Chapter%207

Dr. Ray Chen©
Chapter 7
Poisson’s and Laplace’s Equations

Dr. Ray Chen©
7.1 Poisson’s and Laplace’s Equations
( )
0 region s homogeneou a For =
∂
∂
=
∂
∂
=
∂
∂
= ∇ ⋅ −∇ = ⋅ ∇ = ⋅ ∇
−∇ = = = ⋅ ∇
z y x
V ) E ( D
V E E D D
v
v
ε ε ε
ρ ε ε
ε ρ
v v
v v v
Equation s Poisson'
ε
ρ
v V − = ∇ ⋅ ∇
ε
ρ
v
z y x
z
V
y
V
x
V
)
z
V
(
z
)
y
V
(
y
)
x
V
(
x
V
a
ˆ
z
V
a
ˆ
y
V
a
ˆ
x
V
V
z
z
y
y
x
x
− =
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
= ∇ ⋅ ∇ ∴
∂
∂
+
∂
∂
+
∂
∂
= ∇
∂
∂
+
∂
∂
+
∂
∂
= ⋅ ∇
2
2
2
2
2
2
A A A
A
v

Dr. Ray Chen©
7.1 Poisson’s and Laplace’s Equations
V. of Laplacian the called is
Equation. s Laplace' the is which 0 have we 0 If
2
2
∇
= ∇ = , V ,
v
ρ
true, always is 0 0 If
2
= ∇ = V ,
v
ρ
l cylindrica spherical, cartesian, as such
system, coordinate different in d represente be can
2
∇

Dr. Ray Chen©
7.1 Poisson’s and Laplace’s Equations
2
2
2
2
2
2
2
is equation s Laplace' , coordinate cartesian In
z
V
y
V
x
V
V
∂
∂
+
∂
∂
+
∂
∂
= ∇
2
2
2
2
2
2
1 1
coordinate l cylindrica In
z
V V V
V
∂
∂
+
∂
∂
+








∂
∂
∂
∂
= ∇
φ ρ
ρ
ρ
ρ ρ
2
2
2 2 2
2
2
2
1 1 1
coordinate spherical In
φ θ θ
θ
θ θ ∂
∂
+






∂
∂
∂
∂
+






∂
∂
∂
∂
= ∇
V
sin r
V
sin
sin r r
V
r
r r
V

Dr. Ray Chen©
7.2 Uniqueness Theory
have we condition,
boundary same the under 0 V Equation s Laplace’ the
satisfying and functions potential two are there If
2
2 1
= ∇
V V
2 1
V V =

Dr. Ray Chen©
7.2 Uniqueness Theory
Proof:
( )
0
0 0
Equation s Laplace' to According
2 1
2
2
2
1
2
= − ∇
= ∇ = ∇
V V
V V
Each solution must also satisfy the boundary conditions,
and if we represent the given potential values on the boundaries,
then
0 V V or V V
V boundary) the at
V boundary) the at
2b 1b 2b 1b
2b 2
1b 1
= − = =
=
=
b
V
( V
( V
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Dr. Ray Chen©