Lecture 20 Lecture Notes

# Lecture 20 Lecture Notes - Franklin ch 13.3 7.2.1 Def Green...

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7. Solution to Maxwell's Eqns for arbitrary sources arbitrary current charges find => Greens function Note, Griffith does not follow this approach, but uses the result! Derivation via Green functions is a good motivation! Solution for boundary conditions for "convolution" Green function satisfies: Green function Lec 20 - 5. Mar January 29 08 9:54 PM Lecture Notes Page 1

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Therefore differentiation with respect to ! Boundary conditions: require if in a limited volume Summary: Green function is a solution of a PDE function with -function source term, subject to the desired boundary condition. => put hard work into finding Green function => once Green function known, problem of finding solution for any given source distribution shifted from solving a PDE to integration (convolution between Green function and source) 7.2 Solution of a wave equation via Green functions Jackson, 2nd ed., ch. 6.6 and 12.11

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Unformatted text preview: Franklin, ch. 13.3 7.2.1 Def. Green function Green function satisfies and boundary condition: for Then a solution to only outgoing waves for the same boundary conditions is given by Lecture Notes Page 2 The general solution , including in- and outgoing waves is solution of in & outgoing waves 7.2.2 Construction of Green function via Fourier Transform Then with implicitly defined via Fourier transform results in Therefore and Looks easy enough, but: Integrand has poles at Lecture Notes Page 3 7.2.3 Evaluation of Green function integral Evaluate via Residue Theorem Embedding into complex parameter space: retarded G. funct. advanced G.funct. all paths possible and valid, but leading to different structure of boundary conditions-different Green functions-homogeneous wave equation! => differ only by solutions of Lecture Notes Page 4...
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Lecture 20 Lecture Notes - Franklin ch 13.3 7.2.1 Def Green...

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