March 2014
Intermediate
876 pages
28h 17m
English
Content preview from The Finite Element Method in Electromagnetics, 3rd Edition
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APPENDIX CGREEN’S FUNCTIONS
The Green’s function method is a powerful technique for solving boundary-value problems. Green’s functions were named after George Green (1793–1841), who developed a general method to obtain solutions of Poisson’s equation in potential theory. This method was described in an essay by Green titled “On the application of mathematical analysis to the theories of electricity and magnetism,” published in 1828.
To illustrate the Green’s function method, consider the electric potential produced by a point electric charge q1 placed at r1 in an unbounded homogeneous medium. It is well known from the elementary theory of electricity [1] that this potential at r is given by
(C.1)
where |r − r1| denotes the distance between the points r and r1 and ∊ is the permittivity of the medium. If there is another point charge q2 placed at r2, the potential produced by this charge is
(C.2)
The total potential produced by q1 and q2 is then the linear superposition of ϕ1 and ϕ2:
(C.3)
If there are N point charges in the space, the total potential is given by
(C.4)
where ...
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