UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
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 ) and wants to know what the potential V ( r ) and / or the electric field
intensity E ( r ) are far from the localized charge distribution.
If the localized charge distribution has a net electric charge Qnet, then far away from this
localized charge distribution, the potential V ( r ) to a good approximation will behave very much
like that of a point charge,
V far ( r )
1 Qnet
4πε o r
and
E far ( r ) = −∇V far ( r )
−
1 Qnet
4πε o r 2
d , the characteristic size of the
when the field point – source charge separation distance, r
charge distribution.
However, as the “observer” moves in closer and closer to the localized charge distribution
ρ ( r ′ ) , he/she will discover that increasingly V ( r ) (and hence E ( r ) ) may deviate more and
more from pure point charge behavior, because ρ ( r ′ ) is an extended source charge distribution.
Furthermore, ρ ( r ′ ) may be such that Qnet ≡ 0 , but that does NOT necessarily imply that
V ( r ) = 0 (and E ( r ) =0)!
Example:
A pure, physical electric dipole is a spatially-extended, simple charge distribution where Qnet = 0
but V ( r ) ≠ 0 and E ( r ) = −∇V ( r ) ≠ 0 , as shown in the figure below:
+q
A pure physical electric dipole is
composed of two opposite electric
charges separated by a distance d:
d
θ
r+
P (field point)
r
r−
−q
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
1
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The Potential V ( r ) and Electric Field E ( r ) of a Pure Physical Electric Dipole
“Pure” → Qnet = 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.
ẑ′
Examples of BAD choices of coordinate systems:
a.)
q+
b.)
ẑ
ẑ
Ο′ θ ′
q+
ŷ′
rdipole
Ο
ŷ
ŷ
q−
ϕ
x̂
ϕ'
Ο
q−
x̂′
x̂
Dipole lying in x – y plane has
ϕ -dependence, but (at least it)
is centered at the origin.
Even more mathematically complicated!!
Origin is not conveniently chosen (arbitrary?)
Angle the dipole axis makes with respect to
ẑ & x̂ 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 ẑ axis to be along line separating q+ and q−.
Choose x-y plane to lie mid-way between q+ and q−:
P (field point)
ẑ
r+
n.b. This problem
now has no
ϕ -dependence
+q
θ
r−
d
Ο
x̂
2
r
−q
ŷ
Mathematical expressions obtained for
V ( r ) , E ( r ) = −∇V ( r ) 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!
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
Pure, Physical Electric Dipole:
ẑ
P (field point)
+q
r+
r+′
1
2
Source Charge Locations
r
r+′ = + 12 dzˆ, r+′ = 12 d
d
Ο θ
r−
r−′ = − 12 dzˆ, r−′ = 12 d
ŷ
π −θ
d
1
2
d
r−′
x̂
−q
Law of Cosines:
+q
q+
c2 = a2 + b2 – 2ab cos θ
c
π −θ
−q
P
P
b
a
Ο
Ο
a
−q
b
θ
2
c
2
⎛d ⎞
⎛d⎞
r = ⎜ ⎟ + r 2 − 2 ⎜ ⎟ r cos θ
⎝2⎠
⎝2⎠
⎛d ⎞
⎛d⎞
r = ⎜ ⎟ + r 2 − 2 ⎜ ⎟ r cos (π − θ )
⎝2⎠
⎝2⎠
2
+
2
−
2
2
⎛d ⎞
= ⎜ ⎟ + r 2 − dr cos θ
⎝2⎠
⎛d ⎞
= ⎜ ⎟ + r 2 + dr cos θ
⎝2⎠
2
2
⎛d ⎞
= r + ⎜ ⎟ − rd cos θ
⎝2⎠
⎛d ⎞
= r + ⎜ ⎟ + rd cos θ
⎝2⎠
2
2
Use Principle of Linear Superposition for Total Potential:
VTOT ( r ) = V+ q ( r ) + V− q ( r ) ≡ Vdipole ( r )
V+ q ( r ) =
+q
1
=
4πε o r+ 4πε o
V− q ( r ) =
1 −q
1
=
4πε o r− 4πε o
1
q
r 2 + ( d 2 ) − rd cos θ
2
−q
r 2 + ( d 2 ) + rd cos θ
2
=
+q
4πε o
=
−q
4πε o
1
r 2 + ( d 2 ) − rd cos θ
2
1
r 2 + ( d 2 ) + rd cos θ
2
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
3
UIUC Physics 435 EM Fields & Sources I
∴
Fall Semester, 2007
1
q
4πε o
r + ( d 2 ) − rd cos θ
2
Prof. Steven Errede
1 +q
1 q
−
4πε o r+
4πε o r−
Vdipole ( r ) = V+ q ( r ) + V− q ( r ) =
=
Lecture Notes 8
2
−
⎡
q ⎢
1
=
−
4πε o ⎢ r 2 + ( d 2 )2 − rd cos θ
⎣
1
q
4πε o
r + ( d 2 ) + rd cos θ
2
2
⎤
⎥
2
2
r + ( d 2 ) + rd cos θ ⎥⎦
1
This is an exact analytic mathematical expression for the potential associated with a pure
( Qnet = 0 ) 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, Vdipole ( r ) has no ϕ -dependence.
The exact analytic expression for potential associated with pure physical electric dipole:
⎧
⎫
q ⎪
1
1
⎪
−
Vdipole ( r ) =
⎨
⎬
2
2
4πε o ⎪ r 2 + ( d 2 )2 − rd cos θ
+
+
⎪⎭
r
d
2
rd
cos
θ
( )
⎩
As mentioned earlier, often we are / will be interested only in knowing (approximately)
Vdipole ( r ) when r
d . For example, many kinds of neutral molecules have permanent electric
dipole moments p ≡ qd (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. r = 1μ m = 10−6 m
away from such a molecular dipole, r = 1μ m
In such situations, when r
d ~ 5nm , since d r
0.005 !
d an approximate solution for Vdipole ( r ) 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
d that are of interest to us.
exact analytical expression over the range of distance scales r
Thus for r > d , the exact expressions for the r+ and r− separation distances are:
r+ = r 2 + ( d 2 ) − rd cos θ
2
2
⎛d ⎞ ⎛d ⎞
= r 1+ ⎜
r ⎟ − ⎜ ⎟ cos θ
⎝2 ⎠ ⎝r⎠
2
1⎛d ⎞ ⎛d ⎞
= r 1 + ⎜ ⎟ − ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
4
r− = r 2 + ( d 2 ) + rd cos θ
2
2
⎛d ⎞ ⎛d ⎞
= r 1+ ⎜
r ⎟ + ⎜ ⎟ cos θ
⎝2 ⎠ ⎝r⎠
2
1⎛d ⎞ ⎛d ⎞
= r 1 + ⎜ ⎟ + ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Now if ( d r )
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
1 , then let us define:
1⎛d ⎞
2
⎛d⎞
ε + ≡ ⎜ ⎟ − ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
and:
Then:
1
1
=
r+ r 1 + ε +
and:
with:
ε+
and:
1
1⎛d ⎞
2
⎛d⎞
ε − ≡ ⎜ ⎟ + ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
1
1
=
r− r 1 + ε −
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
1
1
1i3 2 1i3i5 3
= (1 + ε ± ) 2 = 1 − ε ± +
ε± −
ε ± + ... − ... (Valid on the interval: −1 ≤ ε ± ≤ +1 )
2
2i4
2i4i6
1+ ε±
Since ε ± is already <<1, then the higher-order terms ( ε ± ) , ( ε ± ) , ( ε ± ) ,... etc. are incredibly
small (<<<<<1), so negligible error is incurred by neglecting these higher-order terms,
1
i.e. keeping only terms linear in ε ± in the binomial expansion of
, we have:
1+ ε±
2
1
1
=
r+ r 1 + ε +
1
(1 − 12 ε + )
r
and:
3
1
1
=
r− r 1 + ε −
4
1
(1 − 12 ε − )
r
q ⎧1 1⎫
q ⎧1
1
⎫
1
1
⎨ − ⎬
⎨ (1 − 2 ε + ) − (1 − 2 ε − ) ⎬
4πε o ⎩ r+ r− ⎭ 4πε o ⎩ r
r
⎭
q ⎛1⎞ 1
q ⎛1⎞
1
1
=
⎜ ⎟ {( 2 ) ( ε − − ε + )}
⎜ ⎟ 1 − 2 ε+ − 1 + 2 ε− =
4πε o ⎝ r ⎠
4πε o ⎝ x ⎠
V dipole ( r ) =
Then:
{
}
2
1⎛d ⎞ ⎛d ⎞
ε + ≡ ⎜ ⎟ − ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
Now:
2
and:
1⎛d ⎞ ⎛d ⎞
ε − ≡ ⎜ ⎟ + ⎜ ⎟ cos θ
4⎝ r ⎠ ⎝ r ⎠
2
⎧
⎞ ⎤ ⎫⎪
⎞ ⎛ ⎛ 1 ⎞ ⎛ d ⎞2 ⎛ d ⎞
q ⎛ 1 ⎞ ⎪⎛ 1 ⎞ ⎡⎛ ⎛ 1 ⎞ ⎛ d ⎞ ⎛ d ⎞
⎢
⎜
⎜
⎟
Vdipole ( r ) =
⎜ ⎟ ⎨⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ cos θ ⎟ − ⎜ ⎜ ⎟ ⎜ ⎟ − ⎜ ⎟ cos θ ⎟⎟ ⎥ ⎬
4πε o ⎝ r ⎠ ⎪⎝ 2 ⎠ ⎢⎜ ⎝ 4 ⎠ ⎝ r ⎠ ⎝ r ⎠
4 r
r
⎠ ⎝ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎠ ⎦⎥ ⎭⎪
⎣⎝
⎩
Then:
Thus:
Vdipole ( r )
=
⎫
q ⎛ 1 ⎞ ⎛ 1 ⎞ ⎧⎛ d ⎞
⎛d⎞
⎜ ⎟ ⎜ ⎟ ⎨⎜ ⎟ cos θ + ⎜ ⎟ cos θ ⎬
4πε o ⎝ r ⎠ ⎝ 2 ⎠ ⎩⎝ r ⎠
⎝r⎠
⎭
=
⎫
q ⎛ 1 ⎞⎛ d ⎞
q ⎛ 1 ⎞⎛ 1 ⎞⎧ ⎛ d ⎞
⎜ ⎟ ⎜ ⎟ ⎨ 2 ⎜ ⎟ cos θ ⎬ =
⎜ ⎟ ⎜ ⎟ cos θ
4πε o ⎝ r ⎠ ⎝ 2 ⎠ ⎩ ⎝ r ⎠
⎭ 4πε o ⎝ r ⎠ ⎝ r ⎠
q ⎛ 1 ⎞⎛ d ⎞
q ⎛d ⎞
qd ⎛ 1 ⎞
⎜ ⎟ ⎜ ⎟ cos θ =
⎜ 2 ⎟ cos θ =
⎜ ⎟ cos θ
4πε o ⎝ r ⎠ ⎝ r ⎠
4πε o ⎝ r ⎠
4πε o ⎝ r 2 ⎠
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
5
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
p ≡ qd = p
The Magnitude of the Electric Dipole Moment:
Thus, we may also express the potential of a pure physical dipole as:
qd ⎛ 1 ⎞
p ⎛1⎞
Vdipole ( r ) =
(valid for d << r)
⎜ 2 ⎟ cos θ =
⎜ ⎟ cos θ
4πε o ⎝ r ⎠
4πε o ⎝ r 2 ⎠
Note that: Vdipole ( r ) ∼
1
1
whereas Vmonopole ( r ) ∼ (valid for point charge q located at origin)
2
r
r
We define the vector electric dipole moment as: p ≡ qd where the charge-separation distance
vector d points (by convention) from –q to +q:
+q
p ≡ qd
SI Units of p = Coulomb-meters
d d
−q
In our current situation here we see that d = dzˆ :
ẑ
P (field point)
+q
θ
r
d d
p ≡ qd Ο
ŷ
−q
x̂
Thus here if: p = qd = qdzˆ
Then: Vdipole ( r )
but: zˆ = cos θ rˆ then: p = qd = qdzˆ = qd cos θ rˆ = p cos θ rˆ
qd ⎛ 1 ⎞
qd cos θ ⎛ 1 ⎞ p cos θ ⎛ 1 ⎞
⎜ 2 ⎟ cos θ =
⎜ ⎟=
⎜ ⎟
4πε o ⎝ r ⎠
4πε o ⎝ r 2 ⎠
4πε o ⎝ r 2 ⎠
(
)
The potential Vdipole ( r ) associated with an electric dipole moment p p = qd = qdzˆ from a pure,
physical electric dipole oriented with d = dzˆ , for r
Vdipole ( r )
6
p cos θ ⎛ 1 ⎞ p irˆ ⎛ 1 ⎞
⎜ ⎟=
⎜ ⎟ where:
4πε o ⎝ r 2 ⎠ 4πε o ⎝ r 2 ⎠
d is thus given by:
p irˆ = p cos θ = qd cos θ
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The electric field Edipole ( r ) associated with a pure, physical electric dipole,
with electric dipole moment p = qd = qdzˆ is:
Edipole ( r ) = −∇Vdipole ( r ) = Erdipole ( r ) rˆ + Eθdipole ( r ) θˆ + Eϕdipole ( r ) ϕˆ in spherical-polar coordinates.
The components of Edipole ( r ) in spherical-polar coordinates are:
Erdipole ( r ) = −
Eθdipole ( r ) = −
Eϕdipole ( r ) = −
∂Vdipole ( r )
∂r
=
1 2p
cos θ
4πε o r 3
1 ∂Vdipole ( r )
1 p
=
sin θ
∂θ
r
4πε o r 3
1 ∂Vdipole ( r )
=0
r sin θ
∂ϕ
Explicitly, the electric field intensity of a pure, physical electric dipole with electric dipole
moment p = qd = qdzˆ (in spherical-polar coordinates) is:
Edipole ( r ) =
1 2p
1 p
1 p⎡
cos θ rˆ +
sin θθˆ =
2 cos θ rˆ + sin θθˆ ⎤⎦
3
3
4πε o r
4πε o r
4πε o r 3 ⎣
1
1
(c.f. w/ Emonopole ( r ) ∼ 2 for single point charge q at r = 0 ).
3
r
r
Note also that Vdipole ( r ) and Edipole ( r ) have no explicit ϕ -dependence, since the charge
Note that: Edipole ( r ) ∼
configuration for an electric dipole is manifestly axially / azimuthally symmetric
(i.e. charge configuration for electric dipole is invariant under arbitrary ϕ -rotations).
Now: Vdipole ( r ) =
p irˆ ⎛ 1 ⎞
⎜ ⎟ with electric dipole moment p = qdzˆ, and pirˆ = p cos θ = qd cos θ ,
4πε o ⎝ r 2 ⎠
(since zˆ irˆ = cos θ ), and r 2 = x 2 + y 2 + z 2 in Cartesian/rectangular coordinates.
In Cartesian/rectangular coordinates the electric field intensity of a pure, physical electric dipole
with electric dipole moment p = qd = qdzˆ (in spherical-polar coordinates) is:
⎛ ∂
∂
∂
E dipole ( r ) = −∇Vdipole ( r ) = − ⎜ xˆ +
yˆ +
∂y
∂z
⎝ ∂x
⎞
zˆ ⎟ Vdipole ( r ) = Exdipole xˆ + E ydipole yˆ + Ezdipole zˆ
⎠
Transformation from Spherical-Polar → Cartesian Coordinates:
y = r sin θ sin ϕ
xˆ = sin θ cos ϕ rˆ + cos θ cos ϕθˆ − sin ϕϕˆ
yˆ = sin θ sin ϕ rˆ + cos θ sin ϕθˆ + sin ϕϕˆ
z = r cos θ
zˆ = cos θ rˆ − sin θθˆ
x = r sin θ cos ϕ
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
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 = qdzˆ (in Cartesian coordinates) are:
p ⎛ 3xz ⎞
p ⎛ 3sin θ cos θ ⎞
⎜ 5 ⎟=
⎜
⎟
4πε o ⎝ r ⎠ 4πε o ⎝
r3
⎠
p ⎛ 3 yz ⎞
p ⎛ 3sin θ cos θ ⎞
dipole
=
⇐
⎜ 5 ⎟=
⎜
⎟ = Ex
3
4πε o ⎝ r ⎠ 4πε o ⎝
r
⎠
Exdipole =
E ydipole
Ezdipole =
(since charge configuration
of electric dipole is axially /
azimuthally symmetric)
p ⎛ 3z 2 − r 2 ⎞
p ⎛ 3cos 2 θ − 1 ⎞
=
⎜
⎟
⎜
⎟
4πε o ⎝ r 5 ⎠ 4πε o ⎝
r3
⎠
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 = qdzˆ is of
the form:
1 ⎛1⎞
physical
Edipole
(r ) =
⎜ ⎟ ⎡3 ( p irˆ ) rˆ − p ⎤⎦
4πε o ⎝ r 3 ⎠ ⎣
whereas the coordinate-free form of a point electric dipole is of the form:
point
Edipole
(r ) =
1 ⎛1⎞
1
p δ 3 (r )
⎜ 3 ⎟ ⎡⎣3 ( p irˆ ) rˆ − p ⎤⎦ −
4πε o ⎝ r ⎠
3ε o
E − Field Lines & Equipotentials Associated with a Pure, Physical Electric Dipole:
n.b. Equipotentials
are ⊥ to lines of
E ( r ) everywhere!
8
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
We explicitly show here that the electric field associated with a pure physical electric dipole
with electric dipole moment p = pzˆ = qdzˆ can be written in coordinate-free form as:
1 ⎛1⎞
physical
Edipole
(r ) =
⎜ ⎟ ⎡3 ( p irˆ ) rˆ − p ⎤⎦
4πε o ⎝ r 3 ⎠ ⎣
We have already shown (above) that:
1 ⎛ p ⎞⎡
physical
ˆ
Edipole
(r ) =
⎜ 3 ⎟ ⎣ 2 cos θ rˆ + sin θθ ⎤⎦
4πε o ⎝ r ⎠
Now: p = pzˆ and zˆ = cos θ rˆ − sin θθˆ (in spherical-polar coordinates)
Thus: pirˆ = ppˆ irˆ = pzˆ irˆ
(
ẑ
)
r
But: zˆ irˆ = cos θ rˆ − sin θθˆ irˆ = cos θ
ϕ̂
And: rˆirˆ = 1 , θˆirˆ = 0
Thus: p irˆ = p cos θ
( )
θ
θˆ
= ( p i rˆ ) rˆ
And: p = ( pirˆ ) rˆ + piθˆ θˆ = p cos θ rˆ − p sin θθˆ
O
ϕ
So therefore:
⎡⎣3 ( p irˆ ) rˆ − p ⎤⎦ = 3 p cos θ rˆ − p cos θ rˆ + p sin θθˆ
= 2 p cos θ rˆ + p sin θθˆ
ŷ
ϕ̂
x̂
= p ⎡⎣ 2 cos θ rˆ + sin θθˆ ⎤⎦
1 ⎛1⎞
physical
Thus: Edipole
(r ) =
⎜ ⎟ ⎡3 ( p irˆ ) rˆ − p ⎤⎦ Q.E.D.
4πε o ⎝ r 3 ⎠ ⎣
The Potential Vquad ( r ) and Electric Field Equad ( r ) Associated with a
Pure, Linear Physical Electric Quadrupole
We have seen that a pure, physical electric dipole was constructed by:
1. Starting with a monopole electric moment (i.e. charge +Q)
2. “Copying it”
3. Charge-conjugating (+Q → −Q) the “copied” charge
4. Displacing the conjugated charge –Q from the original charge +Q by a separation distance d
Likewise, we can construct a pure, physical, linear electric quadrupole by:
1. Starting with a pure, physical, linear electric dipole
2. “Copying it”
3. Charge-conjugating the charges associated with the “copied” electric dipole
4. Translating the charge conjugated electric dipole along the symmetry axis of the original
electric dipole by amount d, as shown in the figures below:
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
9
UIUC Physics 435 EM Fields & Sources I
1.
2.
ẑ
3.
ẑ
copy
→ +Q
+Q
+Q
Fall Semester, 2007
d
Lecture Notes 8
ẑ
+Q
d
−Q
−Q
Original
Original
4.
Prof. Steven Errede
−Q
d
−Q
Copy
−Q
+Q
Original
Charge-Conjugated Copy
ẑ
−Q
+Q
d
+Q
= −2Q −Q
−Q
Translation of charge-conjugated
copy along axis of original dipole
by amount d.
d
+Q
Pure, Physical, Linear Electric Quadrupole:
P (Field Point)
ẑ
Note that this linear electric quadrupole has
axial / aximuthal symmetry – i.e. because
all charges (+Q, −2Q, +Q) are co-linear
(all on ẑ axis), problem is invariant under
(arbitrary) ϕ -rotations.
ra
+Q
r
d
θ
Ο
−2Q
ŷ
π −θ
d
x̂
⇒ Vquad ( r ) and Equad ( r ) will have no
explicit ϕ -dependence for the linear
electric quadrupole.
rb
n.b. QTOT = 0 for pure electric quadrupole.
+Q
Again, we use the principle of (linear) superposition to obtain Vquad ( r ) :
Vquad ( r ) = VTOT ( r ) = V+ Q ( @ z = + d ) + V−2Q ( @ z = 0 ) + V+ Q ( @ z = − d )
=
10
⎛ r ⎞⎤
1 ⎛ Q 2Q Q ⎞
1 ⎛ Q ⎞ ⎡⎛ r ⎞
+ ⎟=
⎜ −
⎜ ⎟ ⎢⎜ ⎟ − 2 + ⎜ ⎟ ⎥
4πε o ⎝ ra
r
rb ⎠ 4πε o ⎝ r ⎠ ⎣⎝ ra ⎠
⎝ rb ⎠ ⎦
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
ra2 = r 2 + d 2 − 2rd cos θ
Again, using the Law of Cosines:
Lecture Notes 8
and
Prof. Steven Errede
rb2 = r 2 + d 2 + rd cos θ
We obtain:
Vquad ( r ) =
⎫
1 ⎛ Q ⎞⎧
r
r
−2+
⎬ ⇐
⎜ ⎟⎨ 2
4πε o ⎝ r ⎠ ⎩ r + d 2 − 2rd cos θ
r 2 + d 2 + 2rd cos θ ⎭
Exact analytic
expression
Again, for regime where the observation point P is far away from pure, physical, linear electric
⎛r⎞
⎛r⎞
quadrupole, i.e. r >> d, we expand ⎜ ⎟ and ⎜ ⎟ in a binomial (i.e. Taylor) series
⎝ ra ⎠
⎝ rb ⎠
(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. > ( d r ) ) we obtain:
2
2
2
⎛r⎞
⎛d ⎞
⎛ d ⎞ ( 3cos θ − 1)
θ
1
cos
−
+
⎜ ⎟
⎜ ⎟
⎜ ⎟
2
⎝r⎠
⎝r⎠
⎝ ra ⎠
2
2
⎛r⎞
⎛d ⎞
⎛ d ⎞ ( 3cos θ − 1)
1
cos
θ
+
+
⎜ ⎟
⎜ ⎟
⎜ ⎟
2
⎝r⎠
⎝r⎠
⎝ rb ⎠
x = cosθ
Recall that the Ordinary Legendré Polynomials P
P0 ( x ) = 1
→ P0 ( cos θ ) = 1
P1 ( x ) = x
→ P1 ( cos θ ) = cos θ
P2 ( x ) =
( 3x
2
− 1)
2
→ P2 ( cos θ ) =
( x)
are:
Shorthand notation:
P ( cos θ ) = P (θ )
( 3cos θ − 1)
2
2
2
⎛r⎞
∴ ⎜ ⎟
⎝ ra ⎠
2
⎛r⎞
⎛d ⎞
⎛d ⎞
⎛d ⎞
⎛d⎞
P0 (θ ) − ⎜ ⎟ P1 (θ ) + ⎜ ⎟ P2 (θ ) and ⎜ ⎟ P0 (θ ) + ⎜ ⎟ P1 (θ ) + ⎜ ⎟ P2 (θ )
⎝r⎠
⎝r⎠
⎝r⎠
⎝r⎠
⎝ rb ⎠
2
2
⎛ r ⎞⎤
1 ⎛ Q ⎞ ⎡⎛ r ⎞
1 ⎛ Q ⎞ ⎡ ⎛ d ⎞ ( 3cos θ − 1) ⎤
⎢
⎥
∴ Vquad ( r ) =
−
+
=
2
2
⎜ ⎟⎥
⎜ ⎟ ⎢⎜ ⎟
⎜ ⎟ ⎜ ⎟
4πε o ⎝ r ⎠ ⎣⎝ ra ⎠
4
πε
2
r
r
r
⎝
⎠
⎝
⎠
⎢
⎥⎦
o
⎝ b ⎠⎦
⎣
2Qd 2 ⎛ 1 ⎞ ⎛ 3cos 2 θ − 1 ⎞
=
⎟
⎜ ⎟⎜
4πε o ⎝ r 3 ⎠ ⎝
2
⎠
Then for r >> d:
P2 (θ )
Vquad ( r )
Note that:
2Qd 2 ⎛ 1 ⎞ ⎛ 3cos 2 θ − 1 ⎞ 2Qd 2 ⎛ 1 ⎞
⎟=
⎜ ⎟⎜
⎜ 3 ⎟ P2 (θ )
4πε o ⎝ r 3 ⎠ ⎝
2
⎠ 4πε o ⎝ r ⎠
Vquad ( r ) ∼
1
r3
(c.f. with Vmonopole ( r ) ∼ 1 r and Vdipole ( r ) ∼ 1 2 )
r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
11
UIUC Physics 435 EM Fields & Sources I
Note also that:
Vquad ( r ) ∼ P2 (θ )
1
2
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
(c.f. with Vmonopole ( r ) ∼ P0 (θ ) , Vdipole ( r ) ∼ P1 (θ ) )
(3cos θ −1)
= cosθ
=1
2
Note further that: Vquad ( r ) must be proportional to an even power of l, i.e. Pl =even (θ ) because a
pure, physical, linear electric quadrupole has reflection symmetry about the ẑ -axis (i.e. about
θ = π / 2 ) (i.e. a rotation from / by a vector lying in x – y plane e.g. x̂ or ŷ axis).
ẑ
θ → (π − θ )
ẑ
+Qa
−2Q
( 3cos θ − 1)
+Qb
P2 (θ ) =
−2Q
is an even function under
θ → (π − θ ) reflection:
1
2
2
P2 (π − θ ) = + P2 (θ )
+Qb
+Qa
We can also see that Vdipole ( r ) must be proportional to an odd power of l, i.e. Pl =odd (θ ) because a
pure, physical, linear electric dipole has a sign change under reflection symmetry about θ = π / 2
ẑ
θ → (π − θ )
+Q
p = Qdzˆ
ẑ
P1 (θ ) = cos θ
−Q
is an odd function under
θ → (π − θ ) reflection:
p = −Qdzˆ
P1 (π − θ ) = − P1 (θ )
=0
cos (π − θ ) = cos π cos θ + sin θ sin π
−Q
= − cos θ
+Q
Likewise, Vmonopole ( r ) must be proportional to an even power of l:
ẑ
P0 (θ ) = P0 (π − θ ) = 1
12
+Q
θ → (π − θ )
ẑ
+Q
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
As we have seen for the two previous cases, that of:
1. 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)
2. The electric dipole with its accompanying electric dipole moment p ≡ Qd , p = p = Qd
(n.b. p is a vector quantity)
(SI units of p : Coulomb-meters)
3. The electric quadrupole also has an accompanying electric quadrupole moment Q ≡ 2Qdd
(SI units of Q : Coulomb-meters2)
(n.b. Q is a tensor quantity)
Tensor Q ≡ 2Qdd = “double vector”
Q ≡ 2Qdd = 2Qd 2
2-dimensional matrix
Formally speaking, Q is a rank-2 tensor (i.e. a 2-dimensional matrix) - the 9 elements of the
Q tensor (in general) are:
Qxx
Qyz
Qzx
Q = Qxy
Qxz
Qyy
Qyz
Qzy
Qzz
n.b. Q has only six independent components, because Qij = Qji
i.e.
Qxy = Qyx
Qxz = Qzx
Qyz = Qzy
Also, note that: Qxx + Qyy + Qzz = 0 or: Qzz = −(Qxx+Qyy) {i.e. Q is traceless}
The quadrupole moment tensor can also be written in coordinate-free form, e.g. in Cartesian
coordinates as:
n = # discrete charges qi
Q≡
∑ ( 3r r − 1r ) q
n
1
2
i =1
2
i i
i
i
with ri 2 = ri iri
ˆˆ
xx
Unit Dyadic: 1 ≡ 0
0
0
ˆˆ
yy
0
0
0
ˆˆ
zz
For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with
quadrupole moment Q (e.g. oriented along the ẑ -axis):
ẑ
Here, Qxx = Qyy, and since: Qxx + Qyy + Qzz = 0
+Q
d
−2Q
d
+Q
Then: Qzz = −2Qxx = −2Qyy ≡ 2Qd 2 All other Qij vanish (= 0) for i ≠ j
−1 0 0
n.b. conventions / definitions of
linear
2
i.e. Qquad = Qd 0 −1 0
linear
Qquad
differ in different textbooks!!!
0 0 +2
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
13
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with
quadrupole moment Q (oriented along the ẑ -axis), expressed in Cartesian coordinates:
P (Field Point)
ẑ
Q≡
ra
+Q
# discrete charges
∑ ( 3r r − 1r ) q
n
1
2
i =1
2
i i
i
with ri 2 = ri iri
Unit Dyadic:
ˆˆ 0 0
xx
r
d
i
rb
−2Q
d
+Q
ˆˆ
1 ≡ 0 yy
0 0
ŷ
x̂
i = 1: r1 = + dzˆ
q1 = + Q ri = ri
i = 2 : r2 = 0 zˆ
i = 3 : r3 = − dzˆ
q2 = −2Q
q3 = + Q
0
ˆˆ
zz
=0
⎞ 1
1
2 ⎛ =0
ˆˆ − d 2 1 − Q ⎜ 3i 0 zz
ˆˆ − 0i1 ⎟ + Q 3d 2 zz
ˆˆ − d 2 1 = Qd 2 3zz
ˆˆ − 1
Thus: Q = Q 3d 2 zz
⎜
⎟
2
2 ⎝
2
⎠
(
)
for charge 1:
+ Q @ r1 =+ dzˆ
(
for charge 2:
−2Q @ r2 = 0 zˆ
)
)
for charge 3:
+ Q @ r3 =− dzˆ
⎛ 3 zz
ˆˆ − 1 ⎞
ˆˆ − 1 = 2Qd 2 ⎜
∴ Q = Qd 2 3 zz
⎟
⎝ 2 ⎠
2
2Qd 2 ⎛ 1 ⎞ ( 3cos θ − 1) 2Qd 2 ⎛ 1 ⎞
=
Then: Vquad ( r )
⎜ ⎟
⎜ ⎟ P2 ( cos θ )
4πε o ⎝ r 3 ⎠
2
4πε o ⎝ r 3 ⎠
(
(
)
P2 ( cos θ ) =
1
( 3cos2 θ − 1)
2
We can express Vquad ( r ) in a different (but totally equivalent manner), using the fact(s) that:
14
rˆ = sin θ cos ϕ xˆ + sin θ sin ϕ yˆ + cos θ zˆ
zˆ irˆ = rˆi zˆ = cos θ
3 ( rˆi zˆ )( zˆ irˆ ) = 3cos 2 θ
xˆ i xˆ = 1, xˆ i yˆ = 0, xˆ i zˆ = 0
yˆ i xˆ = 0, yˆ i yˆ = 1, yˆ i zˆ = 0
rˆi1irˆ = 1
zˆ i x = 0, zˆ i yˆ = 0, zˆ i zˆ = 1
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
Then for observation/field point P far from quadrupole, i.e. r >> d:
coordinate-free form
(
)
ˆˆ − 1 irˆ ⎤
1 ⎛1⎞
2Qd 2 ⎛ 1 ⎞ ⎡ rˆi 3zz
⎢
⎥
ˆ
ˆ
Vquad ( r ) ≈
⎜ ⎟ r iQir =
⎜ ⎟
4πε o ⎝ r 3 ⎠
4πε o ⎝ r 3 ⎠ ⎢
2
⎥
⎣
⎦
(
=
)
2Qd 2 ⎛ 1 ⎞ ⎡ 3 ( rˆi zˆ )( zˆ irˆ ) − rˆi1irˆ ⎤ 2Qd 2 ⎛ 1 ⎞ ⎡ 3cos 2 θ − 1 ⎤
⎥=
⎜ ⎟⎢
⎜ 3 ⎟⎢
⎥
4πε o ⎝ r 3 ⎠ ⎣⎢
2
2
⎦
⎦⎥ 4πε o ⎝ r ⎠ ⎣
≡ P2 ( cosθ )
=
2Qd 2 ⎛ 1 ⎞
⎜ ⎟ P2 ( cos θ )
4πε o ⎝ r 3 ⎠
axially-symmetric
Vquad ( r ) as given above is valid for a pure, linear, physical electric quadrupole oriented along
the ẑ -axis, for r (observation / field point) >> d.
The potential Vquad ( r ) and electric field intensity Equad ( r ) associated with a pure, physical,
linear electric quadrupole with quadrupole moment Q (oriented along the ẑ -axis) are:
Vquad ( r ) =
2Qd 2 ⎛ 1 ⎞ ⎡ 3cos 2 θ − 1 ⎤
⎜ ⎟⎢
⎥
4πε o ⎝ r 3 ⎠ ⎣
2
⎦
Equad ( r ) = Er rˆ + Eθ θˆ + Eϕϕˆ = −∇Vquad ( r ) , in spherical-polar coordinates:
Er ( r ) = −
Eθ ( r ) = −
Eϕ ( r ) = −
∂V ( r )
∂r
=
3i2Qd 2 ⎛ 1 ⎞ ⎡ 3cos 2 θ − 1 ⎤ 3i2Qd 2 ⎛ 1 ⎞
⎜ ⎟⎢
⎜ ⎟ P2 ( cos θ )
⎥=
4πε o ⎝ r 4 ⎠ ⎣
2
4πε o ⎝ r 4 ⎠
⎦
1 ∂V ( r ) 3i2Qd 2 ⎛ 1 ⎞
=
⎜ ⎟ sin θ cos θ
4πε o ⎝ r 4 ⎠
r ∂θ
1 ∂V ( r )
= 0 ← No ϕ -dependence because charge configuration is manifestly
r sin θ ∂ϕ
axially / azimuthally symmetric (invariant under arbitrary ϕ -rotations)
Explicitly writing out the form of the electric field intensity Equad ( r ) for a pure, linear, physical
electric quadrupole oriented along the ẑ -axis, for r (observation / field point) >> d:
Equad ( r ) =
3i2Qd 2 ⎛ 1 ⎞ ⎡ 3cos 2 θ − 1 ⎤
3i2Qd 2 ⎛ 1 ⎞
ˆ
ˆ
+
r
⎜ ⎟
⎜ 4 ⎟ sin θ cos θθ
⎥
r
πε
4πε o ⎝ r 4 ⎠ ⎢⎣
2
4
⎠
⎦
o ⎝
3i2Qd 2 ⎛ 1 ⎞ ⎡⎛ 3cos 2 θ − 1 ⎞
ˆ⎤
=
⎟ rˆ + sin θ cos θθ ⎥
⎜ 4 ⎟ ⎢⎜
4πε o ⎝ r ⎠ ⎣⎝
2
⎠
⎦
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
15
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
E -field lines & equipotentials associated with a pure, physical, linear electric quadrupole:
n.b. E -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:
1.
2.
3.
4.
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:
1.
2.
ẑ
+Q
ẑ
+Q
d
−2Q
−2Q
16
Original
ẑ
+Q
−Q
d
→
−2Q
−2Q
d
+Q
Original
+Q
ẑ
d
d
+Q
3.
ẑ
copy
→
+
+2Q
d
→
+Q
Copy
+Q
Original
−Q
Charge-Conjugated Copy
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
4.
ẑ
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
ẑ
+Q
Original →
−2Q
−Q
← Charge-Conjugated Copy
+2Q
= 0Q +Q
−Q
−Q
+2Q
−Q
Pure, Linear (Axially/Azimuthally-Symmetric) Physical Electric Octupole:
ẑ
P (Observation / Field point)
+Q
ra
d
2d
Following the methodology as used in previous cases:
rb
−2Q
d
4
1
⎛1⎞
Voctupole ( r ) = ∑ Vi ( r ) ∼ ⎜ 4 ⎟ P3 ( cos θ ) ∗
∗Ο
4πε o
⎝r ⎠ 1 3
i =1
r
(
= 2 5cos θ − 3cosθ
4d
d
Ο
+2Q
x̂
ŷ
rc
rd
)
⎛Ο⎞
1
Eoctupole ( r ) = −∇Voctupole ( r ) ∼ ⎜ 5 ⎟ ∗
⎝ r ⎠ 4πε o
Ο = Octupole Moment ∼ Qddd (Rank-3 tensor)
Ο ~ Qd 3 (SI units: coulomb-meter3)
d
−Q
In general, for lth-order electric multipole,
Note: QTOT = 0
= 0, 1, 2, 3, . . . defining M ≡
th
-order multipole
moment (SI units: coulomb-(meters)b) then the potential associated with a pure, physical, linear
multipole moment is of the form:
M ⎛ 1 ⎞
V (r ) ∼
⎜
⎟ P ( cos θ )
4πε o ⎝ r +1 ⎠
The electric field intensity associated with a pure, physical, linear multipole moment is of the
1 M
E ( r ) = −∇V ( r ) ∼
form:
4πε o r + 2
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
17
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
Multipole Moments, Potential and Electric Field Associated with an
Arbitrary Localized Electric Charge Distribution ρ ( r ′ ) - Outside of ρ ( r ′ )
Suppose we have an arbitrary, but localized electric charge distribution ρ ( r ′ ) somewhere in
space, contained within the volume v′ and bounded by the surface S ′ :
r = r − r′
r = r = r − r′
cos Θ′ = rˆirˆ′ = cosine of opening angle between vectors r and r ′.
Θ′ = opening angle between vectors r and r ′ - very important!
r 2 = r 2 + r ′2 − 2rr ′ cos Θ′ = r 2 + r ′2 − 2r ir ′
Law of Cosines:
If the observation / field point P is far away from electric charge distribution ρ ( r ′ ) such that:
r= r
a = a = maximum distance of ρ ( r ′ ) to origin ϑ then for r >> a (a = max value of r ′ ):
⎡ ⎛ r ′ ⎞2
⎤
⎛ r′ ⎞
r 2 = r 2 ⎢1 + ⎜ ⎟ − 2 ⎜ ⎟ cos Θ′⎥
⎝r⎠
⎢⎣ ⎝ r ⎠
⎥⎦
2
or:
⎛ r′ ⎞
⎛ r′ ⎞
r = r 1 + ⎜ ⎟ − 2 ⎜ ⎟ cos Θ′
⎝r⎠
⎝r⎠
≡ε 1 for r
a
2
⎛ r′ ⎞
⎛ r′ ⎞
Define: ε ≡ ⎜ ⎟ − 2 ⎜ ⎟ cos Θ′ for r >> a (a = max value of r ′ )
⎝r⎠
⎝r⎠
Now: V ( r ) =
18
1
4πε o
∫
v′
ρ ( r′)
r
dτ ′ with:
1 1
−1 2
= (1 + ε )
r r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
Carry out a (full) binomial expansion of 1/r (for r >> a):
1 1
1 ∞ ⎛ −1 2 ⎞ n 1 ⎛ 1
3 2 5 3
−1/ 2
⎞
= (1 + ε )
= ∑⎜
⎟ ε = ⎜1 − ε + ε − ε + ... ⎟
n
r r
8
16
r n=0 ⎝
r⎝ 2
⎠
⎠
⎛ −1 2 ⎞ ( −1) Γ ( n − 12 )
is the binomial coefficient and Γ ( x ) is the gamma function.
where: ⎜
⎟=
n!
Γ ( n)
⎝ n ⎠
Γ ( n − 12 )
and:
= ( − 12 )( − 12 + 1) .... ( − 12 + n − 1) = ( − 12 )( 12 ) .... ( n − 23 )
Γ (n)
n
Then:
2
2
3
3
⎤
1 1 ⎡ 1 ⎛ r′ ⎞ ⎛ r′
5 ⎛ r′ ⎞ ⎛ r′
⎞ 3 ⎛ r′ ⎞ ⎛ r′
⎞
⎞
= ⎢1 − ⎜ ⎟ ⎜ − 2 cos Θ′ ⎟ + ⎜ ⎟ ⎜ − 2 cos Θ′ ⎟ − ⎜ ⎟ ⎜ − 2 cos Θ′ ⎟ + ...⎥
r r ⎢⎣ 2 ⎝ r ⎠ ⎝ r
⎠ 8⎝ r ⎠ ⎝ r
⎠ 16 ⎝ r ⎠ ⎝ r
⎠
⎥⎦
Collecting together like powers of r ′ r :
3
2
3
2
3
⎤
1 1 ⎡ ⎛ r′ ⎞
⎛ r ′ ⎞ ⎛ 3cos Θ′ − 1 ⎞ ⎛ r ′ ⎞ ⎛ 5cos Θ′ − 3cos Θ′ ⎞
′
= ⎢1 + ⎜ ⎟ cos Θ + ⎜ ⎟ ⎜
⎟+⎜ ⎟ ⎜
⎟ + ...⎥
r r⎢ ⎝r⎠
2
2
⎝r⎠ ⎝
⎥⎦
⎠ ⎝r⎠ ⎝
⎠
⎣
Thus we see that:
2
3
⎤
1 1⎡
r′ ⎞
r′ ⎞
r′ ⎞
⎛
⎛
⎛
= ⎢ P0 ( cos Θ′ ) + ⎜ ⎟ P1 ( cos Θ′ ) + ⎜ ⎟ P2 ( cos Θ′ ) + ⎜ ⎟ P3 ( cos Θ′ ) + ...⎥ !!!!
r r ⎢⎣
⎝r⎠
⎝r⎠
⎝r⎠
⎥⎦
Hence:
1 1 ∞ ⎛ r′ ⎞
= ∑ ⎜ ⎟ P ( cos Θ′ ) where Θ′ = opening angle between r and r ′.
r r =0 ⎝ r ⎠
2
1
⎛ r′ ⎞
⎛ r′ ⎞
(where ε ≡ ⎜ ⎟ − 2 ⎜ ⎟ cos Θ′ ) is known as the
1+ ε
⎝r⎠
⎝r⎠
Generating Function for the Legendré Polynomials!!!
This remarkable result occurs because
Then, since V ( r ) =
⎛1⎞
1
4πε o
∫ ρ ( r ′) ⎜⎝ r ⎟⎠ dτ ′ for r >> a
(a = max value of r ′ ), the potential outside
v′
the volume v′ containing the charge distribution ρ ( r ′ ) is given by:
⎛ 1 ⎞ ∞ ⎛ r′ ⎞
⎜ ⎟ ∑ ⎜ ⎟ ρ ( r ′ )P ( cos Θ′ ) dτ ′
4πε o v∫′ ⎝ r ⎠ =0 ⎝ r ⎠
1
Voutside ( r ) =
=
1
4πε o
Then defining: V outside ( r ) =
∞
⎛ 1 ⎞ ′
r ρ ( r ′ ) P ( cos Θ′ ) dτ ′
+1 ⎟ ∫ ( )
⎠ v′
=0
∑ ⎜⎝ r
1 ⎛ 1 ⎞
⎜
⎟ ( r ′ ) ρ ( r ′ ) P ( cos Θ′ ) dτ ′
4πε o ⎝ r +1 ⎠ ∫v′
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
19
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
∞
Prof. Steven Errede
∞
⎛ 1 ⎞
r ′ ρ ( r ′ ) P ( cos Θ′ ) dτ ′
+1 ⎟ ∫ ( )
4πε o =0
⎠ v′
Θ′ = opening angle
Linear superposition of
multipole potentials!!!
between r and r ′.
We obtain (for r >> a): Voutside ( r ) = ∑ V outside ( r ) =
=0
Lecture Notes 8
1
∑ ⎜⎝ r
This expression is known as the Multipole Expansion of Voutside ( r ) in powers of 1/r.
It is valid / useful when r >> a (a = max value of r ′ ). Note that this is an exact expression.
Having obtained Voutside ( r ) , we can then obtain Eoutside ( r ) = −∇Voutside ( r ) , and thus we see that:
∞
∞
Eoutside ( r ) = ∑ E outside ( r ) = −∑ ∇V outside ( r )
=0
i.e.
E outside ( r ) = −∇V outside ( r )
=0
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 ′ ) (i.e. r >> a (a = max value of r ′ )) the electrostatic potential
Voutside ( r ) and associated electric field Eoutside ( r ) = −∇Voutside ( r ) are linear superpositions of
multipole electrostatic potentials V outside ( r ) and multipole electric fields E outside ( r ) respectively,
each arising from the
th
electric multipole moment M associated with the localized electric
charge distribution ρ ( r ′ ) !!!
Order of
Electrostatic Potential
Electric Multipole
V outside ( r )
E
=0
Monopole
=
1 ⎛Q⎞
⎜ ⎟
4πε o ⎝ r 2 ⎠
M0 = Q (total/net
charge, coulombs)
(scalar)
M 1 = Qd = p
(coulomb-meters)
(vector)
M 2 = 2Qdd = Q
(coulomb-meters2)
(rank-2 tensor)
M 3 ∼ Qddd = Ο
(coulomb-meters3)
(rank-3 tensor)
M 4 ∼ Qdddd = S
(coulomb-meters4)
(rank-4 tensor)
P0 = 1
1 ⎛Q⎞
=
⎜ ⎟
4πε o ⎝ r ⎠
Electric Field
( r ) = −∇V outside ( r )
outside
=1
Dipole
∼
1 ⎛ Qd ⎞
⎜
⎟
4πε o ⎝ r 2 ⎠
∼
1 ⎛ Qd ⎞
⎜
⎟
4πε o ⎝ r 3 ⎠
=2
Quadrupole
∼
1 ⎛ Qd 2 ⎞
⎜
⎟
4πε o ⎝ r 3 ⎠
∼
1 ⎛ Qd 2 ⎞
⎜
⎟
4πε o ⎝ r 4 ⎠
=3
Octupole
1 ⎛ Qd 3 ⎞
∼
⎜
⎟
4πε o ⎝ r 4 ⎠
1 ⎛ Qd 3 ⎞
∼
⎜
⎟
4πε o ⎝ r 5 ⎠
=4
Sextupole
1 ⎛ Qd 4 ⎞
∼
⎜
⎟
4πε o ⎝ r 5 ⎠
1 ⎛ Qd 4 ⎞
∼
⎜
⎟
4πε o ⎝ r 6 ⎠
..........
..........
............
Order
Multipole
∼
th
20
1 ⎛ Qd ⎞
⎜
⎟
4πε o ⎝ r +1 ⎠
∼
1 ⎛ Qd ⎞
⎜
⎟
4πε o ⎝ r + 2 ⎠
Electric Multipole
Moment M
............
M ∼ Q (r ) = M
(coulomb-metersb)
(rank- tensor)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
Thus we see that:
→ The higher-order multipole fields fall off 1/r faster than those associated with next lower
order multipole.
→ Must get in closer and closer to charge distribution ρ ( r ′ ) in order to sense / observe / detect
the higher-order moments!
We can write the electrostatic potential yet another way:
For r >> a (a = max value of r ′ )
∞
Voutside ( r ) = ∑ Vl
l =0
outside
⎡
1 ⎢1
1
1
ρ ( r ′ )dτ ′ + 2 r i ∫ r ′ρ ( r ′ ) dτ ′ + 3
(r ) =
∫
r v′
r
4πε o ⎢ r v′
⎢⎣
∫
(3 ( rˆir′)
v′
⎡
⎤
⎢Q
⎥
pirˆ rˆiQirˆ
1 ⎢ Net
⎥
=
+
+
+
V
r
.....
Thus, we see that: outside ( )
r2
r3
4πε o ⎢ r
⎥
⎢ monopole dipole quadrupole
⎥
term
term
⎣ term
⎦
with: QNet ≡ ∫ ρ ( r ′ ) dτ ′ ,
v′
p ≡ ∫ r ′ρ ( r ′ ) dτ ′ and
Q≡∫
v′
v′
(3 ( rˆir′)
2
− r ′2
2
2
− r ′2
2
)ρ ( r′) dτ ′
)ρ ( r′) dτ ′ + ....⎤⎥
⎥
⎥⎦
…..
Recall / note: rˆir ′ = r ′irˆ = r ′ cos Θ′ where Θ′ = opening angle between r and r ′.
The multipole expansion of Voutside ( r ) which contains the opening angle Θ′ between r (field
point) and r ′ (source point) can be rewritten in terms of ( θ and ϕ ) for r and ( θ ′ and ϕ ′ ) for r ′
using the so-called Addition Theorem for Spherical Harmonics:
Θ′ = opening angle between r and r ′
ẑ
S ′ (source
point)
r′
P (field point)
Θ′
θ′
θ
r
Spherical Harmonics Addition Theorem:
ŷ
2π − ϕ ′
⎛ 4π ⎞ + *
P ( cos Θ′ ) = ⎜
⎟ ∑ Y m (θ , ϕ ) Y m (θ ′, ϕ ′ )
⎝ 2 + 1 ⎠ m =−
n.b. complex conjugate
ϕ
ϕ′
x̂
Then:
∞
∞
( r ′)
1
1
=
= ∑ +1 P ( cos Θ′ ) = ∑
r r − r ′ l =0 r
=0
+
⎛ 4π ⎞ ( r ′ ) *
Y m (θ , ϕ ) Y m (θ ′, ϕ ′ )
⎟
+ 1 ⎠ r +1
∑ ⎜⎝ 2
m =−
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
21
UIUC Physics 435 EM Fields & Sources I
=
Thus:
⎛ 1 ⎞ ′
r ρ ( r ′ ) P ( cos Θ′ ) dτ ′
+1 ⎟ ∫ ( )
⎠ v′
=0
⎛ 1 ⎞ + ⎛ 4π ⎞ ′
*
∑⎜ ⎟ ∑ ⎜
⎟ ( r ) ρ ( r ′ ) Y ,m (θ , ϕ ) Y , m (θ ′, ϕ ′ ) dτ ′
4πε o =0 ⎝ r +1 ⎠ ∫v′ m =− ⎝ 2l + 1 ⎠
∞
1
∞
+
∑V
= 0 m =−
where: V outside
(r ) =
m
Thus:
=
Prof. Steven Errede
∑ ⎜⎝ r
4πε o
=∑
Voutside ( r ) =
Lecture Notes 8
∞
1
Voutside ( r ) =
Fall Semester, 2007
outside
m
(r )
1 ⎛ 4π ⎞ ( r ′ )
*
⎜
⎟ ∫ +1 ρ ( r ′ ) Y , m (θ , ϕ ) Y ,m (θ ′, ϕ ′ ) dτ ′
4πε o ⎝ 2 + 1 ⎠ v′ r
+
⎛ 4π ⎞ ⎛ 1 ⎞ ′
′
r
ρ
r
Y *,m (θ , ϕ ) Y ,m (θ ′, ϕ ′ ) dτ ′
(
)
(
)
∑⎜
∑
⎟⎜
⎟
4πε o =0 ⎝ 2l + 1 ⎠ ⎝ r +1 ⎠ ∫v′
m =−
∞
1
⎡
⎤
⎛ 4π ⎞ ⎛ 1 ⎞ + *
Y ,m (θ , ϕ ) ⎢ ∫ ( r ′ ) ρ ( r ′ )Y , m (θ ′, ϕ ′ ) dτ ′⎥
+1 ⎟ ∑
⎠ m =−
=0
⎣ v′
⎦
∞
1
4πε o
∑ ⎜⎝ 2l + 1 ⎟⎠ ⎜⎝ r
The Yl ,m (θ , ϕ ) are the Spherical Harmonics; θ and ϕ are the polar & azimuthal angles for r ,
the vector from the origin to the field point, P and θ ′ and ϕ ′ are the polar & azimuthal angles
for r ′ , the vector from the origin to the source point, S ′ .
We can then define q m - the Electric Multipole Moment of order
& m:
q m ≡ ∫ ( r ′ ) ρ ( r ′ ) Y m (θ ′, ϕ ′ ) dτ ′
v′
Because of the properties of the Y , m (θ , ϕ ) , namely that:
Y − m (θ , ϕ ) = ( −1) Y *,m (θ , ϕ )
We see that:
Y m (θ , ϕ ) =
2 ( + 1)( − m ) !
P ( cos θ ) eimϕ
4π ( + m ) !
q − m = ( −1) q*,m
1 ⎛ 4π ⎞ 1 *
Y , m (θ , ϕ ) q m
⎜
⎟
4πε o ⎝ 2 + 1 ⎠ r +1
∞
+
1 ∞ + ⎛ 4π ⎞ 1 *
r
Y ,m (θ , ϕ ) q , m
=
Then: Voutside ( r ) = ∑ ∑ V outside
(
)
∑∑⎜
,m
⎟
4πε o =0 m =− ⎝ 2 + 1 ⎠ r +1
= 0 m =−
=
Thus: V outside
,m
Again, Eoutside ( r ) = −∇Voutside ( r ) which by the principle of linear superposition becomes:
∞
=∑
+
∑
= 0 m =−
i.e.
22
∞
E outside
( r ) = −∑
,m
+
∑ ∇V
= 0 m =−
outside
,m
(r )
E outside
( r ) = −∇V outside
(r )
,m
,m
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The main advantage of using these seemingly more complex expressions for V outside
(r )
,m
involving the Y *m (θ , ϕ ) and Y m (θ ′, ϕ ′ ) spherical harmonics is that they are directly connected to
a right-handed xˆ − yˆ − zˆ coordinate system. The earlier expression for Voutside ( r ) involving the
P ( cos Θ′ ) Legendré Polynomials, it must be kept in mind at all times that Θ′ = opening angle
between field point r and source point r ′ .
The explicit derivation of Voutside ( r ) using the Addition Theorem for Spherical Harmonics:
Voutside ( r ) =
⎛ 4π ⎞ ⎛ 1 ⎞ + *
∑⎜
⎟⎜
⎟ ∑ Y m (θ , ϕ ) ∫ ( r ′ ) ρ ( r ′ ) Y m (θ ′, ϕ ′ ) dτ ′
4πε o =0 ⎝ 2 + 1 ⎠ ⎝ r +1 ⎠ m =−
v′
∞
1
≡q m
(electric multipole moment of order & m )
thus makes it explicitly clear that Voutside ( r ) = fcn ( r , θ , ϕ ) only – all source variable
( r ′,θ ′, ϕ ′ ) dependence has been integrated out, in carrying out the integral over the volume v′ !!!
Thus Voutside ( r ) is fully capable of correctly/exactly describing many other kinds of multipole
moments we have not yet discussed, e.g.:
A. Pure Physical Electric Dipole(s) Lying in the x-y Plane:
a.
b.
ẑ
d/2
d/2
−Q
ẑ
−Q d/2
d/2 +Q
ŷ
c.
ẑ
−Q
d/2
d/2
ŷ
ŷ
ϕ
+Q
x̂
x̂
(x-axis)
x̂
(y-axis)
(x-y plane)
B. Pure, Physical Electric Dipole Randomly Oriented in Space:
ẑ
−Q
d/2
(3-D dipole)
θ
ŷ
ϕ
x̂
d/2
+Q
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
23
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
C. Pure Physical, but Non-Colinear Electric Quadrupoles:
a.
b.
ẑ
c.
ẑ
+Q
d/2
d/2
+Q
−Q d/2
d/2
d/2 −Q
+Q
x̂
−Q
−Q d/2
ŷ
−Q
d/2
d/2
+Q
+Q
d/2 d/2
ŷ
ŷ
+Q d/2 d/2
x̂
(x-y plane)
ẑ
−Q
x̂
(y-z plane)
(x-z plane)
D. Pure Physical, but Non-Colinear Electric Octupoles:
Cube Centered on (x,y,z) = (0,0,0)
The Choice of Origin of Coordinates Does Matter!!!
Note that the choice of origin of coordinates in the electric multipole expansion of Voutside ( r )
does matter – can affect e.g. determination of electric dipole moment, p if QNET ≠ 0 !!
A point charge Q located at the origin of coordinates Ο (x,y,z) = (0,0,0) is a pure electric
monopole. However, a point charge Q located some distance d along d̂ from the origin is no
longer a pure electric monopole! The monopole moment Q = QTOT does not change, but V0 ( r )
(where
= 0) does change, because V ( r ) =
1 ⎛Q⎞
⎜ ⎟ is not quite correct – the exact potential
4πε o ⎝ r ⎠
is V ( r ) =
1 ⎛Q⎞
⎜ ⎟ and r ≠ r; however r
4πε o ⎝ r ⎠
r when r >> r ′ .
24
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
For higher electric moments, if (and only if) QTOT = QNET = 0, then (pure) electric
moment M (where > 0) is independent of choice of origin of coordinate system.
If net / total charge QNET = QTOT ≠ 0, then the higher-order electric moment(s)
M (where > 0) can be made to vanish if one chooses the origin or coordinates to be
located at the charge-weighted center of charge, then r ′ = 0 .
-
q m = ∫ ( r ′ ) ρ ( r ′ ) Y m (θ , ϕ ) dτ ′ = 0 if r ′ = 0
v′
−2Q
+Q
d
−Q
+Q
−Q
d
=
ẑ +
p1 = Qdzˆ
d
Ο (origin)
ẑ
p2 = −Qdzˆ
Note here that: p = p1 + p2 = 0!!!
If the origin is displaced from the center of charge for electric dipole by an amount a :
e.g.
r *′ = r ′ + a where a = vector displacement of origin of coordinate system,
p* = ∫ r *′ ρ ( r ′ )dτ ′ → p′* = ∫ ( r ′ + a )ρ ( r ′ ) dτ ′
v′
v′
= ∫ r ′ρ ( r ′ ) dτ ′ + ∫ aρ ( r ′ ) dτ ′
then:
v′
v′
= p + a ∫ ρ ( r ′ ) dτ ′ = p + QNet a = p + porigin
v′
= QNet ( = QTot )
- If QNET ≠ 0, then p* = p + porigin ≠ p because the origin-dependent electric dipole moment,
porigin ≡ QNet a ≠ 0 !!!
If QNET ≠ 0, then the choice of origin does matter; because the electric dipole moment p
depends on the choice of origin !!!
If QNET ≠ 0, then higher-order electric multipole moments must be accompanied by explicitly
specifying the choice of origin of coordinates!!!
- Iff QNET = 0, then p* = p , i.e. p is independent of choice or origin of coordinate system.
+Q
d
−5Q
Origin, Ο
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
25
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The Potential for a Pure Physical Electric Quadrupole (in Cartesian Coordinates)
Not Necessarily With Colinear Charges
The potential for a pure, physical electric quadrupole (not necessarily with collinear charges) can
be written in Cartesian coordinates as:
1 3 3 1 ⎛ xi x j ⎞
2
Vquad ( r ) =
∑∑
⎜ 5 ⎟ ∫ ( 3 xi x j − r ′ δ ij ) ρ ( r ′ ) dτ ′
4πε o i =1 j =1 2 ⎝ r ⎠ v′
or as: Vquad ( r ) =
1
4πε o
3
1 ⎛ xi x j
5
j =1
⎝ r
3
∑∑ 2 ⎜
i =1
⎞
⎟ Qij
⎠
)
(
with elements of the quadrupole moment tensor Qij ≡ ∫ 3xi′ x′j − r ′2δ ij ρ ( r ′ ) dτ ′
v′
with r ′2 = x′2 + y′2 + z ′2 = x1′2 + x2′ 2 + x3′ 2
and where the summations i = 1, 2, 3 and j = 1, 2, 3 represent sums over the {1,x ,2,3
y , z } components
i, j = 2: x2 ≡ y and
respectively; i.e. i, j = 1: x1 ≡ x
and where δ ij = Kroenecker δ − function { == 10 ifif ii≠= jj }
i, j = 3: x3 ≡ z
⎛ Q11 Q12
⎜
The 9 elements of the quadrupole moment tensor Q are the Qij’s: Q = ⎜ Q21 Q22
⎜Q
⎝ 31 Q32
Where:
Q13 ⎞
⎟
Q23 ⎟
Q33 ⎟⎠
sum of diagonal
elements =0
3
∑Q
ii
= 0 i.e. Q11 + Q22 + Q33 = 0 (i.e. Q is a traceless rank-2 tensor / 3 × 3 matrix)
i=1
and also: Qij = Q ji for i ≠ j , i.e. Q12 = Q21 , Q13 = Q31 and Q23 = Q32 .
In general, if r = xiˆ + yjˆ + zkˆ and r ′ = x′iˆ + y′ˆj + z ′kˆ then:
Vquad ( r ) =
+
26
1 ⎛ 1 ⎞⎧
⎜ ⎟ ⎨3xy x′y′ρ ( r ′ ) dτ ′ + 3 zx ∫ x′z ′ρ ( r ′ ) dτ ′ + 3 yz ∫ y′z ′ρ ( r ′ ) dτ ′
4πε o ⎝ r 5 ⎠ ⎩ ∫v′
v′
v′
⎫
1
1
1
3x 2 − 1) ∫ x′2 ρ ( r ′ ) dτ ′ + ( 3 y 2 − 1) ∫ y′2 ρ ( r ′ ) dτ ′ + ( 3z 2 − 1) ∫ z ′2 ρ ( r ′ ) dτ ′⎬
(
2
2
2
v′
v′
v′
⎭
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The 9 elements of the quadrupole moment tensor Q (in Cartesian coordinates) are thus:
Mean square of xixj (multiplied by q).
Qxx = ∫ x′2 ρ ( r ′ ) dτ ′ = qx′
2
= q x′
2
v′
Qyy = ∫ y′2 ρ ( r ′ ) dτ ′ = q y′
2
2
= q y′
v′
Qzz = ∫ z ′2 ρ ( r ′ ) dτ ′ = qz ′
2
= q z′
2
v′
Qxy = ∫ x′y′ρ ( r ′ ) dτ ′ = qx′y′ = q x′y′ = Qyx
⇐
n.b. The Quadrupole Moment Tensor
Q has only 6 independent components
v′
Qyz = ∫ y′z ′ρ ( r ′ ) dτ ′ = q y′z ′ = q y′z ′ = Qzy
v′
Qzx = ∫ z ′x′ρ ( r ′ ) dτ ′ = qz ′x′ = q z ′x′ = Qxz
v′
Then:
Vquad ( r ) =
1 ⎛ 1 ⎞⎡
1
1
1
2
2
2
⎜ 5 ⎟ ⎢3 xyQxy + 3 yzQyz + 3 xzQxz + ( 3 x − 1) Qxx + ( 3 y − 1) Qyy + ( 3 z − 1) Qzz
4πε o ⎝ r ⎠ ⎣
2
2
2
A relationship exists between multipole moments expressed using spherical-polar coordinates
q m and those expressed using Cartesian coordinates Qij . The first few of these are given below:
q00 =
1
q
4π
q20 =
1 5
Q33
2 4π
q10 =
3
pz
4π
q21 = −
1 15
m
( Q13 − iQ23 ) with q −m = ( −1) q*m
3 8π
q11 = −
3
( px − ip y )
8π
q22 =
1 15
( Q11 − 2iQ12 − Q22 )
12 2π
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
27
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 8
Prof. Steven Errede
The Energy / Work Associated With a Charge Distribution ρ ( r ′ ) Located at (or Near) the
Origin of the Coordinate System in an External Electric Field Eext ( r )
For r >> a (a = max value of r ′ ), the energy / work associated with a charge distribution in an
external field Eext ( r ) is given by:
W = QVext ( r = 0 ) − p i Eext ( r = 0 ) −
=−
r =0
∫refpt. . Eext i dl
∂E ext
1 3 3
j
Q
∑∑
ij
6 i =1 j =1
∂xi
− ....
xi = 0
Eext ( r = 0 ) = −∇Vext ( r = 0 )
Where the summations i = 1, 2, 3 and j = 1, 2, 3 represent sums over the
respectively; i.e. i, j = 1: x1 ≡ x
i, j = 2: x2 ≡ y
And: Qij ≡ ∫ ( 3 xi′x′j − r ′2δ ij ) ρ ( r ′ ) dτ ′ with
{ } components
1, 2,3
x, y, z
i, j = 3: x3 ≡ z
r ′2 = x′2 + y′2 + z ′2 = x1′2 + x2′ 2 + x3′ 2
v′
3
∑Q
And with: Qij = Q ji , and
i =1
ii
= Q11 + Q22 + Q33 = Qxx + Qyy + Qzz = 0
Note: The multipole expansion method for Voutside ( r ) =
1
4πε o
∞
⎛ 4π ⎞
+
⎛ 1 ⎞ *
Y θ ,ϕ ) q m
+1 ⎟ m (
⎠
∑ ⎜⎝ 2l + 1 ⎟⎠ ∑ ⎜⎝ r
=0
m =−
with q m = ∫ ( r ′ ) Ylm (θ ′, ϕ ′ ) ρ ( r ′ ) dτ ′ is analogous to the taking of an inner product!!!
v′
It can then be seen that the electric multipole moments q m are the strengths (i.e. coefficients)
associated with the
(
, m ) -order multipoles of the electric charge distribution ρ ( r ′ ) !!!
th
Electrostatic Forces and Torques Acting on Multipole Moments of the Charge Distribution
The net force and torque acting on the charge distribution as an expansion in multipole moments
are given below:
⎡1 3 3
⎤
∂E ext
j ( r = 0)
F ( r ) = qE ( r = 0 ) + ∇ p i E ( r )
+ ∇ ⎢ ∑∑ Qij
+ ....
⎥
r =0
∂xi
⎢⎣ 6 i =1 j =1
⎥⎦ x =0
i
(
τ ( r ) = ( p × E ( r ))
)
⎞ ∂ ⎛ 3
⎞⎤
1 ⎪⎧ ⎡ ∂ ⎛ 3
ext
0
+ ⎨⎢
=
−
Q
E
r
Q3 j E ext
(
)
⎜
⎟
⎜
∑
∑
2j j
j ( r = 0) ⎟⎥
r =0
3 ⎪⎣⎢ ∂x3 ⎝ j =1
⎠ ∂x2 ⎝ j =1
⎠ ⎥⎦ r =0
⎩
⎡ ∂ ⎛ 3
⎞ ∂ ⎛ 3
⎞⎤
0
+ ⎢ ⎜ ∑ Q3 j E ext
=
r
) ⎟ − ⎜ ∑ Q1 j E extj ( r = 0 ) ⎟ ⎥
j (
⎢⎣ ∂x1 ⎝ j =1
⎠ ∂x3 ⎝ j =1
⎠ ⎥⎦ r =0
⎡ ∂ ⎛ 3
⎞ ∂ ⎛ 3
⎞ ⎤ ⎪⎫
ext
ext
+⎢
⎜ ∑ Q1 j E j ( r = 0 ) ⎟ −
⎜ ∑ Q2 j E j ( r = 0 ) ⎟ ⎥ ⎬ + ....
⎠ ∂x1 ⎝ j =1
⎠ ⎦⎥ r =0 ⎪⎭
⎣⎢ ∂x2 ⎝ j =1
28
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.