1
PHYS809Class9Notes
Separation of variables in polar coordinates
To illustrate the method of separation of variables, consider the two
dimensional problem in which two semiinfinite conducting planes at
potential
V
meet along their edges at an angle
α
. The geometry suggests
that polar coordinates will be useful for solving Laplace’s equation for the
potential between the two planes. In this coordinate system,
2
2
2
1
1
0.
r
r
r
r
r
φ
φ
θ
∂
∂
∂
+
=
∂
∂
∂
(1.1)
To separate the variables, we assume that the potential is the product of a function of
r
and a function
of
θ
,
(
)
( )
(
)
,
.
r
R r
φ
θ
θ
=
Θ
(1.2)
On dividing by the potential, Laplace’s equation leads to
2
2
1
.
r d
dR
d
r
R dr
dr
d
θ
Θ
= 
Θ
(1.3)
Each side of the equation is a function of a single variable.
This is possible only if the two sides are equal
to the same constant. For reasons that will become apparent, we will set this constant to be
ν
2
, where
ν
is nonnegative. The equations for
R
and
Θ
are now
2
,
d
dR
r
r
R
dr
dr
ν
=
(1.4)
and
2
2
2
,
d
d
ν
θ
Θ
= 
Θ
(1.5)
which have solutions
,
0,
ln ,
0,
ar
br
R
a
b
r
ν
ν
ν
ν

+
≠
=
+
=
(1.6)
and
sin
cos
,
0,
,
0,
c
d
c
d
νθ
νθ
ν
θ
ν
+
≠
Θ =
+
=
(1.7)
where
a
,
b
,
c
and
d
are constants.
In general, the solution is a linear superposition of terms of form
r
θ
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(
)
(
)
(
)(
)
sin
cos
,
0,
ln
,
0.
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 Fall '11
 MacDonald
 Cartesian Coordinate System, Electric Potential, Electric charge, Polar coordinate system, charge density

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