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

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Chapter 14: Radiation by Moving Charges 2 2 2 2 2 2 2 1 2 4 1 converted to Gaussian unit system, see p.782 for conversion formulae. 4 (6.15) (6.16) c t c c t π πρ ∇ Φ − Φ = − = − Review of Basic Equations : A A J 2 2 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 c t in f t t t d x dt G t t f t ψ ψ π ψ ψ + = − ′ ′ = + x x x x x x 2 2 2 2 1 etarded Green's function ( , , , ) / (6.44) is the solution of (with outgoing-wave b.c.) ( , , , ) 4 ( ) ( ) (6.41) [ ( )] ( ) c t c G t t t t G t t t t δ πδ δ + + ′ ′ = = − x x x x x x 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 t t t t d x dt t t Note δ ψ ρ = ′ ′ Φ = ′ ′ Φ x x x x x x J x A x A E 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.
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x ( ) t r o (at ) e t i (at ) e t i ( ) t v ( ) t n ( ) R t orbit of e [ ( , ), ( , )] at point of observation t t Φ A x x i 3 1 ( , ) ( , ) Rewrite (9.2): ( , ) ( , ) ( , ), ( , ) due to a point charge ( carrie [ )] ( c c t t t t d x dt t t t t e e δ ρ ρ = Φ x x x x Lienard - Wiechert Potentials for a Point Charge : A x J x x x x J x [ ] [ ] ( ) ( ) ( ) ( ) ( ) s a sign) moving along the orbit ( ) at the velocity ( ) ( ) / can be written ( , ) ( ) ( , ) ( ) ( ) ( , ) ( , ) [ ] [ ] R t c R t c t R t R t t t t t t t d t dt t e t t e t t t e dt t e dt δ δ ρ δ δ + + = = = Φ = = β r v r x x r J x v x r x A x (1) where ( ) ( ) and ( ) ( ) . R t t t t c = = x r 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 R t c f t t t f t R t R t R t R t t t t t t t t e dt e dt t e dt e dt f t t R t c δ δ δ δ + + Φ = = = = + β β x A x [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2) Using ( ) ( ) , we obtain ( , ) ( , ) [ ] [ ] [ ] i d dt d i dx g x ret d dt ret x R t f t t R t f t f x e g x f x a dx e t t δ = Φ = = β x A x (3) where [ ] implies that quantities in the bracket are to be evaluated at the retarded time [ ( ) ].
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