Poisson equation
•
The Poisson equation can be written,
∇
2
u
(
~
r
) =
ρ
(
~
r
)
•
For example, in two dimensions,
u
(
x
,
y
) and
ρ
(
x
,
y
)
∂
2
∂
x
2
+
∂
2
∂
y
2
u
(
x
,
y
) =
ρ
(
x
,
y
)
•
For the case of electrostatics, using Gaussian units we have
∇
2
φ
(
~
r
) =

4
πρ
(
~
r
)
•
For a point charge,
∇
2
φ
(
~
r
) =

4
πδ
(
~
r

~
r
0
), and we find
φ
(
~
r
) =
1

~
r

~
r
0

•
The function
G
(
~
r
,~
r
0
) =
1

~
r

~
r
0

is the Green function for the
Poisson equation
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Green function for the Poisson equation
•
The Green function
G
(
~
r
,~
r
0
) =
1

~
r

~
r
0

solves the Poisson equation
for a point source,
∇
2
G
(
~
r
,~
r
0
) =

4
πδ
(
~
r

~
r
0
)
•
Now we notice that
ρ
(
~
r
) =
R R R
ρ
(
~
r
0
)
δ
(
~
r

~
r
0
)
d
τ
, and then with
the volume integral extending over the region of nonzero charge
density,
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 Spring '03
 Staff
 Electrostatics, Fundamental physics concepts, Green's function, Laplace's equation, Poisson's equation

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