7.1 、 potentials of electromagnetic field, gauge invariance

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. 特征：与1/r 正比的电磁场！ 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.

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

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

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: sinceand ， and can not change E and B, so l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. condition 3)．Two typical gauges To reduce arbitrariness of potential, we give some constraint ---Gauge fixing。 Symmetry or explicit physical interpretation l Coulomb gauge transverse (横场)， longitudinal (纵场)。 is determined by instantaneous distribution of charge density (similar to static coulomb field)

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

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

10. 4.a) Under coulomb gauge So satisfies Poisson equation as in static case. instantaneous interaction? 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。 l wave properties 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. l highly symmetric and independent to each other

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

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

14. The general solution for 1D wave equation is For radiation Outward spherical wave Inward wave If point charge is placed at Compared with static potential , we have： 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. 证：令 2.Show the solution 、 satisfies Lorenz condition

17. 电荷守恒定律 0

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

19. §7.3 Electric dipole radiation lwe 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. 随时间正弦或余弦变化 Charge/current： let ，so 1. General formula for radiation field Substitute into retarded solution（ ） Compared with static case, there is an additional phase factor

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

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

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

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

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

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

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

28. using In spherical coordinates, Let along axis 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. Average energy flux vector (平均能流密度矢量) 角分布 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。 4．Energy flux, angular distribution and power of radiation

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 At rest frame At Lab frame where Lienard-Wiechert potential

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

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

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

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