1. 7.1 、 potentials of electromagnetic field, gauge invariance 7.2 、 d ’ Alembert equation and retarded potential 7.3 、 electric dipole radiation 7.4 、 EM radiation from arbitrary motion charge Ch 7 Radiation of Electromagnetic Waves

2. 1. What is EM radiation EM field is excited by time-dependent charge and currents. It may propagate in form of waves. The problem is usually solved in terms of potentials. 2 ． It is a boundary value problem Source (charge and current) excites EMF, EMF in turn affects source distribution --- boundary value problem! For convenience, our discussions are limited to a simple case – Distribution of source is known. 特征：与 1/r 正比的电磁场！

3. §7.1 vector potential and scalar potential potentials are slightly different from the static cases since ， we can introduce vector potential as the static field, 1.a ） vector potential

4. Since ， scalar potential can not be defined as before 1.b ） scalar potential Define scalar func

5. 2) ． Gauge invariance Potentials are not uniquely determined, they differ by a gauge transformation. Gauge: Given a set of give identical electric and magnetic fields

6. Prove: since and ， and can not change E and B, so 规范不变性 规范不变性：在规范变换下物理规律满足的动力学方程保持不变 的性质（在微观世界是一条物理学基本原理）。

7. Coulomb gauge condition 3) ． Two typical gauges transverse ( 横场 ) ， longitudinal ( 纵场 ) 。 is determined by instantaneous distribution of charge density (similar to static coulomb field) To reduce arbitrariness of potential, we give some constraint --- Gauge fixing 。 Symmetry or explicit physical interpretation

8. Function satisfies Lorenz gauge condition satisfy manifest relativistic covariant equations Function satisfies Prove prove ： Ludwig Lorenz

9. Prove ： substitute ， into Maxwell eqs And using 4) ． D’Alembert equation

10. So satisfies Poisson equation as in static case. instantaneous interaction? 4.a) Under coulomb gauge 4.b) Under Lorenz gauge

11. 洛仑兹规范下的达朗贝尔方程是两个波动方程， 因此由它们求出的 及 均为波动 形式，反映了电磁场的波动性。 wave properties highly symmetric and independent to each other Get one, get 2nd for free. Solution of d’alembert eq under Lorenz gauge indicates that EM interaction takes time. To study radiation, we use Lorenz gauge.

12. §7.2 Retarded potential A ssume is known. We first solve point charge problem ， then use superposition to get general solution A ssume is known. We first solve point charge problem ， then use superposition to get general solution 1. Solve d’Alembert equation

13. Assume point charge at origin ， ， symmetry indicates is independent of, so d’Alembert eq for scalar potential is, so d’Alembert eq for scalar potential is as ＊ let let

14. Outward spherical wave Inward wave The general solution for 1D wave equation is Compared with static potential, we have ： For radiation If point charge is placed at 容易证明上述解的形式满足波动方程＊式

15. For contineous charge distribution Since satisfies identical equation as, so the solution ：

16. 2.Show the solution 、 satisfies Lorenz condition 证：令

17. 0 电荷守恒定律

18. The value of the retarded potential at, depends on charge/current distribution at. physical excitation at reaches observation point by. And the speed of signal traveling in vacuum is c. 3 ． Physical interpretation Electromagnetic interaction takes time!

19. §7.3 Electric dipole radiation we limit our discussion to charge distribution of periodic motion. Furthermore, size of charge distribution is much smaller than the distance between charge and the observation point. 电磁波是从变化的电荷、电流系统辐射出来的。 Antenna with high frequency alternative electric current Non-uniform moving charged particles

20. Substitute into retarded solution （ ） let ， so 1. General formula for radiation field Compared with static case, there is an additional phase factor Charge/current ： 随时间正弦 或余弦变化

21. Similarly, Satisfy Lorenz condition Electromagnetic fields are （ ）

22. Assume source to field point distance (size of charge/current distribution), so Perform power expansion around 2 ． Multiple expansion Where is unit vector along, 1) ． Power expansion for small size of source (R is distance between center of coordinate and field point)

23. Since, so in denominator can be neglected. But it maybe important in phase, because is not necessary small, compared with Keep first two terms, we have

24. if, The radiation field for is The first term dominates Electric dipole radiation

25. Under ， ， we can further divide into three cases according to and 2) ． 与 的关系 a ） （近区）, Time of propagation EM field is similar to the static case. c ） （远区，即辐射区） EM waves propagates away from the source. interested b ） （感应区） Very complicate.

26. 1) ． Re-express in terms of dipole moment dipole radiation 3 ． Electric dipole radiation 2) ． Electric and magnetic fields here

27. Consider 远区 ， ，即 ， so ( ) Magnetic induction

28. In spherical coordinates, Let along axis using 电场线是经面上的闭合线

29. Discussion: （ 1 ） E oscillates along longitude and B along latitude lines. Direction of propagating, E and B are orthogonal to each other (right-hand). （ 2 ） E,B are proportional to ， so they are propagating spherical waves. They are transverse （ TEM 波） and maybe regarded as plane wave as. （ 3 ） without （ ）, it can be shown that E is no longer perpendicular to k, electric lines are not close, but magnetic lines are still close （ TM 波）.

30. 4 ． Energy flux, angular distribution and power of radiation Average power 1) 。与球半径无关，能量可以传播到无穷远。 2) 。与电磁波的频率 4 次方成正比。 Average energy flux vector ( 平均能流密度矢量 ) 角分布

31. Example: Short antenna 短天线辐射能力不强。 通常天线长度与波长同数量级， 不能用简单的偶极辐射公式。

32. 例：半波天线（长度为半波长） 必须直接用推迟势计算 天线电流要与外场联合作为边值问题求解，一般较复杂。 对于细长直天线，电流分布应是驻波，两端是波节。如 辐射电磁场

33. The energy flux 要得到高度定向的辐射，可利用天线阵的干涉效应。 张角

34. §7.4 Magnetic dipole radiation and electric quadrupole radiation 电四极辐射 磁偶极辐射

35. §7.5 Radiation from a localized charge in arbitrary motion (Bo-p124) method1 ： use the retarded potential 粒子看作小体积电荷分布，直接积分

36. method2 ： using Lorentz transformation Lienard-Wiechert potential At rest frame where At Lab frame

37. Derivative was performed with respective to t ， x. However

39. 在与加速度垂直方向辐射最强！ Q ：圆周运动和直线运动时的辐射方向？

40. Radiation field from a relativistic charged particle 当 v 趋于光速，辐射集中于朝前方向，张角为 Radiation power Radiation field for arbitrary moving charge particle

41. Acceleration is parallel to velocity ( 轫致辐射） Bremsstrahlung and Synchrotron radiation Angular distribution where Radiation power

42. Acceleration is perpendicular to velocity （同步辐射） 辐射功率与粒子 能量平方正比 ! Angular distribution Power where

43. e.g. BEPC E= 2.8GeV ， =5479 高能加速器设计与 同步辐射光源