Electrostatic Field due to an
Arbitrary Charge Distribution
EE 141 Lecture Notes
Topic 10
Professor K. E. Oughstun
School of Engineering
College of Engineering & Mathematical Sciences
University of Vermont
2014
Motivation
E - Field Outside an Arbitrary Charge Distribution
Consider an arbitrary (but fixed) charge distribution with density ̺(r′ )
occupying a region τ ′ in free-space and extending to a maximum
′
distance rmax
from the origin O of a coordinate system.
z
P(x,y,z)
r‘’
P‘(x’,y’,z’)
r
τ‘
r‘
V
O
x
1r
y
It is assumed that the origin O is positioned either within the charged
region τ ′ or is within close proximity to it.
E - Field Outside an Arbitrary Charge Distribution
The absolute electrostatic potential V (r) = V (x, y , z) at some fixed
′
observation point P(x, y , z) at a distance r > rmax
from O is given by
ZZZ
1
̺(x ′ , y ′ , z ′ ) 3 ′
V (x, y , z) =
d r
(1)
4πǫ0
r ′′
τ′
where
i1/2
h
2
2
2
r ′′ = (x − x ′ ) + (y − y ′) + (z − z ′ )
(2)
is the distance from the elemental source point at P ′ (x ′ , y ′ , z ′ ) to the
field point at P(x, y , z).
Because the source point P ′ (x ′ , y ′, z ′ ) is taken to be near to the
origin O whereas the field point P(x, y , z) is taken to be removed
from the origin, one may then expand the quantity 1/r ′′ in a Taylor
series about the origin (x ′ , y ′ , z ′ ) = (0, 0, 0) as follows:
E - Field Outside an Arbitrary Charge Distribution
1
r ′′
=
∂
1 ∂
∂
1
′
′
′
+
x
+
y
+
z
r ′′ ∂x ′
∂y ′
∂z ′ r ′′ O
O
2 1
∂
1
∂
∂
+
x′ ′ + y ′ ′ + z′ ′
+ · · · , (3)
2!
∂x
∂y
∂z
r ′′ O
where O =⇒ (x ′ , y ′ , z ′ ) = (0, 0, 0), and
2
2
2
2
′ ∂
′ ∂
′2 ∂
′2 ∂
′2 ∂
′ ∂
+
y
+
z
+
y
+
z
=
x
x
∂x ′
∂y ′
∂z ′
∂x ′2
∂y ′2
∂z ′2
2
2
∂
′ ′
′ ′ ∂
+2x y
+ 2x z
∂x ′ ∂y ′
∂x ′ ∂z ′
∂2
+2y ′z ′ ′ ′ .
∂y ∂z
E - Field Outside an Arbitrary Charge Distribution
The first term in the Taylor series expansion (3) is given by
1 1
1 =
=
.
r ′′ r ′′ ′ ′ ′
r
O
For the second term in
∂ 1 =
∂x ′ r ′′ O
∂ 1 =
∂y ′ r ′′ O
∂ 1 =
∂z ′ r ′′ O
(x ,y ,z )=(0,0,0)
the Taylor series expansion (3), one has that
1 ∂r ′′ x − x ′ x
ℓ
− ′′2 ′ = ′′3 = 3 = 2 ,
r ∂x r
r
r
O
O
y
m
=
,
r3
r2
z
n
= 2,
3
r
r
where ℓ ≡ x/r , m ≡ y /r , and n ≡ z/r are the cosines of the angles
between the position vector r and the x, y , and z-axes, respectively.
E - Field Outside an Arbitrary Charge Distribution
For the third term in the Taylor series expansion (3), one has that
∂ 2 1 ∂ x − x ′ −r ′′3 − 3(x − x ′ )r ′′2 ∂r ′′ /∂x ′ =
=
∂x ′2 r ′′ ∂x ′ r ′′3 r ′′6
O
O
3ℓ2 − 1
3(x 2 /r 2 ) − 1
=
,
=
r3
r3
3m2 − 1
=
,
r3
∂ 2 1 ∂y ′2 r ′′ O
2
3n2 − 1
∂ 1 =
,
∂z ′2 r ′′ r3
O
and
O
E - Field Outside an Arbitrary Charge Distribution
∂ 2 1 ∂x ′ ∂y ′ r ′′ O
∂ 2 1 ∂x ′ ∂z ′ r ′′ O
2
∂
1 ∂y ′ ∂z ′ r ′′ O
∂ 2 1 ∂ x − x ′ (x − x ′ ) ∂r ′′ =
=
= −3
∂y ′ ∂x ′ r ′′ ∂y ′ r ′′3 r ′′4 ∂y ′ O
O
O
′
′ (x − x )(y − y ) xy
ℓm
= 3
=
3
=
3
,
r ′′5
r5
r3
O
2
∂
ℓn
1
=
=
3
,
∂z ′ ∂x ′ r ′′ r3
O
2
∂
mn
1 =
=
3
.
∂z ′ ∂y ′ r ′′ r3
O
E - Field Outside an Arbitrary Charge Distribution
With the Taylor series expansion (3) of 1/r ′′ about the origin O, the
electrostatic potential (1) at the field point P(x, y , z) becomes
( ZZZ
1
1
̺(x ′ , y ′, z ′ )d 3 r ′
V (x, y , z) =
4πǫ0 r
τ′
ZZZ
1
(ℓx ′ + my ′ + nz ′ )̺(x ′ , y ′, z ′ )d 3 r ′
+ 2
r
′
Z Z Zτ
1
+ 3
3ℓmx ′ y ′ + 3ℓnx ′z ′ + 3mny ′ z ′
r
′
τ
1
1
+ (3ℓ2 − 1)x ′2 + (3m2 − 1)y ′2
2
2
)
1 2
+ (3n − 1)z ′2 ̺(x ′ , y ′ , z ′ )d 3 r ′ + · · ·
2
= V0 (x, y , z) + V1 (x, y , z) + V2 (x, y , z) + · · · .
(4)
E - Field Outside an Arbitrary Charge Distribution:
The Monopole Term V0(r)
The first (or zeroth-order) term in this expansion is called the
monopole term because it is simply the electrostatic potential one
would have at P if the entire charge distribution were concentrated
at the origin O, where
ZZZ
Q
1
̺(x ′ , y ′, z ′ )d 3 r ′ =
.
(5)
V0 (x, y , z) =
4πǫ0 r
4πǫ0 r
τ′
|
{z
}
net charge Q in τ ′
The monopole term is zero only if the total net charge Q in the
region τ ′ is zero. If it is nonvanishing, then it is the dominant term in
the series as r → ∞ because it decreases only as r −1 .
E - Field Outside an Arbitrary Charge Distribution:
The Dipole Term V1(r)
The second (or first-order) term in the expansion (4) is called the
dipole term and may be written as
p · r̂
,
(6)
V1 (x, y , z) =
4πǫ0 r 2
r̂ ≡ 1̂x ℓ + 1̂y m + 1̂z n
(7)
denoting the unit vector along the radial line from the origin O to the
observation (or field) point P(x, y , z), where
ZZZ ZZZ
′
′
′
′
′ ′
3 ′
p≡
1̂x x + 1̂y y + 1̂z z ̺(x , y , z )d r =
r′ ̺(r′ )d 3 r ′
τ′ |
τ′
{z
}
r′
(8)
is the dipole moment of the charge distribution with respect to O.
Hence, V1 (r) = V1 (x, y , z) is the electrostatic potential at P(x, y , z)
due to an effective dipole at the origin O with dipole moment p.
E - Field Outside an Arbitrary Charge Distribution:
The Dipole Term V1(r)
The dipole moment of an extended charge distribution can also be
defined as
p = Q r¯′ ,
(9)
where Q is the total net charge in the distribution, as given in Eq.(5),
and where r¯′ is a vector extending from the origin O to the charge
centroid of the distribution, given by
RRR ′ ′ 3 ′
ZZZ
1
′ r ̺(r )d r
τ
′
¯
=
r′ ̺(r′ )d 3 r ′ .
(10)
r = RRR
′ )d 3 r ′
Q
̺(r
′
′
τ
τ
Notice that if Q = 0, then r¯′ → ∞ and p, as given by Eq. (9), is
indeterminate; however, Eq. (8) always determines the dipole
moment unambiguously. When Q = 0, the dipole moment is
independent of the choice of origin. If Q 6= 0, the dipole moment of
the charge distribution can always be made to vanish by choosing the
origin O at the centroid of the charge distribution (r¯′ = 0).
E - Field Outside an Arbitrary Charge Distribution:
The Quadrupole Term V2(r)
The third (or second-order) term in the expansion (4) is called the
quadrupole term and may be written as
"
ZZZ
ZZZ
1
′ ′
′
3 ′
3ℓm
x y ̺(r )d r + 3ℓn
x ′ z ′ ̺(r′ )d 3 r ′
V3 (r) =
4πǫ0 r 3
′
′
τ
τ
ZZZ
+3mn
y ′z ′ ̺(r′ )d 3 r ′
′
τ
ZZ Z
1 2
+ (3ℓ − 1)
x ′2 ̺(r′ )d 3 r ′
2
′
τ Z
ZZ
1
2
+ (3m − 1)
y ′2 ̺(r′ )d 3 r ′
2
τ′
#
ZZZ
1 2
z ′2 ̺(r′ )d 3 r ′ . (11)
+ (3n − 1)
2
τ′
E - Field Outside an Arbitrary Charge Distribution:
The Quadrupole Term V2(r)
These six integrals define the quadrupole moment of the charge
distribution, where
ZZZ
(12)
Qxx ≡
x ′2 ̺(r′ )d 3 r ′ = Qx ′2 ,
′
τ
ZZZ
Qyy ≡
y ′2 ̺(r′ )d 3 r ′ = Qy ′2,
(13)
′
τ
ZZZ
(14)
Qzz ≡
z ′2 ̺(r′ )d 3 r ′ = Qz ′2 ,
′
τ
ZZZ
Qxy = Qyx ≡
x ′ y ′ ̺(r′ )d 3 r ′ = Qx ′y ′ ,
(15)
′
τ
ZZZ
Qxz = Qzx ≡
x ′ z ′ ̺(r′ )d 3 r ′ = Qx ′ z ′ ,
(16)
τ′
ZZZ
(17)
Qyz = Qzy ≡
y ′ z ′ ̺(r′ )d 3 r ′ = Qy ′z ′ .
τ′
E - Field Outside an Arbitrary Charge Distribution:
The Quadrupole Term V2(r)
With these results, Eq. (11) becomes
"
1
V3 (r) =
3ℓmQxy + 3ℓnQxz + 3mnQyz
4πǫ0 r 3
1
1
1
+ (3ℓ2 − 1)Qxx + (3m2 − 1)Qyy + (3n2 − 1)Qzz
2
2
2
#
(18)
E - Field Outside an Arbitrary Charge Distribution:
Quadrupole Term V2(r) - Cylindrical Symmetry
If the charge distribution in τ ′ possesses cylindrical symmetry about
the z-axis, for example, then
Qxy = Qyz = Qxz = 0,
Qxx = Qyy .
It is then convenient to define a single (scalar) quadrupole moment
Q of the charge distribution as
ZZZ
Q ≡ 2(Qzz − Qxx ) =
(3z ′2 − r ′2 )̺(r′ )d 3 r ′ ,
(19)
τ′
where r ′2 = x ′2 + y ′2 + z ′2 .
E - Field Outside an Arbitrary Charge Distribution:
Quadrupole Term V2(r) - Cylindrical Symmetry
Under these conditions and with this definition, Eq. (18) becomes
#
"
1 2
1
1
V3 (r) =
(3ℓ + 3m2 − 2)Qxx + (3n2 − 1) Qzz
|{z}
4πǫ0 r 3 2
2
1
Q+Qxx
2
"
#
1
1
3
=
(ℓ2 + m2 + n2 ) −1 Qxx + (3n2 − 1)Q
{z
}
4πǫ0 r 3 2 |
4
1
2
=
Q 3 cos2 θ − 1
Q 3n − 1
=
4πǫ0 4r 3
4πǫ0
4r 3
where n ≡ z/r = cos θ.
(20)
E - Field Outside an Arbitrary Charge Distribution:
The Multipole Expansion of the Potential
The multipole expansion (4) of the electrostatic potential (1) at the
field point P(x, y , z) due to the charge distribution ̺(r′ ) in the region
τ ′ about the origin O becomes
V (x, y , z) = V0 (x, y , z) + V1 (x, y , z) + V2 (x, y , z) + · · ·
p · r̂
Q 3 cos2 θ − 1
Q
+
+
+···
=
4πǫ r
4πǫ r 2
4πǫ r 3
4
| {z0 }
| {z0 } | 0
{z
}
monopole term
dipole term
quadrupole term
(21)
in the cylindrically symmetric case about the z-axis.
Average E - Field Inside a Sphere Containing an
Arbitrary Charge Distribution
Consider first determining the average electric field intensity inside a
sphere of radius R containing a point charge Q situated a distance r ′
from the center O of the sphere with the z-axis taken along the line
passing through the point charge Q and the origin O.
z
dτ1
θ
Q
r‘’
P
R
r‘
O
dτ2
Average E - Field Inside a Sphere Containing an
Arbitrary Charge Distribution
By symmetry, the average E-field over the spherical volume must be
along the z-axis. The average E - field over the spherical region is
then given by
ZZZ
1
Ez d 3 r
(22)
Ez =
τ
τ
where τ = 34 πR 3 denotes the volume of the sphere.
This integral can be separated into two parts, one over the spherical
shell τ1 between r ′ and R, and one over the sphere τ0 of radius r ′ :
ZZZ
ZZZ
1
1
3
Ez =
(23)
Ez d r +
Ez d 3 r .
τ
τ
τ0
τ1
{z
}
|
0
Average E - Field Inside a Sphere Containing an
Arbitrary Charge Distribution
The integral over the spherical shell τ1 vanishes because:
d Ω intercepts volume elements d τ1 & d τ2 in each spherical shell.
Ez decreases as the square of the distance from Q whereas d τ
increases as the square of this distance. ∴ product is constant.
Ez is positive at d τ1 while it is negative at d τ2 , the two
contributions to the integral then canceling.
z
dτ1
θ
Q
r‘’
P
R
r‘
O
dτ2
Average E - Field Inside a Sphere Containing an
Arbitrary Charge Distribution
The remaining integral over the inner sphere τ0 may then be
evaluated using spherical polar coordinates with origin at the position
of the point charge Q. At the point P,
Q
Q
Ez = E · 1̂z =
1̂r ′′ · 1̂z =
cos θ,
′′2
4πǫ0 r
4πǫ0 r ′′2
so that
ZZZ
Z 2π Z π Z −2r ′ cos θ
Q cos θ ′′2
3
r sin θdr ′′ d θd φ
Ez d r =
′′2
4πǫ
r
′′
0
τ0
φ=0 θ=π/2 r =0
#
"Z
Z 2π
Z π
−2r ′ cos θ
Q
=
dφ
sin θ cos θ
dr ′′ d θ
4πǫ0 0
}
| {z } π/2
| 0 {z
2π
−2r ′ cos θ
Z
Qr ′ π
Qr ′
= −
cos2 θ sin θd θ = −
.
ǫ0 π/2
3ǫ0
Average E - Field Inside a Sphere Containing an
Arbitrary Charge Distribution
Hence
1
Ez = 4 3
πR
3
Qr ′
−
3ǫ0
=−
Qr ′
.
4πǫ0 R 3
(24)
Because Qr ′ is the dipole moment of the charge Q relative to the
origin O of the sphere, then
E=−
p
4πǫ0 R 3
(25)
For an arbitrary charge distribution ̺(r) inside the sphere, the
electrostatic field is the superposition of the fields due to the
individual charge elements ̺(r)d 3 r . The average E - field within the
sphere is then given by Eq. (25) with p given by the dipole moment
of the arbitrary charge distribution within the sphere of radius R, as
given by Eq. (8).