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

Example: constV 0

odd is if

4

even is if 0

0 nn

V

nCn

Contributions of the first terms of the Fourier sum at x=0.

a) n=1, b) n<6, c) n<11, d) n<101

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

Jean Bapitiste Joseph Fourier 21 March 1768 – 16 May 1830

Example 3.4

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

Example 3.6

Example 3.8

Multipole Expansion

Approximate potential at large distance

Dipole:2

0

cos

4

1)(

r

qdV

r

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.

The electric field of a dipole along the z-axis.

)ˆsinˆcos2(4

),(3

rEr

pr

odip