Appendix C

Scalar Green's Function and the Solution to Helmholtz Equation

Solutions to the vector Helmholtz equations (A.11) and (A.14), and the scalar Helmholtz equations (A.12) and (A.15), respectively, could be found by using the solution to the scalar Helmholtz equation [1,2]. If the Laplace operator is written using the Cartesian coordinate system,

${\mathrm{\nabla }}^{2}\stackrel{\to }{A}={\mathrm{\nabla }}^{2}{A}_{x}{\stackrel{ˆ}{e}}_{x}+{\mathrm{\nabla }}^{2}{A}_{y}{\stackrel{ˆ}{e}}_{y}+{\mathrm{\nabla }}^{2}{A}_{z}{\stackrel{ˆ}{e}}_{z},$

(C.1)

and using the fact that within homogeneous space the magnetic and electric vector potentials $\stackrel{\to }{A}$ and $\stackrel{\to }{F}$ are collinear with electric and magnetic current densities ...

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