢量场 ，

(2) 对于矢量场 ，

(3) ， ，则

3. Vector Calculus

(4) ， ，则

o

o

o

Conclusions

• 在矩坐标系中，矢量函数对某一变量的偏导数（或导数）仍是矢量，其各个分量等于原矢量函数各分量对该变量的偏导数（或导数）（∵坐标单位矢量是常矢量）。简言之，只需将坐标单位矢量提到微分符号外即可。
• 在圆柱坐标系或球坐标系中，由于某些坐标单位矢量不是常矢量，对矢量函数求偏导数（或导数）时，不能直接将坐标单位矢量提到微分符号之外。

∴ 各坐标单位矢量对空间坐标变量的偏导数为

∴ 各坐标单位矢量对空间坐标变量的偏导数为

, ∵ dx is a differential element

1.6 Differential Elements of Length, Surface, and Volume

1.6.1 Rectangular Coordinate System

(1) differential volume element

：面元面积，其值可认为无限小

：面元法线方向的单位矢量

(2) differential surface area element: 面积很小的有向曲面

• 开曲面上的面元：假设开曲面由封闭曲线l 围成，选定绕行l 的方向，运用右手螺旋法则，大拇指所指方向为面元方向
• 闭曲面上的面元：封闭曲面的外法线方向

O

1.6.2 Cylindrical Coordinate System

The differential volume is bounded by six surfaces:

(1) differential volume element

(2) differential surface area element

1.6.3 Spherical Coordinate System

Q

The differential volume is bounded by six surfaces:

(1) differential volume element

(2) differential surface area element

volume of a sphere of radius a :

surface area of a sphere of radius a :

• 单位矢量 表示方向；
• 面积(线)元非负 ；

• 积分限的选取。

∵ 从A 点到 B 点， df < 0，

1.7 Line, Surface, and Volume Integrals

The basic laws of electromagnetic fields are often expressed in terms of integrals of field quantities over various curves, surfaces, and volumes in a region.

Examples:

• In static electric field, the potential difference between point a and point b is defined as
• The current through a conductor is defined as the surface integral of volume current density,

the value of f within the length segment

1.7.1 The Line Integral

1. the line integral of a scalar field f from point a to point b along a curve c in three-dimensional space

vector

2. the scalar line integral of a vector field

scalar

3. the vector line integral of a vector field

vector

When c is a closed curve, the integral sign is denoted as∮.

the value of f over the elemental surface

1.7.2 The Surface Integral

1. the surface integral of a scalar field f

vector

2. the scalar surface integral of a vector field

scalar

3. the vector surface integral of a vector field

vector

1.7.3 The Volume Integral

1. the scalar volume integral of a scalar field

2. the volume integral of a vector field

1.8 The Gradient of a Scalar Function

1.8.1 the equivalence surface of a scalar field

For example, isothermalsurface of a temperature field (温度场的等温面) , equipotential surface of a electric potential field (电位场的等位面) .

f (x, y, z)= C1

f = C2

f = C3

notice

• 根据标量场的定义，空间中每一点上只对应场函数的一个确定值。因此，充满整个标量场所在空间的众多等值面互不相交，即场中的一个点只能在一个等值面上。

f = C2

f = C1

1.8.2 the directional derivative of a scalar field

1. Definition

• 方向导数是函数f (x, y, z) 在给定点沿某一方向对

2. 矩坐标系中方向导数 的计算公式

a ,b ,g 分别是 与 x 轴、y 轴和 z 轴的夹角。

1.8.3 the gradient of a scalar function

f = C2

f = C1

1. Definition of the Gradient

• 在 上的标投影＝ 函数 f (x, y, z) 沿 方向的方向导数
• 当 与 方向相同时， f (x, y, z) 沿 方向的方向导数最大

（当 ）

is defined as the gradient of f (x, y, z)

2. properties of the gradient

∴ 在点 M，f 的梯度垂直于过点 M 的等值面

del 或 nabla

3. Hamilton operator ▽

4. basic formulae of the gradient

0

Example: Using the expression for the gradient of a scalar function, verify that .

0

Solution:

Example: , is the distance vector from point to point ( x, y, z ) .