Electrodynamics, chap14

# Electrodynamics, chap14 - Chapter 14 Radiation by Moving...

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Chapter 14: Radiation by Moving Charges 2 22 2 2 1 2 4 1 converted to Gaussian unit system, see p.782 for conversion formulae. 4 (6.15) (6.16) ct c π πρ ∇Φ− Φ=− ∇− = Review of Basic Equations : AA J 2 2 1 3 general form of 4 ( , ) (6.32) (6.15) and (6.16) Solution of (6.32) with outgoing-wave b.c.: ( , ) ( , ) ( , , , ) ( , ), (6.45) where the r in ft tt d x d t G t t f t ψψ + ⎡⎤ = ⎢⎥ ⎣⎦ ′′ =+ ∫∫ x xx x x x 2 2 1 etarded Green's function ( , , , ) / (6.44) is the solution of (with outgoing-wave b.c.) ( , , , ) 4 ( ) ( ) (6.41) [( ) ] () c Gt t t t t t t δ πδ + + =− = x x x x free-space inhomogeneous wave equations

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From R. M. Eisberg, "Fundamentals of Modern Physics" 3 1 Apply (6.45) (assuming 0) to (6.15) & (6.16) (, ) (,) (9.2) ) : We need both and to specify and , u [) ] ( in c c tt t t dx d t t t Note δ ψ ρ − ′ = ′′ Φ ⎧⎫ = ⎨⎬ ⎩⎭ Φ ∫∫ xx x x Jx Ax AE B nless the source has harmonic time dependence (as in Chs. 9 and 10). A Qualitative Picture of Radiation by an Accelerated Charge : Review of Basic Equations ( continued ) E -field lines surrounding a stationary charge. A fraction of E -field lines showing the effect of charge acceleration.
x () t r o (at ) et i (at ) i t v t n R t orbit of e [ ( , ), ( , )] at point of observation tt Φ A xx i 3 1 (,) (, ) Rewrite (9.2): ) ( , ), ( , ) due to a point charge ( carrie [) ] ( c c t t dx d t t t e e δ ρ − ′ ′′ ⎧⎫ = ⎨⎬ Φ ⎩⎭ ∫∫ Lienard - Wiechert Potentials for a Point Charge : Ax Jx x x xJ x [] s a sign) moving along the orbit ( ) at the velocity ( ) ( ) / can be written ) ( ) ) ( ) ( ) Rt c c t d t d t te t t t d t d t ρδ +− = =− Φ= = β rv r r v x r x (1) w h e r e a n d () . t t t c = xr β v 14.1 Liénard-Wiechert Potentials and Fields for a Point Charge

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14.1 Liénard-Wiechert Potentials … ( continued ) () (,) Rewrite (1): , where ( ) ( ) . [] R t c Rt c ft t tf t tt t t te d t e d t d t e d t t Rt c δ ′′ +− Φ= = == ≡+ ∫∫ β β x Ax (2) U s i n g , w e o b t a i n i d dt d i dx gx ret d dt ret x f t t fx e f x ad x e t t −= = β x (3) where [ ] implies that quantities in the bracket are to be evaluated at the retarded time [ ( ) ]. What information is needed : in order to find ? ret Question R t c t =− is the solution of ( ) a. i i x =
x () t r o (at ) et i (at ) i t v t n R t orbit of e [ ( , ), ( , )] at point of observation tt Φ A xx i 1 2 1 2 22 2 ()2() 2 2 ( ) ( ) ( is a fixed position, indep. of time) 2( ) ( ) [] dd dt dt d dR t d dt dt dt t t xt t t t ′′ −⋅ + == + = + =− xr r r r x r v ( ) ( ) (4) ( ) 1 ( ) ( ) ( 0) (5) Sub. (5) into Rt c dt dt t ft t t t κ ⇒= + = > vn β n (1 ) 1 (3) gives the Lienard-Wiechert potentials (,) (14.8) ret e R ret R e t t ⎡⎤ Φ= ⎢⎥ ⎣⎦ = β β n β n x Ax 14.1 Liénard-Wiechert Potentials … ( continued )

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x () t r o (at ) et i (at ) i
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