Electrodynamics

REN Xincheng, Postdoctorate , Professor

Tel ： 2331505; 18329918078

Email: xchren@yau.edu.cn

Chapter 1. The universal law of Chapter 1. The universal law of electromagnetic phenomenaelectromagnetic phenomena

Electromagnetic field is a form of material existence, it has a specific law of motion and material properties with other charged materials interact with a certain form. This chapter describes the universal law of electromagnetic fields and electromagnetic fields interact with charged matter, To sum up the main contents are:

1 ） Maxwell's equations are established (Three forms), namely, the partial differential equation satisfied field quantity are studied (E 、 H, In the macroscopic dynamics, they decided the nature of the electromagnetic field.)

2 ） The establishment of electromagnetic field energy, mom

entum, and energy flow, momentum flow expression.

Coulomb's law

Biot-Sa Fire Law Four basic experimental laws

Charge conservation law

Faraday's law of electromagnetic induction

Energy and momentum conservation law.

In this chapter, the basis for discussion problem are:

§1.1 Charge and electric field (Electrostatic field) Coulomb's law (the law of the force between two

charges in vacuum)

0

The introduction of electric field /1 212 122

0

/ Gauss Theorem (1)1 ˆ

4 0 Circle Theorem (2)

SE F q

L

E dS qq q

F rr E dl

Attention: 1 ） They are the basic equation satisfied electrostatic field 2 ） The relationships of each quantity in Formula . 3 ） Circulation of field quantity E is zero, can not say that E

of each points in circulation circuit is zero, but only say that electrostatic field is a conservative force field.

These are the results obtained in electromagnetic, the differential form of above basic equation of the electrostatic field is derived.

VS

dVSdE 0

1

0)1

(0

V

dVE

)'1(0

E

Consider the case of continuous distribution charge (Suppose the charge density as ρ), then the Gauss theorem can be rewritten as:

VS

dVESdE

Using Gaussian formula( ), then obtain

The above formula is for arbitrary size, so there is

Similarly, the formula (2’) is obtained using Stokes formula from (2)

)'2(0 E

This is differential form of the basic equation of electrostatic field. Illustration ： 1 ） The equation of differential form describe the local nature of t

he field, the equation of integral form describes the overall nature of the field. The equation of differential form describes the nature of electrostatic field effectively ；

2 ） (1’) indicate that the field divergence at a point in space only with the point charge density, charge only stimulate the adjacent field, while the far field through the field passing out their own internal role, the charge is the source of electric field, electrostatic field is active field.

3 ） (2’) indicate that electrostatic field is irrotational filed. Example (P.7)

§1.2 Current and magnetic field (Magetostatic field) Biot-Sa Fire Law

int 03

'

( ')( ) ' ( )

4Bis roduced

V

J x rB x dV A

r

)(

)(

4 311220

12

BlIdFdr

rldIldIFd

0

0 (3) Gauss theorem of magnetic field

(4) Ampere circle theorem

S

L

B dS

B dl I

x

'x0

r

Attention:

1) The field point, the source points and distances of equation (A);

2) The nature of magetostatic field is obtained from equation (3), (4), magetostatic field is non-conservative force field.

The differential equations satisfied magetostatic field is derived

0

0 (3')Magetostatic field is a rotating and passive field

(4 ')

B

B J

'

30 '

)'(

4)(

V

dVr

rxJxB

'

0

'3

0 '1

)'(4

')'(4 VV

dVr

xJdVr

rxJ

AdVr

xJV

]'

1)'(

4[

'

0

一、 Be derived using the same method with the electrostatic field

二、 Be derived using the Biot-Sa Fire Law directly (Familiar with the algebra )

So that ( ) 0 namely 3' formularB A

（ ）

AAAB

2)()(

]'1

)'([4 '

0 V

dVr

xJA

'

0 '1

)'(4 V

dVr

xJ

'

0 '1

')'(4 V

dVr

xJ

'

0

'

0 ')'('1

4'

)'('

4 VV

dVxJr

dVr

xJ

0')'('1

4

)'('

4 '

0

'

0 VS

dVxJrr

xJSd

And

'

202 '1

)'(4 V

dVr

xJA

'

30 ')'(

4 V

dVr

rxJ

3 3 30 0; 0 0, '

r r rr r namely x x is not zero

r r r

onl y i n

'

30

'3

0 '4

)('')(

4 SV r

rSd

xJdV

r

rxJ

0

'

( )' ( ' ' ' )

4 S

J xd Select S is spherethat surround x x r and dS reversed

，

)(0 xJ

0 3(4 ') ( 4 ( ))

rB J Obtained incidentally r

r

namel y

School work exercises: p.33-34: 1 、 2 、 3 、 6

§1.3 Maxwell's equations in vacuum The basic rules of constant field are summarized by experimental laws in the a

bove two section. However, in the study on the alternating field, people’s kno

wledge have a leap to the electric and magnetic fields. It founded: not only the

charge excite electric field, magnetic field excite current, and varying electric a

nd magnetic fields can excite each other, electric and magnetic fields compose

unified ensemble-electromagnetic field.

Compared with the constant field, the new rule of varying electromagnetic fiel

d reflects mainly.

1 ） Varying magnetic field excite electric field, （ Faraday’s law of electrom

agnetic induction ） 2 ） Varying electric field excite magnetic field, （ Maxwell displacement cu

rrent hypothesis ）

一、一、 Electromagnetic induction lawElectromagnetic induction law The induced voltage in a closed circuit loop is equal to the

varying rate of magnetic flux. That

the surrounded line is unchanged

S S

d BB dS dS

dt t

L S

E dl E dS

i nduced i nduced

( ) 0S

BE dS

t

i nduced

At the same time, We believe that the induced electromotive force is due to changes in the magnetic flux generated in the loop due to induced electric fields, So there are

So there are

The above formula is satisfy to any surface S, So there are

BE

t

i nduced

coulomb coulomb 0E E E and E

i nduced

)''2(t

BE

This formula shows the general electric field is a rotation field.

二、二、 Charge conservation law and displacemeCharge conservation law and displacement current hypothesisnt current hypothesis

1 、 Charge conservation lawCharge conservation law Charge conservation law is one of the most basic experi

ments law of nature, its differential form is

0

t

J

0 0Jt

（steady condi t i on）

This formula is also called the continuity equation. For the costent current

2 、 Displacement current hypothesis In theory, the inside of general electromagnetic equations

and it should be compatible with the charge conservation law, that is, no contradiction between them. According it, we modify the fourth equation in the steady field to make it applicable to the general situation.

00)(0)(

BBt

E

0)( 00

t

JB

On (2'') formula taking the divergence at both ends

On (4 ') formula taking the divergence at both ends

And the left of the above formula is equal to zero, thus contradictions emerge. This shows that: (4 ') formula is only applicable to costent field, when there are changes in charge density, this equation will not set up. The displacement current is introduced from this point b

y Maxwell. In order to be compatible with the charge conservation law, envisag

ed the fourth equation should be amended to:

0 ( ) ( is the displacement current that need to assume )D DB J J J

t

EJ

t

E

tJJ DD

00 )(

Taking the divergence on both sides of the above formula and

using the charge conservation law, we will have:

Attention:

1 ） The similarities and differences of displacement current and conduction current;

2 ） The essence of displacement current is that varying electric field can induce magnetic field inevitably. (However, this conclusion does not have experiment basis at that time.)

三、三、 Maxwell's equations in Vacuum (Free spacMaxwell's equations in Vacuum (Free spac

e)e) According to the top discussion, we can obtain a set of self-

consistent equations.

)(

0

/

0

00

0

000

0

SL

S

SL

S

SdEdt

dIldB

SdB

SdBtd

dldE

qSdE

t

EJB

Bt

BE

E

积分形式

This is Maxwell's equations that has been widely accepted today.

Attention ： 1 ） There is not contradiction in the above equation is only a necessary condi

tion for correctness and can not guarantee that the above equation is correct. Today, Maxwell's equations as electromagnetic theory is accepted as a general rule, it is not because of its no contradiction, but because its reasoning has been validated by the subsequent large number of experiments. (Predict the existence of electromagnetic waves, and was confirmed by Hertz experiment. )

2 ） The other equation of Maxwell's equations that did not change relative to the constant field equation as a general electromagnetic law, in fact, was given a new meaning. (Different from the constant field).

The two equations describing the electric field, and now regard them as a general rule, they contain a number of the content that the original does not have. First, it shows that electric field distribution depends only on the charge distribution and magnetic field changes. Other methods that generated electric field are no longer have.

Secondly, in the case that there is a varying of charge density, the divergence

of the electric field strength is still proportional to the local charge density at

that time, but the induced electric field has not divergence. These are new

results. For the law of magnetic field, firstly, magnetic field is produced only two ways,

namely by the current generation and the electric field induced by the changes

in production; Secondly, the magnetic field produced by these two methods are

the vortex field; again, no dispersion of the magnetic field is independent with

current whether the static. These conclusions are not from past experience.

Thus, from one speaking, the correctness of these conclusions is the need for

new practices to prove, on the other hand, these new results deepen people's

understanding of the electromagnetic field.

四、四、 Lorentz forceLorentz force There is close contact between electromagnetic fields and charged matter, in

addition to Maxwell's equations (including charge conservation law ) that

reflecting the charge system excite field and the motion ( electromagnetic field

excitation each other) in inner of electromagnetic, but also formula reflecting

interaction law between the fields and the charge system, which have been

reflected under certain conditions in Coulomb's law and Ampere's law.

dVBJBlIdFdEqF

, In the electromagnetic field, if the distribution of charge is continuously, the

force acting on unit volume of the charge system (namely the density of force )

is:

The above formula is generally applicable ， this inference is der

ived by Lorentz, so this force is called Lorentz force. On charged

particles of motion, Lorentz force is:

BqEqF

BJEf

§1.4 The electromagnetic properties of

medium In principle, all problems of electrodynamics are solved by the Maxwell’s equ

ation and the Lorentz force formula. The problems of polarization and magne

tization of medium is derived by quantum mechanics considering the structur

e model of matter on this basis, however, this derivation depends heavily on p

eople's understanding of matter microscopic structure and dynamics machine

processed, can not be completely accurate yet. Therefore, in the macroscopic

electrodynamics, in addition to the basic Maxwell’s equation and the Lorentz

force formula, we also need to add some phenomenological experiment equati

ons related electromagnetic properties of medium.

Due to polarization and magnetization of the medium, there will appear bound charge (polarization charge) , magnetization current and polarization current (current generated by polarization charge varying with time) in the medium.

PMP JJ

、、 Bound charge (polarization charge) , magnetization current and polarization cur

rent can also stimulate electromagnetic field, considering their interaction, Maxwell’s equation can be extended to the situation with medium. (basis: When the scope of the research problem down to the atom, the inside of atom can also be seen as a vacuum--the scale of nucleus is much smaller than the scale of atoms, this is called the vacuum model of medium). Namely, medium polarization and magnetization can be use the bound charge, polarization current and magnetization current to describe equivalently, but their amplitude and direction are described by the polarizability and the magnetization. Based on this understanding, we first review the law of medium polarization and magnetization in the electromagnetics, and finally get Maxwell’s equation when medium exists.

一、 Polarization of dielectric medium and magnetization of magnetic medium The physical quantity describing dielectric polarization is the electric

polarization vector p, defined as )(

0

limLnq

V

p

VP i

displacement polarization non-polar moleculepolarization

orientation polarization polar molecule

（ ）（ ）

lim

0imM

V V

Namely, vector sum of the electric moment of molecule per unit volume

The physical quantity describing magnetization properties of magnetic medium is

the magnetization, defined as

Namely, vector sum of the within the magnetic moment of molecule per unit

volume. Analysis on situation before and after magnetization.

二、 The relationship between polarization charge and electric polarization Here, as an example, we discuss the displacement polarization

only.

l qq

E

SdPSdlnq

So, quantity of electric charge piercing from the entire surface are:

S

SdP

gauss theoremPV V

S

dV P dS PdV

PP

According to the charge conservation law, the negative charge remaining in

column equal to the positive charge piercing from surface numerically.

Thus available This is the relationship between the polarization charge density and polarizabili

ty. Attention: 1 ） After non uniform polarization, the bound charges occur inner of whole m

edium generally; 2 ） For the homogeneous medium polarization, bound charges occur in the vi

cinity of the free charge and the medium interface (physical interface) only.

三、 The relationship between

magnetization current and magnetization To consider any curved surface of medium S, its boundary line is L, to discuss the

magnetization current through the curved surface. magnetizing current is the

macroscopic expression of molecular current, the relationship between molecular

current and the curved surface S can be divided into the following three types:

intersection once, intersection twice, and do not intersect. Obviously, the molecule

current that intersection once with the surface can contribute to current through the

surface S.

LS

SL

M SdMldMI

M M

S

M

S S

and I J dS

So there J dS M dS

MJM

ld

a

The above formula is set up for any curved surface S, so there

四、四、 Polarization currentPolarization current Be studied before, due to polarization, the quantity of electric charge through

any curved surface S of the medium are

PJ

P

S

Q P dS

S

PP Sd

t

P

dt

dQI

P P

S

and I J dS

t

PJ P

The quantity of electric charge flowing through the surface S per unit time,

namely, the polarization current through the surface

So there is

五、 Maxwell’s equations with medium

To the type into the generalized form of Maxwell's equations,

by the arrangement and the introduction

, ,P M P

PP J M J

t

0Electric displacement vector D E P

0

Magnetic field intensity vectorB

H M

Maxwell’s equations with medium are obtained.

In this equations, E and B are macroscopic physical quantity

describing electromagnetic field, D and H are auxiliary physical

quantity only for convenience. The relationship between these

auxiliary quantity and base quantity is given as their definition.

0 0

S

L S

S

L S

D dS qD

dB E dl B dSEd tt

B B dSD

H J dt H dl I D dSdt

i ntegral f orm

For general medium, there is no simple relationship of P with E, M and B, which

determines the equations of dielectric properties is very complex. But for isotropi

c linear medium and isotropic nonferromagnetic material, there are

)1(/)1(

0

00

M

e

M

e

BHED

HMEP

olarizability and magnetic susceptibility separatelye Mand i s p

EJ

In addition, in conductor medium, there is the Ohm's law that describing the

nature of the medium.

Maxwell's equations, the equation of the nature of the medium, and Lorentz force

formula constitute a perfect set, in principle, can deal with all electromagnetic

problems.

Discussion

0

0

2)The ralationshipbetweenDand E

Theestablishment of D E is need tocondition

are both equal, the above formula is established

// E

E

→The establishment condition that equal is homogeneous medium or

（ inhomogeneous and when is not established ）

There are a number of conductors in homogeneous medium;

1)The nature equation of general medium and the application scope of special case

EJ

applies to no external electromotive force, no external magnetic field, lo

w frequency and proper temperature (superconducting phenomena occurs in con

dition of low temperature).

4) The difference of homogeneous medium with homogeneous polarization

The former characterizes that the physical nature of the medium is uniform,

reflects the ε of all points are same; while the latter characterizes that the nature

of dielectric polarization is uniform, reflecting the P of all points are same.

For example: in the case of point charge is in a homogeneous medium, is

homogeneous dielectric, but not homogeneous polarization.

School work exercises: P.35 9

3) The characteristics of homogeneous dielectric polarization and inhomogeneous di

electric polarization

0 homogeneous ( 1) , inhomogeneous 0 is not necessarily zero.P f f P

，

§1.5 Electromagnetic field§1.5 Electromagnetic field boundary value boundary value relationrelation The differential form of Maxwell’s equations can be applied to the inside of

any continuous medium. At the interface of two medium, in general, have

emerged the distribution of surface charge and surface current, so that

physical quantities (field quantities) have a jump in this place. Therefore, the

differential form of Maxwell’s equations can not be applied in this case. For

example: 1 2

0E 0B

Hence, we need to find that the relationship between the varying of field

quantity near the both sides of interface and the distribution of surface charge

and surface current , which is boundary relations to be discussed in this sectio

n, and which is the equation of Maxwell’s equations at the interface.

The Integral form of Maxwell’s equations is used to describe the overall

nature of the electromagnetic field of a certain region, so it can be used to

deal with the electromagnetic field when the medium is not continuous. So

the base that researching boundary relations is the integral form of Maxwell's

equations. When the two flux equations is used, integral volume can take into

the shape of oblate tank; when the two circulation equations is used, integral

loop can take into the shape of narrow band. The two medium are known as

medium 1 and medium 2, the normal unit vector of interface is defined from

medium 1 to medium 2. The following boundary relations is obtained:

where

)(0)(0)(

)(

12

12

12

12

HHnBBnEEnDDn

/ arg

/

0(macroscopy) ontains

a large number of molecular layer in microscopic

h q S is thedensity of surfacech e

hJ I l is thelinecurrent density

h

，butc

S

n

2E

1E

h

1介质

2介质

The following, it is illustrated simply that the boundary value relations is obtain

ed from the integral. Look at the above scheme.

Interface charge is fixed ， when

hSNSnDSnD 12

The Flux through the side.

0h ，

Sqh /0 lim 0when h Dis ited N canbeignored

， ，

）（ 12 DDn

012 ）（ BBn

for this reason, the surface charge density is introduced.

Meanwhile

So there

In the following, consider the tangential component.

Similarly available

As shown below, the circuit of narrow and long shape is taken on both sides of the interface.

n 't

t

1H

2Hl

h

ntt

'

hlt

DhltJeltHtH

')( 12

hJ 0

tntntHHt

)(

')( 12

Since is optional is also

the unit vector of any director of interface

l t

，then the di recti on of

，so there

2 1 //( )H H n

2 1 // 2 1( ) ( )n H H n H H

The fourth formula of Maxwell’s equations will be used in this circuit, have

Take the left multiplication cross n on both sides of the formula, and notes

and 0n

)( 12 HHn

0)( 12 EEn

( , )( , )

0differential quotient of field quantity with time

n

n

Surface quantitySize quantity J

)(0)(0)(

)(

12

12

12

12

HHnBBnEEnDDn

Similarly available

The comparison for three forms of Maxwell’s equations.

1 ） The case of application is different; 2 ） Corresponding relationship

In addition, this corresponding relationship has universal significance.

For example )( 12 PPnP PP

0)(0 12

t

JJnt

J

take volume integral on both sides and using the Gaussian formula

take surface integral on both sides and using Stokes formula

Differential form transform the integral form.

It can be seen, it can be to grasp the differential form of Maxwel

l’s equations. School work exercises: p.35-36: 8 、 11 、 12

§1.6 The energy and the energy flow of electromagnetic field Electromagnetic field is a existence form of matter, it has the universal nature

of the matter (with an internal motion, energy and momentum, etc.), In

addition, there are special properties (different motion forms, can be measured,

but can not visible, can access, and so on) compared with other matter.

1.1.TheThe general typegeneral type of energy conservation law of the field of energy conservation law of the field with the charge systemwith the charge system

The energy conservation law is considered a universal law of physics. In fact,

whenever a new physics domain is involved, to the applicability of the energy

conservation law, there is no a transcendental answer. After recognizing the

electromagnetic interaction of electromagnetic field to the objects containing the

charge, whether the energy conservation law is to set up, it is also need to be re-

examined from experiment and theorem.

Two physical quantity describing the energy of

electromagnetic field

1. energy density

2. energy-flux density

TheThe general typegeneral type of energy conservation law of the field with of energy conservation law of the field with

the charge systemthe charge system

V

dVf

V

dwdV

dt

S

S d

VV

S

wdVdt

ddVfdS

St

wf

V

df dV wdV

dt

Lorentz force ( )f E B E J E

t

DHJ

t

DEHE

)(

)()()( HEEHHE

t

DEEHHE

)()(

t

BE

)()(t

BH

t

DEHE

2. The expressions of the energy density and the energy

flux density

By comparison, the below formula can be obtained.

)()(t

BH

t

DEHE

t

wS

HES

t

BH

t

DE

t

w

)/(2

10

220 BEw

The case of the charge distribution in vacuum

Discussion

00

,B

H D E

0

1S E B

1( )

2w E D H B

The electromagnetic energy and energy flow in medium

Then we completed the discussion that energy conservation under electromag

netic interaction. We can know from the discussion, as long as the Maxwell’s

equations and the Lorentz force formula is correct, then the energy conservati

on is a inevitable result, and the expression of energy and energy flow density

of field was completely determined by them.

w E D H B

,D E B H

3. The transform energy of electromagnetic field3. The transform energy of electromagnetic field

Example （ P32 ）School work exercises: p.36 14