Today in Physics 217: multipole expansion

[Pages:17]Today in Physics 217: multipole expansion

Multipole expansions Electric multipoles and their moments

? Monopole and dipole, in detail ? Quadrupole, octupole, ... Example use of multipole expansion as approximate solution to potential from a charge distribution (Griffiths problem 3.26)

q

q -q

-q

q

-q

qq

-q

q -q

-q q q -q

21 October 2002

Physics 217, Fall 2002

1

Solving the Laplace and Poisson equations by sleight of hand

The guaranteed uniqueness of solutions has spawned several creative ways to solve the Laplace and Poisson equations for the electric potential. We will treat three of them in this class: Method of images (9 October).

Very powerful technique for solving electrostatics problems involving charges and conductors. Separation of variables (11-18 October) Perhaps the most useful technique for solving partial differential equations. You'll be using it frequently in quantum mechanics too. Multipole expansion (today) Fermi used to say, "When in doubt, expand in a power series." This provides another fruitful way to approach problems not immediately accessible by other means.

21 October 2002

Physics 217, Fall 2002

2

Multipole expansions

Suppose we have a known charge distribution for which we want to know the potential or field outside the region where the charges are. If the distribution were symmetrical enough we could find the answer by several means: direct calculation using Gauss' Law; direct calculation Coulomb's Law; solution of the Laplace equation, using the charge

distribution for boundary conditions.

But even when is symmetrical this can be a lot of work. Moreover, it may give more precise information on the potential or field than is actually needed.

Consider instead a direct calculation combined with a series expansion...

21 October 2002

Physics 217, Fall 2002

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Multipole expansions (continued)

If the reference point for potential is

(and can be) at infinity, then

V

(r

)

=

V

d r

,

where r2 = r2 + r2 - 2rr cos

=

r2

1

+

r 2 r

-

2

r r

cos

r2 (1+ ) .

If point P is far away from the charge distribution, then 1.

P

r r

d

(r)

V

r

21 October 2002

Physics 217, Fall 2002

4

Multipole expansions (continued)

So consider

1 r

=

1 r

1 . But first recall this infinite series: 1+

(1 +

x )s

=

n=0

s!

n!(s - n)!

xn

= 1 + s x + s(s - 1) x2 + s(s - 1)(s - 2) x3 +...

1!

2!

3!

where x < 1 and s is any real number. (This is one form of the

binomial theorem.) Then

1 r

=

1 r

1

-

1 2

+

3 8

2

-

5 16

3

+ ...

21 October 2002

Physics 217, Fall 2002

5

Multipole expansions (continued)

1 r

=

1 r

1 -

1 2

r r

r r

-

2

cos

+

3 8

r r

2

r r

-

2

cos

2

-

5 16

r r

3

r r

-

2

cos

3

+ ...

=

r r

2

+

4 cos2

-

4

r r

cos

=

1 r

1 +

r r

cos

-

1 2

r r

2

+

3 2

r r

2

cos2

-

3 2

r r

3

cos

+

3 8

r r

4

-

5 16

r r

3

r r

3

+

4r r

cos2

-

4

r r

2

cos

r r

3

-2

r r

2

cos

-

8 cos3

+

8

r r

cos2

+ ...

21 October 2002

Physics 217, Fall 2002

6

Multipole expansions (continued)

Collect terms with the same powers of r r , and ignore

higher powers than (r r )3 , for now:

1 r

=

1 r

1

+

r r

cos

+

r r

2

3 2

cos2

-

1 2

+

r r

3

5 2

cos3

-

3 2

cos

+ ...

=

1 r

P0

( cos

)

+

r r

P1

( cos

)

+

r r

2

P2

( cos

)

+

r r

3

P3

(

cos

)

+

...

21 October 2002

Physics 217, Fall 2002

7

Multipole expansions (continued)

Thus,

1 r

=

1 r

n=0

r r

n

Pn

( cos

)

V

(r)

=

1 r

n=0 V

(

r)

r r

n

Pn

( cos

) d

=

1 r

V

(r) d +

1 r2

V

(r) r cos d

Monopole,

Dipole

+

1 r3

V

( r ) r 2

3 2

cos2

-

1 2

d

Quadrupole

+

1 r4

V

( r ) r 3

5 2

cos3

-

3 2

cos

d

+

...

Octupole

21 October 2002

Physics 217, Fall 2002

8

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