The Method of Moments in Electromagnetics

A Brief Review of Electromagnetics 9 around the source. Substituting (2.34) for ´r r ¼ µ,weget Ö Ö¡Ö · ¾ ÖÎ ½ (2.35)Ö ÖÎTo evaluate the the first term, we use the divergence theorem to write Ö ÖÖ¡Ö Î n ¡Ö Ë (2.36)ÖÖOn the sphere, n r, thereforeÎËr ¡Ö´ Öµ Ë Öwhich is Ö ¾ Ö ÖTaking the limit as ¼,weget ÖÐѼ ¾ Ö ÖThe second term is ¾ ¼ ÖÖËËÖ´ Öµ Ë (2.37)ÖÖÖ(2.38) (2.39) Ö ¾ Ö ¾ Ö Ö Ö (2.40)By inspection, this integral tends to zero as ¼. Therefore,¼and ½´r r ¼ µ r r¼ r r ¼ (2.41)(2.42)which is the electrodynamic Green’s function in three dimensions.2.3.2 Two-Dimensional Green’s FunctionThe scalar Helmholtz equation in two dimensions can be written asÖ ¾ ´ ¼ µ· ¾ ´ ¼ µ Æ´ ¼ µ (2.43)The solutions to the above for the homogeneous case are the Hankel functions ofthe first and second kinds of order zero. Because we know the solution comprises