Electrodynamics

1. Electrodynamics REN Xincheng, Postdoctorate , Professor Tel：2331505; 18329918078 Email: xchren@yau.edu.cn

2. Chapter 1. The universal law of electromagnetic 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.)

3. 2）The establishment of electromagnetic field energy, momentum, and energy flow, momentum flow expression. • In this chapter, the basis for discussion problem are: • Energy and momentum conservation law.

4. §1.1 Charge and electric field(Electrostatic field) • Coulomb's law (the law of the force between two charges in vacuum) • Attention: • 1）They are the basic equation satisfied electrostatic field • 2）The relationships of eachquantityin Formula. • 3）Circulation of field quantity E iszero, can not say that E of each points in circulation circuit is zero, but only say that electrostatic field is a conservative force field.

5. These are the results obtained in electromagnetic, the differential form of above basic equation of the electrostatic field is derived. • Consider the case of continuous distribution charge (Suppose the charge density as ρ), then the Gauss theorem can be rewritten as: • UsingGaussian formula( ), then obtain • The above formula is for arbitrary size, so there is

6. Similarly, the formula (2’) is obtained using Stokes formula from (2) • This is differential form of the basic equation of electrostatic field. • Illustration： • 1）The equation of differential form describe the local nature of the 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)

7. §1.2 Current and magnetic field (Magetostatic field) • Biot-Sa Fire Law • 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.

8. 一、 Be derived using the same method with the electrostatic field • The differential equations satisfied magetostatic field is derived • 二、Be derived using the Biot-Sa Fire Law directly (Familiar with the algebra)

9. And • So that

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

11. §1.3 Maxwell's equations in vacuum • The basic rules of constant field are summarized by experimental laws in the above two section. However, in the study on the alternating field, people’s knowledge 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 and 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 field reflects mainly. • 1）Varying magnetic field excite electric field, （Faraday’s law of electromagnetic induction ） • 2）Varying electric field excite magnetic field, （Maxwell displacement current hypothesis ）

12. 一、Electromagnetic induction law • The induced voltage in a closed circuit loop is equal to the varying rate of magnetic flux. That • 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 thereare

13. The above formula is satisfy to any surface S, So there are • This formula shows the general electric field is a rotation field.

14. 二、Charge conservation law and displacement current hypothesis • 1、 Charge conservation law • Charge conservation law is one of the most basic experiments law of nature, its differential form is • This formula is also called the continuity equation. For the costent current • 2、Displacement current hypothesis • In theory, the inside of general electromagnetic equations

15. 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. • 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 by Maxwell. • In order to be compatible with the charge conservation law, envisaged the fourth equation should be amended to:

16. 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.)

17. 三、Maxwell's equations in Vacuum (Free space) • According to the top discussion, we can obtain a set of self-consistent equations. • This is Maxwell's equations that has been widely accepted today.

18. Attention： • 1）There is not contradiction in the above equation is only a necessary condition 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.

19. 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.

20. 四、Lorentz 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. • 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:

21. The above formula is generally applicable，this inference is derived by Lorentz, so this force is called Lorentz force. On charged particles of motion, Lorentz force is:

22. §1.4 The electromagnetic properties of medium • In principle, all problems of electrodynamics are solved by the Maxwell’s equation and the Lorentz force formula. The problems of polarization and magnetization of medium is derived by quantum mechanics considering the structure model of matter on this basis, however, this derivation depends heavily on people'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 equations related electromagnetic properties of medium.

23. 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. • Bound charge (polarization charge) , magnetization current and polarization current 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.

24. 一、Polarization of dielectric medium and magnetization of magnetic medium • The physical quantity describing dielectric polarization is the electric polarization vector p, defined as • Namely, vector sum of the electric moment of molecule per unit volume • The physical quantitydescribing 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 aftermagnetization.

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

26. According to the charge conservation law, the negative charge remaining in column equal to the positive charge piercing from surface numerically. • So, quantity of electric charge piercing from the entire surface are: • Thus available • This is the relationship between the polarization charge density and polarizability. • Attention: • 1）After non uniform polarization, the bound charges occur inner of whole medium generally; • 2）For the homogeneous medium polarization, bound charges occur in the vicinity of the free charge and the medium interface (physical interface) only.

27. 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.

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

29. 四、Polarization current • Be studied before, due to polarization, the quantity of electric charge through any curved surface S of the medium are • The quantity of electric charge flowing through the surface S per unit time, namely, the polarization current through the surface • So there is

30. 五、Maxwell’s equations with medium • To the type into the generalized form of Maxwell's equations, by the arrangement and the introduction • Maxwell’s equations with medium are obtained.

31. 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.

32. 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 isotropic linear medium and isotropic nonferromagnetic material, there are • 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.

33. applies to no external electromotive force, no external magnetic field, low frequency and proper temperature (superconducting phenomena occurs in condition of low temperature). • →The establishment condition that equal is homogeneous medium or（inhomogeneousand when is not established） 1)The nature equation of general medium and the application scopeof special case • Discussion • There are a number of conductors in homogeneous medium;

34. 3) Thecharacteristics of homogeneous dielectric polarization and inhomogeneous dielectric polarization 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 allpoints 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

35. §1.5 Electromagnetic fieldboundary value relation • 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: • 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 section, and which is the equation of Maxwell’s equations at the interface.

36. 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:

37. where • The following, it is illustrated simply that the boundary value relations is obtained from the integral. • Look at the above scheme.

38. The Flux through the side. • Interface charge is fixed，when • for this reason, the surface charge density is introduced. • Meanwhile • So there • Similarly available • In the following, consider the tangential component.

39. The fourth formula of Maxwell’s equations will be used in this circuit, have • As shown below, the circuit of narrow and long shape is taken on both sides of the interface. • Take the left multiplication crossn on both sides of the formula, and notes

40. and • 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.

41. For example • Differential form transform the integral form. • It can be seen, it can be to grasp the differential form of Maxwell’s equations. • School work exercises: p.35-36: 8、11、12

42. §1.6 The energy and the energy flowof 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.Thegeneral typeof energy conservation law of the field with 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.

43. Two physical quantity describing the energy of electromagnetic field • 1. energy density • 2. energy-flux density • Thegeneral typeof energy conservation law of the field with the charge system

45. 2. The expressions of the energy density and the energy flux density

46. By comparison, the below formula can be obtained. • Discussion • The case of the charge distribution in vacuum

47. The electromagnetic energy and energy flow in medium • Then we completed the discussion that energy conservation under electromagnetic interaction. We can know from the discussion, as long as the Maxwell’s equations and the Lorentz force formula is correct, then the energy conservation is a inevitable result, and the expression of energy and energy flow density of field was completely determined by them.

48. 3. The transform energy of electromagnetic field Example（P32） School work exercises: p.36 14