电磁波辐射（PPT讲稿）Radiation of Electromagnetic Waves

Prove: since A andA',and p can not change E and B. soB=VxA=V×A+V×Vv=V×A A=A+Vy E=-VP-aA aA OA V(q+) at at at at at aA at ●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)

Prove: since and ， and can not change E and B, so A  A    B A A A =  =  +  =    A = A +    t A t t t A t A E   −   = −  +   −     = −  −    = −  −     ( )      A t   = − −  t   = −    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）

3. Two typical gauges To reduce arbitrariness of potential, we give some constraint-- Gauge fiⅸing。 Symmetry or explicit physical interpretation ● Coulomb gauge condition V·A=0 transverse (横场),V0 longitudinal(纵场)。 o is determined by instantaneous distribution of charge density (similar to static coulomb field)

l Coulomb gauge condition  A = 0  3)．Two typical gauges transverse (横场)， longitudinal (纵场)。 is determined by instantaneous distribution of charge density (similar to static coulomb field) A    To reduce arbitrariness of potential, we give some constraint --- Gauge fixing。 Symmetry or explicit physical interpretation

Function satisfies Prove V A'=VA+VVy=oVV=0 ● Lorenz gauge Ludwig Lorenz ondition v4+ a9=o dt Function y satisfies prove:v.A't 1 a =V·4+ V Vu I a1o at t 0 (V·A+ +(V2v c2 at2 satisfy manifest relativistic covariant equations

 0 2 Function satisfies   = l Lorenz gauge condition 0 1 2 =    + c t A   A,  satisfy manifest relativistic covariant equations  Function satisfies  0 1 2 2 2 2 =    − c t   ) 0 1 ) ( 1 ( 2 2 2 2 2 =   +  −    + c t c t A      A =  A +   = 0   0 2 Prove   = 0 1 2 2 2 2 =    − c t   2 2 2 2 2 1 1 1 c t c t A c t A   −   =  +  +      +       prove： Ludwig Lorenz

D'Alembert equation 824 10 V(V·A+-x) at Vp+e(VA=_p Prove: substitute E=V×A E=vat Maxwell egs OE V×B=￡μoat ·E And using V×(V×A)=V(V·A)-V2A

2 2 2 2 2 0 2 0 1 1 ( ) ( ) A A A J c t c t A t         − −    + = −     +   = −  Prove：substitute ， into Maxwell eqs And using B A   =  t A E   = − −    0 0 0 0 ,     +   =    = J E t E B     A A A    2  ( ) = (  ) −  4)． D’Alembert equation

4. a Under coulomb gauge 02 V c2 at2 So o satisfies Poisson equation as in static case instantaneous interaction? 4. b Under lorenz gauge VA c- at c2 at2

2 2 2 2 2 0 2 0 1 1 A A J c t c t         − −  = −    = − So satisfies Poisson equation as in static case. instantaneous interaction?  4.a) Under coulomb gauge 4.b) Under Lorenz gauge 2 2 2 2 2 2 2 2 0 0 1 1 A A J c t c t         − = −  − = −  