Chapter 4
Boundary Value Problems in Spherical and Cylindrical Coordinates
4.1
Laplace’s Equation in spherical coordinates
Laplace’s equation in spherical coordinates,
(
r, θ, φ
)
,
has the form
1
r
2
∂
∂r
·
r
2
∂
Φ
(
r, θ, φ
)
¸
−
1
r
2
L
·
L
Φ
(
r, θ, φ
)=0
.
(4.1)
where
−
L
·
L
≡
1
sin
θ
∂
∂θ
·
sin
θ
∂
Φ
(
r, θ, φ
)
¸
+
1
sin
2
θ
∂
2
∂φ
2
(4.2)
Multiplying Eq. 4.1 by
r
2
,
dividing by
Φ
(
r, θ, φ
)=
R
(
r
)
Y
(
θ,φ
)
and rewriting,
1
R
∂
·
r
2
∂
R
(
r
)
¸
=
1
Y
L
·
L
Y
(
)
.
(4.3)
This equation must be satis
f
ed for all
(
r, θ, φ
)
As
r, θ, φ
are independent variables, Eq. 4.3 can only be satis
f
ed if both
sides equal a constant which we choose this constant to be
c
(
c
+1)
.
Then,
L
·
L
Y
(
c
(
c
Y
(
)
(4.4)
and
d
dr
·
r
2
d
dr
R
(
r
)
¸
−
c
(
c
R
(
r
(4.5)
Eq. 4.4 can be further separated using
Y
(
V
(
θ
)
W
(
φ
)
:
1
V
(
θ
)
sin
θ
∂
·
sin
θ
∂
V
(
θ
)
¸
+
1
V
(
θ
)
sin
2
θc
(
c
V
(
θ
−
1
W
∂
2
2
W
(4.6)
Again, since
θ
,
φ
are independent variables both sides must equal a constant (the second separation constant) which we
take to be
−
m
2
∂
2
2
W
(
φ
−
m
2
W
(
φ
)
or
(4.7a)
W
m
(
φ
A
m
e
imφ
+
B
m
e
−
imφ
(4.7b)
and
sin
θ
∂
·
sin
θ
∂
V
(
θ
)
¸
+[sin
2
(
c
−
m
2
]
V
(
θ
(4.8)
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