3. 3 separation of variables we seek a solution of the form cartesian coordinatescylindrical...
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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.
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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
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)()(),( :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),(
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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
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Example: constV 0
odd is if
4
even is if 0
0 nn
V
nCn
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Contributions of the first terms of the Fourier sum at x=0.
a) n=1, b) n<6, c) n<11, d) n<101
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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
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Jean Bapitiste Joseph Fourier 21 March 1768 – 16 May 1830
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Example 3.4
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Example 3.5
An infinitely long metal pipe is grounded, but one end is maintained at a given potential.
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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
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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
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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
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The first Legendre polynomials
8/)33035()(
2/)35()(
2/)13()(
)(
1)(
244
33
22
1
0
xxxP
xxxP
xxP
xxP
xP
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Example 3.6
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Example 3.8
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Multipole Expansion
Approximate potential at large distance
Dipole:2
0
cos
4
1)(
r
qdV
r
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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
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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
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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
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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
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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.
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The electric field of a dipole along the z-axis.
)ˆsinˆcos2(4
),(3
rEr
pr
odip