Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
to transform the second term on the right hand side of Eq. (10.62) and obtain
…r† ˆ 1 Z
G…r; r 0 †…r 0 †dr 0
"
Z
‡ ‰…r 0 †r 0 G…r; r 0 †G…r; r 0 †r 0 …r 0 †Š dS 0 …10:63†
or
…r† ˆ 1 Z
G…r; r 0 †…r 0 †dr 0
"
Z
‡ …r 0 † @
@n 0 G…r; r 0 †G…r; r 0 † @
@n 0 …r 0 † dS 0 ;
…10:64†
where n 0 is the outward normal to dS 0 . The Green's function G…r; r 0 † can be found
from Eq. (10.60) subject to the appropriate boundary conditions.
If the potential vanishes on the surface S 0 or @=@n 0 vanishes, Eq. (10.64)
reduces to
…r† ˆ 1 Z
G…r; r 0 †…r 0 †dr 0 :
…10:65†
"
On the other hand, if the surface S 0 encloses no charge, then Poisson's equation
reduces to Laplace's equation and Eq. (10.64) reduces to
Z
…r† ˆ …r 0 † @
@n 0 G…r; r 0 †G…r; r 0 † @
@n 0 …r 0 † dS 0 : …10:66†
The potential at a ®eld point r due to a point charge q located at the point r 0 is
Now
…r† ˆ 1 q
4" jr r 0 j :
r 2 1
jr r 0 ˆ4…r r 0 †
j
(the proof is left as an exercise for the reader) and it follows that the Green's
function G…r; r 0 †in this case is equal
G…r; r 0 †ˆ 1 1
4" jr r 0 j :
If the medium is bounded, the Green's function can be obtained by direct solution
of Eq. (10.60) subject to the appropriate boundary conditions.
To illustrate the procedure of the Green's function technique, let us consider a
simple example that can easily be solved by other methods. Consider two
grounded parallel conducting plates of in®nite extent: if the electric charge density
between the two plates is given, ®nd the electric potential distribution between
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