3. 3 Separation of Variables
We seek a solution of the form
)()()(),,( zhygxfzyxV Cartesian coordinates
)()()(),,( zhgsfzsV Cylindrical coordinates
)()()(),,( hgrfrV Spherical coordinates
Not always possible! Usually only for the appropriate symmetry.
Example 3.3
0
problem
ldimensiona-Two
2
2
2
2
y
V
x
V
Special boundary conditions (constant potential on planes):
xVivyVyViii
axViixVi
for 0)(),(),0()(
,0),()(,0)0,()(
0
)()(),( :Ansatz yYxXyxV
22
22
2
2 1 and
1k
dy
Yd
Yk
dx
Xd
X
Special choice of the separation constants to be able to fulfill the boundary conditions.
Boundary conditions (i, ii, iv):
)cossin)((),( kyDkyCBeAeyxV kxkx
a
nkkyCeyxV kx with,sin),(
Boundary condition (iii):
1
0 )sin(),0()(n
n a
ynCyVyV
a
n dya
ynyV
aC
0
0 )sin()(2
Fourier sum
Fourier coefficients
1
)sin(),(n
a
xn
n a
yneCyxV
superposition
Set of functions is called
n
nn ygyfCygcomplete )(function any for )()( if
a
nn nndyyfyforthogonal0
' 'for 0)()( if
a
n dyyfnormal0
2 1)( if
a
n dyyfygnormalorthofor0
n )()(C sets
normal-ortho is )sin(2
)(a
yn
ayfn
Example 3.5
An infinitely long metal pipe is grounded, but one end is maintained at a given potential.
Spherical Coordinates Use for problems with spherical symmetry.
0sin
1)(sin
sin
1)(
12
2
2222
2
V
r
V
rr
Vr
rr
Laplace’s equation:
Boundary conditions on the surface of a sphere, origin, and infinity.
Solution as a product
(((),,( rRrV
Assume azimuthal symmetry
Solution as a product
((),( rRrV
Separation constant )1(21 llCC
Radial equation Rlldr
dRr
dr
d)1()( 2
Solution 1( ll BrArrR
Angular equation
sin)1()(sin lld
d
d
d
Solutions Legendre polynomials )(cos( lP
The second solution can (usually) be excluded because it becomes infinite at
Rodrigues formula 3,2,1,0,)1(!2
1)( 2
lx
dx
d
lxP l
l
ll
Orthogonality
1
1
'
0
'
' if 12
2
' if 0)()(
sin)(cos)(cos
lll
lldxxPxP
dPP
ll
ll
The first Legendre polynomials
8/)33035()(
2/)35()(
2/)13()(
)(
1)(
244
33
22
1
0
xxxP
xxxP
xxP
xxP
xP
Potential of a general charge distribution at large distance
')'(1
4
1)(
0
dV rrr
Warning! The integral dependson the direction of r.
01
0
')'()'(cos)'(1
4
1)(
nn
nn
dPrr
V
rr
Addition theorem for Legendre polynomials:
lml
lml
ml
ln
ml
l
lm
ml
mll
dYrr
Y
lV
YYl
P
rr
,
,0
*1
0
*
')'()','()'(),(
)12(
1)(
)','(),(12
4)(cos
cos''
rr
rr
Spherical harmonics:
imlm
ml eP
lY )(cos
4
12),(
solutions for 3D separation
Angular distributionat large distance
The monopole and Dipole Terms
monopoler
QV
omon 4
1)( r
2
ˆ
4
1)(
rV
odip
rpr
dipole
dipole moment
n
iiiqd
1
' ,')'(' rprrp
A quadrupole has no dipole moment.
physical dipole
drrp qqq ''
“pure” dipole is the limit
constqqd dp,,0
Dipole moments are vectors and add accordingly.
21 ppp
In general, multipole momentsdepend on the choice of the coordinate system.
Has a dipole moment.
app Q If Q=0 the dipole moment does not depend on the coordinate system.