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©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1
LECTURE NOTES 8
POTENTIAL APPROXIMATION TECHNIQUES:
THE ELECTRIC MULTIPOLE EXPANSION
AND MOMENTS OF THE ELECTRIC CHARGE DISTRIBUTION
There are often situations that arise where an “observer” is far away from a localized charge
distribution ( )
ρ r
G
and wants to know what the potential ( )
V r
G
and / or the electric field
intensity ( )
E r
G
are far from the localized charge distribution.
If the localized charge distribution has a net electric charge Q net
, then far away from this
localized charge distribution, the potential V ( r )
G
to a good approximation will behave very much
like that of a point charge,
( )
net
far
o
Q
V r
G
r
and ( ) ( )
2
net
far far
o
Q
E r V r
G JK
G G
r
when the field point – source charge separation distance, r d ,the characteristic size of the
charge distribution.
However, as the “observer” moves in closer and closer to the localized charge distribution
( )
ρ r ′
G
, he/she will discover that increasingly ( )
V r
G
(and hence ( )
E r
G
G
) may deviate more and
more from pure point charge behavior, because ( )
ρ r ′
G
is an extended source charge distribution.
Furthermore, ( )
ρ r
G
may be such that 0
net
Q ≡ , but that does NOT necessarily imply that
( )
V r = 0
G
(and ( )
E r
G
G
Example:
A pure, physical electric dipole is a spatially-extended, simple charge distribution where 0
net
Q =
but ( )
V r ≠ 0
G
and ( ) ( )
E r = −∇ V r ≠ 0
G JK
G G
, as shown in the figure below:
P (field point)
r
A pure physical electric dipole is d θ
composed of two opposite electric r −
charges separated by a distance d :
− q
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2
The Potential ( )
V r
G
and Electric Field ( )
E r
G
G
of a Pure Physical Electric Dipole
“Pure” → Q net
= 0 “ Physical ” → Spatially extended electric eipole d ≠ 0 , d > 0
{n.b. ∃ “point” electric dipoles with d = 0, e.g. neutral atoms & molecules…}
First, let us be very careful / wise as to our choice of coordinate system. A wrong choice of
coordinate system will unnecessarily complicate the mathematics and obscure the physics we are
attempting to learn about the nature / behavior of this system.
Examples of BAD choices of coordinate systems: z ˆ
q
+
a.) q
+
z ˆ b.)
z ˆ
Ο ′ θ ′ y ˆ′
dipole
r
G
Ο Ο ϕ'
y ˆ y ˆ
ϕ q
−
x ˆ′
x ˆ q
−
x ˆ
Dipole lying in x – y plane has Even more mathematically complicated!!
ϕ -dependence, but (at least it) Origin is not conveniently chosen (arbitrary?)
is centered at the origin. Angle the dipole axis makes with respect to
z ˆ & x ˆ axes must be described by two
angles - θ and ϕ.
Smart / wise choice of coordinate system: Exploit intrinsic symmetry of problem.
Physical electric dipole has axial symmetry – choose z ˆ axis to be along line separating q
+
and q
−
Choose x - y plane to lie mid-way between q
+
and q
−
z ˆ P (field point)
G
r
G
d
−
G
r
y
x − q
Mathematical expressions obtained for
( ) ( ) ( )
V r , E r = −∇ V r
G JK
G G G
for this choice
of coordinate system for the physical
electric dipole can be explicitly and
rigorously related to more complicated
/ tedious mathematical expressions for
a.) and b.) above – via coordinate
translations & rotations!
n.b. This problem
now has no
ϕ -dependence
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 4
( ) ( ) ( )
( ) ( )
( ) ( )
2 2
2 2
2 2
2 2
2 cos 2 cos
2 cos 2 cos
dipole q q
o o
o o
o
q q
V r V r V r
q q
r d rd r d rd
q
r d rd r d rd
G G G
r r
This is an exact analytic mathematical expression for the potential associated with a pure
( )
net
Q = physical electric dipole with charges + q and – q separated from each other by a
distance d. Note further that, because of the judicious choice of coordinate system and the
intrinsic (azimuthal) symmetry, ( )
dipole
V r
G
has no ϕ -dependence.
The exact analytic expression for potential associated with pure physical electric dipole:
( )
( ) ( )
2 2
2 2
2 cos 2 cos
dipole
o
q
V r
r d rd r d rd
G
As mentioned earlier, often we are / will be interested only in knowing (approximately)
( )
dipole
V r
G
when r d
G
. For example, many kinds of neutral molecules have permanent electric
dipole moments p ≡ qd
G
G
(Coulomb-meters) and (obviously) for such molecules, the dipole’s
separation distance d is (typically) on the order of ~ few Ångstroms, i.e. d ~Ο (5Å)
{1 Å ≡ 10
− 10
m = 10 nm (1 nm = 10
− 9
m )}. So even if the field point P is e.g.
6
r 1 μ m 10 m
−
G
away from such a molecular dipole, r = 1 μ m d ~ 5 nm
G
, since d r 0.
G
In such situations, when r d
G
an approximate solution for ( )
dipole
V r
G
which has the benefit
of reduced mathematical complexity, will suffice to give a good / reasonable physical
description of the intrinsic physics, accurate e.g. to 1% (or better) when compared directly to the
exact analytical expression over the range of distance scales r d
G
that are of interest to us.
Thus for r > d
G
, the exact expressions for the r
and r −
separation distances are:
( )
2
2
2
2
2 cos
1 cos
1 cos
r d rd
d d
r r
r
d d
r
r r
r ( )
2
2
2
2
2 cos
1 cos
1 cos
r d rd
d d
r r
r
d d
r
r r
−
r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5
Now if ( )
d r 1 , then let us define:
2
cos
d d
r r
and:
2
cos
d d
r r
−
Then:
r 1 ε
r
and:
r 1 ε
− −
r
with: ε 1
and: ε 1
−
Now if ε 1
and ε 1
−
, we can use the Binomial Expansion (a specific version of the more
generalized Taylor Series Expansion) of the expression:
( )
1
2 3
2
−
± ± ± ±
±
i i i
i i i
(Valid on the interval: 1 ε 1
±
Since ε
±
is already <<1, then the higher-order terms ( ) ( ) ( )
2 3 4
± ± ±
etc. are incredibly
small (<<<<<1), so negligible error is incurred by neglecting these higher-order terms,
i.e. keeping only terms linear in ε
±
in the binomial expansion of
±
, we have:
( )
1
2
r r
r
and: ( )
1
2
r r
−
− −
r
Then:
( ) ( ) ( )
1 1
2 2
dipole
o o
o
q q
V r
r r
q
x
G
r r
1
2
{ }
( )( ) { }
1 1
2 2
o
q
r
− − +
Now:
2
cos
d d
r r
and:
2
cos
d d
r r
−
Then:
( )
2
dipole
o
q d
V r
πε r r
G
2
cos
d d
r r
cos
cos cos
o
o
d
r
q d d
r r r
q
r
cos cos
o
d q d
r r r
Thus: ( )
2 2
cos cos cos
dipole
o o o
q d q d qd
V r
r r r r
G
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7
The electric field ( )
dipole
E r
G
G
associated with a pure, physical electric dipole,
with electric dipole moment p = qd = qdz ˆ
G
G
is:
( ) ( ) ( ) ( ) ( )
dipole dipole dipole
dipole dipole r
E r V r E r r E r E r
θ ϕ
G JK
G G G G G
in spherical-polar coordinates.
The components of ( )
dipole
E r
G
G
in spherical-polar coordinates are:
( )
( )
3
cos
dipole dipole
r
o
V r
p
E r
r r
G
G
( )
( )
3
sin
dipole dipole
o
V r
p
E r
r r
θ
G
G
( )
( ) 1
sin
dipole dipole
V r
E r
r
ϕ
G
G
Explicitly, the electric field intensity of a pure, physical electric dipole with electric dipole
moment
p = qd = qdz
G
G
(in spherical-polar coordinates) is:
( )
3 3 3
cos ˆ sin 2 cos ˆ sin
dipole
o o o
p p p
E r r r
r r r
G
G
Note that: ( )
3
dipole
E r
r
G
G
∼ ( c.f. w / ( )
2
monopole
E r
r
G
G
∼ for single point charge q at r = 0
G
Note also that ( )
dipole
V r
G
and ( )
dipole
E r
G
G
have no explicit ϕ -dependence, since the charge
configuration for an electric dipole is manifestly axially / azimuthally symmetric
(i.e. charge configuration for electric dipole is invariant under arbitrary ϕ -rotations).
Now: ( )
2
dipole
o
p r
V r
πε r
G
G i
with electric dipole moment p = qdz ˆ,
G
and p r ˆ = p cos θ= qd cos θ,
G
i
(since z r ˆ i ˆ=cos θ), and
2 2 2 2
r = x + y + z in Cartesian/rectangular coordinates.
In Cartesian/rectangular coordinates the electric field intensity of a pure, physical electric dipole
with electric dipole moment p = qd = qdz ˆ
G
G
(in spherical-polar coordinates) is:
( ) ( ) ( )
dipole dipole
dipole
dipole dipole dipole
x y z
E r V r x y z V r E x E y E z
x y z
G JK
G G G
Transformation from Spherical-Polar → Cartesian Coordinates:
sin cos sin cos cos cos sin
sin sin ˆ sin sin ˆ cos sin sin ˆ
cos ˆ cos sin
x r x r
y r y r
z r z r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 8
It is a straight-forward exercise to show that the electric field components associated with a pure
physical electric dipole with electric dipole moment p = qd = qdz ˆ
G
G
(in Cartesian coordinates) are:
5 3
5 3
2 2 2
5 3
3 3sin cos
3 3sin cos
3 3cos 1
dipole
x
o o
dipole dipole
y x
o o
dipole
z
o o
p xz p
E
r r
p yz p
E E
r r
p z r p
E
r r
In coordinate-free form, it is also a straight-forward exercise (try it!!!) to show that the electric
field intensity of a pure physical electric dipole with electric dipole moment p = qd = qdz ˆ
G
G
is of
the form:
( ) ( )
3
physical
dipole
o
E r p r r p
πε r
G
G G G
i
whereas the coordinate-free form of a point electric dipole is of the form:
( ) ( ) ( )
3
3
point
dipole
o o
E r p r r p p r
r
G
G G G G G
i
E −
G
Field Lines & Equipotentials Associated with a Pure, Physical Electric Dipole:
(since charge configuration
of electric dipole is axially /
azimuthally symmetric)
n.b. Equipotentials
are ⊥ to lines of
E r ( )
G
G
everywhere!
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 10
- z ˆ 2. z ˆ 3. z ˆ
copy
+ Q + Q → + Q + Q − Q
d d d
− Q − Q − Q − Q + Q
Original Original Copy Original Charge-Conjugated Copy
- z ˆ
+ Q − Q
d
= −2Q − Q + Q
− Q
d Translation of charge-conjugated
copy along axis of original dipole
Pure, Physical, Linear Electric Quadrupole:
z ˆ P (Field Point)
a
r
G
Note that this linear electric quadrupole has
- Q axial / aximuthal symmetry – i.e. because
r
G
all charges (+ Q , − 2 Q , + Q ) are co-linear
d (all on ˆ z axis), problem is invariant under
Ο θ (arbitrary) ϕ -rotations.
− 2 Q
y ⇒ ( )
quad
V r
G
and ( )
quad
E r
G
G
will have no
π − θ explicit ϕ -dependence for the linear
d x ˆ electric quadrupole.
b
r
G
TOT
Q = for pure electric quadrupole.
Again, we use the principle of (linear) superposition to obtain ( )
quad
V r
G
( ) ( ) ( ) ( ) ( )
2
quad TOT Q Q Q
o a b o a b
V r V r V z d V z V z d
Q Q Q Q r r
πε r r r πε r r r
G G
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11
Again, using the Law of Cosines:
2 2 2
2 cos
a
r = r + d − rd θ and
2 2 2
cos
b
r = r + d + rd θ
We obtain:
( )
2 2 2 2
2 cos 2 cos
quad
o
Q r r
V r
r
r d rd r d rd
G
Again, for regime where the observation point P is far away from pure, physical, linear electric
quadrupole, i.e. r >> d , we expand
a
r
r
and
b
r
r
in a binomial (i.e. Taylor) series
(as was done previously for the case of a pure, physical electric dipole).
Neglecting terms in these expansions that are higher order than linear (i.e. > ( )
2
d r ) we obtain:
( )
2 2
3cos 1
1 cos
a
r d d
r r r
( )
2 2
3cos 1
1 cos
b
r d d
r r r
Recall that the Ordinary Legendré Polynomials ( )
P
x cos
P x
= θ
A
are:
( ) ( )
0 0
P x = 1 → P cos θ = 1
( ) ( )
1 1
P x = x → P cos θ =cosθ
( )
( )
( )
( )
2 2
2 2
3 1 3cos 1
cos
x
P x P
( ) ( ) ( )
2
0 1 2
a
r d d
P P P
r r r
and ( ) ( ) ( )
2
0 1 2
b
r d d
P P P
r r r
( )
( )
2 2
2 2
3
3cos 1
1 1
2 1 3cos 1
quad
o a b o
o
Q r r Q d
V r
r r r r r
Qd
r
G
Then for r >> d :
( )
( )
( )
2
2 2 2
2 3 3
2 1 3cos 1 2 1
P
quad
o o
Qd Qd
V r P
r r
θ
θ
θ
πε πε
G
Note that: ( )
3
quad
V r
r
G
∼ (c.f. with ( ) ( ) 2
and
monopole dipole
V r V r
r r
G G
Exact analytic
expression
Shorthand notation:
( ) ( )
P cos θ = P θ
A A
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13
As we have seen for the two previous cases, that of:
- The electric monopole, with its accompanying electric monopole moment, the electric charge
Q (n.b. Q is a scalar quantity) (SI units of Q : Coulombs)
- The electric dipole with its accompanying electric dipole moment p ≡ Qd , p = p = Qd
G
G G
(n.b. p
G
is a vector quantity) (SI units of p
G
: Coulomb-meters)
- The electric quadrupole also has an accompanying electric quadrupole moment Q ≡ 2 Qdd
I GG
(n.b. Q
I
is a tensor quantity) (SI units of Q
I
: Coulomb-meters
2
Tensor Q ≡ 2 Qdd
I GG
= “double vector”
2
Q ≡ 2 Qdd = 2 Qd
I
2-dimensional matrix
Formally speaking, Q
I
is a rank-2 tensor (i.e. a 2-dimensional matrix) - the 9 elements of the
Q
I
tensor (in general) are:
Q
xx
Q
yz
Q
zx
n.b. Q
I
has only six independent components, because Q ij
= Q
ji
Q
I
= Q
xy
Q
yy
Q
zy
i.e. Q xy
= Q
yx
Q
xz
Q
yz
Q
zz
Q
xz
= Q
zx
Q
yz
= Q
zy
Also, note that: Q xx
+ Q
yy
+ Q
zz
= 0 or: Q zz
= −( Q
xx
+ Q
yy
) {i.e. Q
I
is traceless }
The quadrupole moment tensor can also be written in coordinate-free form, e.g. in Cartesian
coordinates as:
n = # discrete charges q
i
( )
2
1
2
1
n
i i i i
i
Q r r r q
=
∑
I I
GG
with
2
i i i
r = r r
G G
i
x ˆ ˆ x 0 0
Unit Dyadic: 1 ≡
I
0 y ˆˆ y 0
zz
For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with
quadrupole moment Q
I
(e.g. oriented along the z ˆ -axis):
z
Here, Q xx
= Q
yy
, and since: Q xx
+ Q
yy
+ Q
zz
+ Q
d Then: Q zz
= − 2 Q
xx
= − 2 Q
yy
≡ 2 Qd
2
All other Q ij
vanish (= 0) for i ≠ j
− 2 Q − 1 0 0
d i.e.
linear 2
quad
Q = Qd
I
+ Q 0 0 +
n.b. conventions / definitions of
linear
quad
Q
I
differ in different textbooks!!!
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 14
For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with
quadrupole moment Q
I
(oriented along the ˆ z -axis), expressed in Cartesian coordinates:
z P (Field Point) # discrete charges
a
r
G
( )
2 1
2
1
n
i i i i
i
Q r r r q
=
∑
I I
GG
with
2
i i i
r = r r
G G
i
G
Unit Dyadic:
d
b
r
G
x ˆ ˆ x 0 0
− 2 Q y ˆ 1 ≡
I
0 y ˆˆ y 0
d 0 0 zz ˆˆ
+ Q
x
1
i = 1: r = + dz ˆ
G
1
q = + Q
i i
r = r
G
2
i = 2 : r = 0 z ˆ
G
2
q = − 2 Q
3
i = 3 : r = − dz ˆ
G
3
q = + Q
Thus: ( )
1
2 2
for charge 1:
Q r dz
Q Q d zz d Q zz
=+
G
HG I
i
0
=
I
i
( ) ( )
3
2
0
2 2 2
for charge 3:
for charge 2:
2 @ 0 ˆ
Q r dz
Q r z
Q d zz d Qd zz
=
=−
− =
G
G
I I
( )
2 2
zz
Q Qd zz Qd
I
HG I
Then: ( )
( )
( )
2
2 2
2 3 3
3cos 1
cos
quad
o o
Qd Qd
V r P
r r
G
( ) ( )
2
2
cos 3cos 1
P θ = θ−
We can express ( )
quad
V r
G
in a different (but totally equivalent manner), using the fact(s) that:
r = sin θ cos ϕ x + sin θ sin ϕ y +cosθ z
z r i = r z i =cos θ
x x i = 1, x y i = 0, x z i = 0
( )( )
2
3 r z i z r i =3cos θ
y x i = 0, y y i = 1, y z i = 0
r 1 r = 1
I
i i
z x i = 0, z y i = 0, z z i = 1
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 16
E
G
-field lines & equipotentials associated with a pure, physical, linear electric quadrupole:
n.b. E
G
-field lines ⊥ to equipotentials everywhere in space
Higher-Order Pure, Linear Physical Electric Multipoles
The next higher order pure, linear physical multipole is known as the pure, linear physical
electric octupole. We can construct / create it (as before) by:
- Starting with a pure, linear, physical electric quadrupole
- “Copying it”
- Charge-conjugating ( Q → − Q ) the charges associated with the “copied” electric quadrupole
- Translating the charge-conjugated electric quadrupole along the symmetry axis of the original
electric quadrupole, this time by an amount 2d:
z 2.
z
z 3.
z
z
copy
+ Q + Q → + Q + Q − Q
d d d
− 2 Q − 2 Q → − 2 Q − 2 Q + +2 Q
d d d
+ Q + Q → + Q + Q − Q
Original Original Copy Original Charge-Conjugated Copy
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17
- z ˆ z ˆ
+ Q − Q
Original → ← Charge-Conjugated Copy
− 2 Q +2 Q
= 0Q + Q − Q
− Q
+2 Q
− Q
Pure, Linear (Axially/Azimuthally-Symmetric) Physical Electric Octupole:
z ˆ
P (Observation / Field point)
+ Q
a
r
G
d
b
r
G
Following the methodology as used in previous cases:
2 d − 2 Q
d r
G
( ) ( ) ( )
1 3
2
4
3 4
1
5cos 3cos
cos
octupole i
i o
V r V r P
r
θ θ
=
= −
∑
I
G G
G
4 d d Ο
c
r
G
y ( ) ( )
5
octupole octupole
o
E r V r
r πε
I
G JK
G G
G
+2 Q
d
r
G
I
G
= Octupole Moment Qddd
GGG
∼ (Rank-3 tensor)
x d
3
Ο~ Qd
I
G
(SI units: coulomb-meter
3
− Q Note: Q TOT
In general, for l
th
-order electric multipole, A = 0, 1, 2, 3,... defining
th
M ≡
A
A -order multipole
moment (SI units: coulomb-(meters)
b
) then the potential associated with a pure, physical, linear
multipole moment is of the form:
( ) ( )
1
cos
o
M
V r P
r
θ
πε
A
A A A
G
The electric field intensity associated with a pure, physical, linear multipole moment is of the
form: ( ) ( )
2
o
M
E r V r
πε r
A
A A A
G JK
G G
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19
Carry out a (full) binomial expansion of 1/r (for r >> a ):
( )
1/ 2 2 3
0
n
n
r r n r
∞
−
=
∑
r
where:
( ) ( )
( )
1
2
n
n
n n n
is the binomial coefficient and ( )
Γ x is the gamma function.
and:
( )
( )
( )( ) ( ) ( )( ) ( )
1
2
1 1 1 1 1 3
2 2 2 2 2 2
n
n n
n
Then:
2 2 3 3
1 2 cos 2 cos 2 cos ...
r r r r r r
r r r r r r r
r
Collecting together like powers of r ′ r :
3
2 3 2 3
1 1 3cos 1 5cos 3cos
1 cos ...
r r r
r r r r
r
Thus we see that:
( ) ( ) ( ) ( )
2 3
0 1 2 3
cos cos cos cos ...
r r r
P P P P
r r r r
r
Hence: ( )
0
cos
r
P
r r
∞
=
∑
A
A
A
r
where
Θ = opening angle between r and r.
G G
This remarkable result occurs because
(where
2
2 cos
r r
r r
) is known as the
Generating Function for the Legendré Polynomials!!!
Then, since ( ) ( )
o v
V r r d
r
′
∫
G G
for r >> a ( a = max value of r
G
), the potential outside
the volume v
containing the charge distribution ( )
ρ r
G
is given by:
( ) ( ) ( )
( ) ( ) ( )
0
1
0
cos
cos
outside
o v
o v
r
V r r P d
r r
r r P d
r
ρ τ
πε
ρ τ
πε
∞
= ′
∞
= ′
∑
∫
∑ ∫
A
A
A
A
A A
A
G G
G
Then defining: ( ) ( ) ( ) ( )
1
cos
outside
o v
V r r r P d
r
′
∫
A
A A A
G G
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 20
We obtain (for r >> a ): ( ) ( ) ( ) ( ) ( )
1
0 0
cos
outside
outside
o v
V r V r r r P d
r
∞ ∞
= = ′
∑ ∑
∫
A
A A A
A A
G G G
Linear superposition of Θ′ = opening angle
multipole potentials!!! between r and r ′.
G G
This expression is known as the Multipole Expansion of ( )
outside
V r
G
in powers of 1/ r.
It is valid / useful when r >> a ( a = max value of r
G
). Note that this is an exact expression.
Having obtained ( )
outside
V r
G
, we can then obtain ( ) ( )
outside outside
E r = −∇ V r
G JK
G G
, and thus we see that:
( ) ( ) ( )
0 0
outside outside
outside
E r E r V r
∞ ∞
= =
∑ A ∑ A
A A
G JK
G G G
i.e. ( ) ( )
outside outside
E r = −∇ V r
A A
G JK
G G
Linear superposition of multipole electric fields!!!
Thus, we see that, for observation / field point distances far away from the (arbitrary) localized
electric charge distribution ( )
ρ r ′
G
(i.e. r >> a ( a = max value of r ′
G
)) the electrostatic potential
( )
outside
V r
G
and associated electric field ( ) ( )
outside outside
E r = −∇ V r
G JK
G G
are linear superpositions of
multipole electrostatic potentials ( )
outside
V r
A
G
and multipole electric fields ( )
outside
E r
A
G
G
respectively,
each arising from the
th
A electric multipole moment M
A
associated with the localized electric
charge distribution ( )
ρ r ′
G
Order of
Electric Multipole
Electrostatic Potential
( )
outside
V r
A
G
Electric Field
( ) ( )
outside outside
E r = −∇ V r
A A
G JK
G G
Electric Multipole
Moment M
A
A = 0
Monopole
P
0
o
Q
πε r
2
o
Q
πε r
M
0
= Q (total/net
charge, coulombs)
(scalar)
A = 1
Dipole
2
o
Qd
πε r
3
o
Qd
πε r
1
M = Qd = p
G
G
(coulomb-meters)
(vector)
A = 2
Quadrupole
2
3
o
Qd
πε r
2
4
o
Qd
πε r
2
M = 2 Qdd = Q
GG
I
(coulomb-meters
2
(rank-2 tensor)
A = 3
Octupole
3
4
o
Qd
πε r
3
5
o
Qd
πε r
3
M Qddd = Ο
GGG
I
G
(coulomb-meters
3
(rank-3 tensor)
A = 4
Sextupole
4
5
o
Qd
πε r
4
6
o
Qd
πε r
4
M Qdddd = S
GGGG I
I
(coulomb-meters
4
(rank-4 tensor)
............................................
th
A Order
Multipole 1
o
Qd
πε r
A
A
2
o
Qd
πε r
A
A
( )
M Q r = M
A
A
HJG
G
(coulomb-meters
b
(rank- A tensor)
