Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF POISSON'S EQUATION
the plates. The electric potential distribution is described by solving Poisson's
equation
subject to the boundary conditions
(1) …0† ˆ0;
(2) …1† ˆ0:
r 2 ˆ="
We take the coordinates shown in Fig. 10.3. Poisson's equation reduces to the
simple form
d 2
dx 2 ˆ " :
…10:67†
Instead of using the general result (10.64), it is more convenient to proceed
directly. Multiplying Eq. (10.67) by G…x; x 0 † and integrating, we obtain
Z 1
0
Z
G d2
1
dx 2 dx ˆ
0
…x†G
dx:
"
…10:68†
Then using integration by parts gives
Z 1
G d2
dx 2 dx ˆ G…x; x 0 † d…x†
1
dx
0
0
Z 1
0
dG d
dx dx dx
and using integration by parts again on the right hand side, we obtain
Z 1
G d2
dx 2 dx ˆG…x; x 0 † d…x†
1
" ‡ dG
1 Z #
1
dx dx d2 G
dx 2 dx
0
ˆ G…0; x 0 † d…0†
dx
G…1; x 0 † d…1†
0
0
0
dx Z 1
0
d2 G
dx 2 dx:
Figure 10.3.
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