Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF POISSON'S EQUATION
®eld at a point x due to a unit point source at x 0 , then the total ®eld at x due to a
distributed source …x 0 † is the integral of G over the range of x 0 occupied by the
source. The function G…x; x 0 † is the well-known Green's function. We now apply
this technique to solve Poisson's equation for electric potential (Example 10.2)
r 2 …r† ˆ 1 " …r†;
…10:59†
where is the charge density and " the permittivity of the medium, both are given.
By de®nition, Green's function G…r; r 0 † is the solution of
r 2 G…r; r 0 †ˆ…r r 0 †;
…10:60†
where …r r 0 † is the Dirac delta function.
Now, multiplying Eq. (10.60) by and Eq. (10.59) by G, and then subtracting,
we ®nd
…r†r 2 G…r; r 0 †G…r; r 0 †r 2 …r† ˆ…r†…r r 0 †‡ 1 " G…r; r 0 †…r†;
and on interchanging r and r 0 ,
or
…r 0 †r 02 G…r 0 ; r†G…r 0 ; r†r 02 …r 0 †ˆ…r 0 †…r 0 r†‡ 1 " G…r 0 ; r†…r 0 †
…r 0 †…r 0 r† ˆ…r 0 †r 02 G…r 0 ; r†G…r 0 ; r†r 02 …r 0 † 1 " G…r 0 ; r†…r 0 †;
…10:61†
the prime on r indicates that di€erentiation is with respect to the primed coordinates.
Integrating this last equation over all r 0 within and on the surface S 0
which encloses all sources (charges) yields
…r† ˆ 1 Z
G…r; r 0 †…r 0 †dr 0
"
Z
‡ ‰…r 0 †r 02 G…r; r 0 †G…r; r 0 †r 02 …r 0 †Šdr 0 ; …10:62†
where we have used the property of the delta function
Z ‡1
1
f …r 0 †…r r 0 †dr 0 ˆ f …r†:
We now use Green's theorem
ZZZ
ZZ
f r 02 r 02 f †d 0 ˆ
… f r 0
r 0 f †dS
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