Chapter 7
7.13
7.1
Laplace’s Equation: The Potential Produced by Surface Charge
Problems
Finding Charge From Potential
The potential in a spherical region r < R is '(x, y, z) = '0 (z/R)3 . Find a volume charge density
Ω(r, µ) in the region r < R and a surface charge density æ(µ) on the surface r = R which together
produce this potential. Express your answers in terms of elementary trigonometric functions.
7.2
A Periodic Array of Charged Rings
Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge
Q and radius R. There is one ring in each of the planes z = 0, z = ±b, z = ±2b, etc. Exploit
the Fourier expansion in Example 1.6 to find the potential everywhere in space. Check that your
solution makes sense in the limit that the cylindrical variable Ω ¿ R, b. Hint: If IÆ (y) and KÆ are
modified Bessel functions,
IÆ0 (y)KÆ (y) ° IÆ (y)KÆ0 (y) = 1/y.
7.3
Two Electrostatic Theorems
Use the orthogonality properties of the spherical harmonics to prove the following for a function
'(r) which satisfies Laplace’s equation in and on an origin-centered spherical surface S of radius
R:
(a)
R
dS '(r) = 4ºR2 '(0)
S
(b)
Z
S
7.4
dSz'(r) =
Ø
4º 4 @' ØØ
R
3
@z Ør=0
Make a Field Inside a Sphere
Find the volume charge density Ω and surface charge density æ which much be placed in and on a
sphere of radius R to produce a field inside the sphere of
E = °2V0
xy
V0
V0
ˆ + 3 (y 2 ° x2 )ˆ
ˆ.
x
y°
z
R3
R
R
There is no other charge anywhere. Express your answer in terms of trigonometric functions of µ
and ¡.
7.5
Green’s Formula
ˆ be the normal to an equipotential surface at a point P . If R1 and R2 are the principal
Let n
radii of curvature of the surface at P . A formula due to George Green relates normal derivatives
ˆ · r) of the potential '(r) (which satisfies Laplace’s equation) at the equipotential surface
(@/@n ¥ n
to the mean curvature of that equipotential surface ∑ = 12 (R1°1 + R2°1 ):
@2'
@'
+ 2∑
= 0.
@n2
@n
Derive Green’s equation by direct manipulation of Laplace’s equation.
7.6
The Channeltron
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
The parallel plates of a channeltron are segmented into conducting strips of width b so the potential can be fixed on the strips at staggered values. We model this using infinite-area plates, a
finite portion of which is shown below. Find the potential '(x, y) between the plates and sketch
representative field lines and equipotentials. Note the orientation of the x and y axes.
M 1
M 1
x
d
y
M 0
M 2
M 2
b
7.7
The Calculable Capacitor
The figure below shows a circle which has been divided into two pairs of segments with equal
arc length by a horizontal bisector and a vertical line. The positive x-axis bisects the segment
labelled “1” and the polar angle ¡ increases counterclockwise from the x-axis as indicated . Now let
the segmented circle be the cross section of a segmented conducting cylinder (with tiny insulating
regions to separate the segments).
x
1
r
I
1D
2
2
O
3
4
(a) Let segment 1 have unit potential and ground the three others. If the angle Æ subtends
segment 1 as viewed from the origin O, show that the charge density induced on the inside
surface of segment 3 is
æ(¡) =
∑
∏
sin( 12 Æ + ¡)
sin( 12 Æ ° ¡)
≤0
+
.
2ºR 1 ° cos( 12 Æ + ¡)
1 ° cos( 12 Æ ° ¡)
(b) Enclose the segmented cylinder by a coaxial, grounded, conducting cylindrical shell whose
radius is infinitesimally larger than R. This guarantees that that no charge is induced on the
outside of segment 3. In that case, show that the cross-capacitance per unit length between
segments 1 and 3 is
≤0
C13 = ° ln 2.
º
The non-trivial fact that C13 depends only on defined constants (and not on R) is exploited
worldwide to “realize” the farad—the fundamental unit of capacitance.
7.8
An Incomplete Cylinder
The figure below shows an infinitely long cylindrical shell from which a finite angular range has
been removed. Let the shell be a conductor raised to a potential corresponding to a charge per unit
length ∏. Find the fraction of charge which resides on the inner surface of the shell in terms of ∏
and the angular parameter p. Hint: Calculate Qin ° Qout .
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
2S
p
7.9
Picht’s Equation
This problem addresses the focusing properties of cylindrically symmetric potentials '(Ω, z) which
satisfy Laplace’s equation.
(a) Let V (z) = '(0, z). Use separation of variables to show that E(Ω, z) º 12 V 00 (z)Ω Ω
ˆ ° V 0 (z)ˆ
z
p
0
000
for points near the symmetry axis where Ω ø |V (z)/V (z)|. This is called the paraxial
regime in charged particle optics.
(b) Regard Ω(z, t) as the trajectory of a particle with charge q and mass m and derive the trajectory
equation
q
Ω¨ = z¨Ω0 + z˙ 2 Ω00 =
ΩV 00 (z).
2m
(c) Use Newton’s second law and an approximate form of conservation of energy (valid when vz
is large) to derive the trajectory equation
d2 Ω
1 V 0 dΩ
Ω V 00
+
+
= 0.
2
dz
2 V dz
4 V
(d) Show that a change of variables to R(z) = Ω(z)V 1/4 (z) transforms the equation in part (c) to
Picht’s equation,
∑ 0
∏2
d2 R
3
V (z)
=
°
R(z)
.
dz 2
16
V (z)
(e) Integrate Picht’s equation and explain why it predicts focusing for particles which enter the
potential parallel to the z-axis.
7.10
A Dielectric Wedge in Polar Coordinates
Two wedge-shaped dielectrics meet along the ray ¡ = 0. The opposite edge of each wedge is held
at a fixed potential by a metal plate. The system is invariant to translations perpendicular to the
diagram.
(a) Explain why the potential '(Ω, ¡) between the plates does not depend on the polar coordinate
Ω.
(b) Find the potential everywhere between the plates.
$ #V1
!1
"1
"2 !
2
$ #V2
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" #0
Chapter 7
7.11
Laplace’s Equation: The Potential Produced by Surface Charge
A Split Conical Conductor
An electron deflector takes the form of an infinite, segmented, conducting cone whose apex is at
the origin and whose opening angle is 2Æ. The symmetry axis inside the cone is the positive z-axis
and the two segments are held at the potentials ± V as shown below.
I
V
D
x
V
D
1. Use a separation of variables argument in spherical coordinates to show that the potential
inside the cone is independent of the radial variable
1. Use the result of part (a) to show that Laplace’s equation can be rewritten as
∫2
@'
@'
@2'
+∫
+
=0
2
@∫
@∫
@¡2
where ∫ = tan 12 µ .
1. Separate variables and show that
'(µ, ¡) =
4V
º
1
X
m=1,3,5,···
(°1)(m°1)/2
m
∑
tan µ/2
tan Æ/2
∏m
cos m¡
1. Exploit the expansion ln(1 ± z) = ±z ° 12 z 2 ± 13 z 3 ° 14 z 4 + · · · to sum the series and show
that
4V
'=
tan°1
º
7.12
Ω
æ
2 tan 12 µ tan 12 Æ
cos ¡ .
tan2 12 µ ° tan2 12 Æ
Practice with Bessel Functions
A grounded metal tube with radius R is coaxial with the z-axis. The bottom of the tube at z = 0
is closed by a circular metal plate held at potential V . The top of the tube is open and extends to
infinity. If J0 (km R) = 0, show that the potential inside the tube is
'(Ω, z) =
7.13
1
2V X exp(°km z) J0 (km Ω)
.
R m=1
km
J1 (km R)
The Capacitance of an OÆ-Center Capacitor
A spherical conducting shell centered at the origin has radius R1 and is maintained at potential V1 .
A second spherical conducting shell maintained at potential V2 has radius R2 > R1 but is centered
at the point sˆ
z where s << R1 .
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
(a) To lowest order in s, show that the charge density induced on the surface of the inner shell is
æ(µ) = ≤0
∑
∏
R1 R2 (V2 ° V1 ) 1
3s
°
cos
µ
.
R2 ° R1
R12
R23 ° R13
(b) To lowest order in s, show that the force exerted on the inner shell is
F=
Z
æ2
ˆ=z
ˆ2ºR12
dS
n
2≤0
Zº
dµ sin µ
0
æ 2 (µ)
Q2
sˆ
z
cos µ = °
.
2≤0
4º≤0 R23 ° R13
(c) Integrate the force in (b) to find the capacitance of this structure to second order in s.
7.14
The Plane-Cone Capacitor
A capacitor is formed by the infinite grounded, plane z = 0 and an infinite, solid, conducting cone
with interior angle º/4 held at potential V . A tiny insulating spot at the cone vertex (the origin
of coordinates) isolates the two conductors.
M V
S
4
M 0
(a) Explain why '(r, µ, ¡) = '(µ) in the space between the capacitor “plates”.
(b) Integrate Laplace’s equation explicitly to find the potential between the plates.
7.15
The Near-Origin Potential of Four Point Charges
Four identical positive point charges sit at (a, a), (°a, a), (°a, °a), and (a, °a) in the z = 0 plane.
Very near the origin, the electrostatic potential can be written in the form
'(x, y, z) = A + Bx + Cy + Dz + Exy + F xz + Gyz + Hx2 + Iy 2 + Jz 2 .
(a) Deduce the non-zero terms in this expansion and the algebraic sign of their coe±cients. Do
not calculate the exact value of the non-zero coe±cients.
(b) Sketch electric field lines and equipotentials in the z = 0 plane everywhere inside the square
and a little bit outside the square. Do not miss any important features of the patterns.
7.16
U-Shaped Electrodes
Two semi-infinite blocks of matter share a common interface as shown below. The matter with
dielectric constant ∑2 is completely surrounded by a æ-shaped electrode which is grounded. The
matter with dielectric constant ∑1 is completely surrounded by a Ω-shaped electrode which is held
at potential V .
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
y
a
N1
N2
x
M V
d
M 0
(a) Determine '(x, y) everywhere between the two electrodes.
(b) Find the polarization charge on the interface when ∑1 is slightly greater than ∑2 and also when
∑1 is slightly less than ∑2 .
(c) Sketch electric field lines when ∑1 ¿ ∑2 and also when ∑1 ø ∑2 .
7.17
The Potential Inside an Ohmic Duct
The z-axis runs down the center of an infinitely long heating duct with a square cross section. For
a real metal duct (not a perfect conductor), the electrostatic potential '(x, y) varies linearly along
the sidewalls of the duct. Suppose that the duct corners at (±a, 0) are held at potential +V and
the duct corners at (0, ±a) are held at potential °V. Find the potential inside the duct beginning
with the trial solution
'(x, y) = A + Bx + Cy + Dx2 + Ey 2 + F xy.
7.18
A Potential Patch By Separation of Variables
The square region defined by °a ∑ x ∑ a and °a ∑ y ∑ a in the z = 0 plane is a conductor held
at potential ' = V . The rest of the z = 0 plane is a conductor held at potential ' = 0. The plane
z = d is also a conductor held at zero potential.
V
2a
d
2a
(a) Find the potential for 0 ∑ z ∑ d in the form of a Fourier integral.
(b) Find the total charge induced on the upper surface of the lower (z = 0) plate. The answer is
very simple. Do not leave it in the form of an unevaluated integral or infinite series.
(c) Sketch field lines of E(r) between the plates.
7.19
Poisson’s Integral Formula
The Poisson integral formula
'(r) =
(R2 ° r2 )
4ºR
Z
dyS
|yS |=R
'(y
¯ S)
|r ° yS |3
|r| < R
gives the potential at any point r inside a sphere if we specify the potential '(y
¯ S ) at every point
on the surface of the sphere. Derive this formula by summing the general solution of Laplace’s
equation inside the sphere using the derivatives (with respect to r and R) of the identity
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
1
X
1
r`
=
P` (ˆ
r · yˆS ).
|r ° yS |
R`+1
`=0
7.20
An Electrostatic Analog of the Helmholtz Coil
A spherical shell of radius R is divided into three conducting segments by two very thin air gaps
located at latitudes µ0 and º ° µ0 . The center segment is grounded. The upper and lower segments
are maintained at potentials V and °V , respectively. Find the angle µ0 such that the electric field
inside the shell will be as nearly constant as possible near the center of the sphere.
M V
T0
R
M 0
T0
M V
7.21
A Conducting Sphere at a Dielectric Boundary
A conducting sphere with radius R and charge Q sits at the origin of coordinates. The space outside
the sphere above the z = 0 plane has dielectric constant ∑1 . The space outside the sphere below
the z = 0 plane has dielectric constant ∑2 .
R
Q
N1
N2
(a) Find the potential everywhere outside the conductor.
(b) Find the distributions of free charge and polarization charge wherever they may be.
7.22
Bumps and Pits on a Flat Conductor
A flat metal plate occupies the z = 0 plane. When raised to a non-zero potential, the plate develops
a uniform surface charge density æ0 and a uniform field E0 = (æ0 /≤0 )ˆ
z in the space z > 0.
(a) Place a hemispherical metal bump of radius R on the plate as shown in part (a) of the figure
below. Ground the plate and bump combination and demand that E(z ! 1) ! E0 . Show
that E for this problem diÆers from E0 by the field of a suitably placed point dipole. Calculate
the charge density induced on the conducting surface.
(b) Replace the hemispherical metal bump by a hemispherical metal crater as shown in part (b) of
the figure below. Ground the plate and crater combination and demand that E(z ! 1) !
E0 . Why is it less straightforward to find the potential for this problem as for the bump
problem? How would you set up to solve for '(r, µ) outside the crater? Numerical results
show that E for the crater problem diÆers from E0 by the field of a dipole placed at the same
point as in part (a). However, the dipole moment is reversed in direction and has a magnitude
only 1/10 as large as the bump problem. Rationalize both of these results qualitatively.
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
z
R
(a)
7.23
R
(b)
A Conducting Slot
The figure shows an infinitely long and deep slot formed by two grounded conductor plates at x = 0
and x = a and a conductor plate at z = 0 held at a potential '0 . Find the potential inside the slot.
" !0
z
" ! "0
x!a
x!0
7.24
x
A Corrugated Conductor
A flat metal plate occupies the z = 0 plane. When raised to a non-zero potential '0 , the plate
develops a uniform surface charge density æ0 and a uniform field E0 = (æ0 /2≤0 )ˆ
z in the space z > 0.
(a) Corrugate the plate slightly so z(x) = b sin kx with kb ø 1 describes the free surface. Demand
that E(z ! 1) ! E0 and show that the charge density induced on the metal surface is
æ(x) º æ(0)[1 + kz(x)].
(b) Discuss the behavior of æ(x) at the peaks and valleys of the surface in connection with the
results of Section 7.10.
7.25
Unisphere Potential
Let '0 be the value of the potential applied to the metallic Unisphere in Section 6.8.1. Outline
a procedure (other than direct integration of the Coulomb integral) which gives the potential at
every point in space. The procedure may be partly numerical.
7.26
Potential of a Cylindrical Capacitor
An infinitely long conducting tube (radius Ω1 ) is held at potential '1 . A second, concentric tube
(radius Ω2 > Ω1 ) is held at potential '2 . Integrate Laplace’s equation and find the capacitance per
unit length.
7.27
Axially Symmetric Potentials
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
Let V (z) be the potential on the axis of an axially symmetry electrostatic potential in vacuum.
Show that the potential at any point in space is
V (Ω, z) =
1
º
Zº
d≥ V (z + iΩ cos ≥).
0
Hint: Show that the proposed solution satisfies Laplace’s equation and exploit uniqueness.
7.28
A Segmented Cylinder
The figure below is a cross section of an infinite, conducting cylindrical shell. Two infinitesimally
thin strips of insulating material divide the cylinder into two segments. One segment is held at
unit potential. The other segment is held at zero potential. Find the electrostatic potential inside
the cylinder. Hint:
Z
2º
(m 6= 0)
d¡ cos m¡ cos n¡ = º±mn
0
y
M 1
D
D
R
M 0
7.29
x
A 2D Potential Problem in Cartesian Coordinates
Two flat conductor plates (infinite in the x and y directions) occupy the planes z = ±d. The x > 0
portion of both plates is held at ' = +'0 . The x < 0 portion of both plates is held at ' = °'0 .
Derive an expression for the potential between the plates using a Fourier integral to represent the
x variation of '(x, z).
z
d
!M0
x
M0
!d
7.30
Target Field in a Dielectric Sphere
An origin-centered sphere with radius R and dielectric constant ∑1 is embedded in an infinite
medium with dielectric constant ∑2 . The electric field inside the sphere is
E1 = (V0 /R2 )(zˆ
x + xˆ
z).
(a) Find the electric field outside the sphere, E2 (x, y, z), assuming that E2 ! 0 as r ! 1.
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
(b) Calculate the density of charge (free or polarization) at the interface between the two media.
7.31
The Two-Cylinder Electron Lens
Two semi-infinite, hollow cylinders of radius R are coaxial with the z-axis. Apart from an insulating
ring of thickness d ! 0, the two cylinders abut one another at z = 0 and held at potentials VL and
VR . Find the potential everywhere inside both cylinders. You will need the integrals
Z 1
Z 1
∏
ds s J0 (∏s) = J1 (∏)
and
2
ds s J0 (xn s)J0 (xm s) = J12 (xn )±nm .
0
0
The real numbers xm satisfy J0 (xm ) = 0.
d
VL
R
7.32
VR
Contact Potential
The x > 0 half of a conducting plane at z = 0 is held at zero potential. The x < 0 half of the plane
is held at potential V . A tiny gap at x = 0 prevents electrical contact between the two halves.
z
$
#
x
! "0
! "V
(a) Use a change of scale argument to conclude that the z > 0 potential '(Ω, ¡) in plane polar
coordinates cannot depend on the radial variable Ω.
(a) Find the electrostatic potential in the z > 0 half-space.
(b) Make a semi-quantitative sketch of the electric field lines and use words to describe the most
important features.
7.33
Circular Plate Capacitor
Consider a parallel plate capacitor with circular plates of radius a separated by a distance 2L.
z
a
"V
2L
!
#V
c 2009 Andrew Zangwill
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Chapter 7
Laplace’s Equation: The Potential Produced by Surface Charge
A paper published in 1983 proposed a solution for the potential for this situation of the form
'(Ω, z) =
Z1
dk A(k)f (k, z)J0 (kΩ),
0
where J0 is the zero-order Bessel function and
A(k) =
2V
sin(ka)
.
1 ° e°2kL ºk
(a) Find the function f (k, z) so the proposed solution satisfies the boundary conditions on the
surfaces of the plates. You may make use of the integral
8
Z1
0∑Ω∑a
< º/2
sin(ka)
dk
J0 (kΩ) =
:
k
sin°1 (a/Ω)
Ω ∏ a.
0
(b) Show that the proposed solution nevertheless fails to solve the problem because the electric
field it predicts is not a continuous function of z when Ω > a.
7.34
A Slightly Dented Spherical Conductor
The surface of a slightly dented spherical conductor is given by the equation r = a[1 + ≤PN (cos µ)]
where ≤ ø 1. Let the conductor be grounded and placed in a constant electric field E0 parallel to
the polar axis, Show that the induced surface charge density is
æ(µ) = æ0 + ≤
Ω
3N ≤0 E
[(N + 1)PN +1 (cos µ) + (N ° 2)PN °1 (cos µ)]
2N + 1
æ
where æ0 is the induced charge density for ≤ = 0. Along the way, confirm and use the fact that the
@Pn ˆ
ˆ =ˆ
normal to the surface is n
r°≤
µ + O(≤2 ). Hint: (2N + 1)PN (x)P1 (x) = N PN °1 (x) + (N +
@µ
1)PN +1 (x).
7.35
A Conducting Duct
Solve the conducting duct problem treated in Section 7.5.1 using the method indicated in the
penultimate paragraph of that section.
7.36
The Force on an Inserted Conductor
A set of known constants Æn parameterizes the potential in a volume r < a as
'ext (r, µ) =
1
X
Æn
n=1
≥ r ¥n
R
Pn (cos µ).
ˆ point along µ = 0 and insert a solid conducting sphere of radius R < a at the origin. Show
Let z
that the force exerted on the sphere when it is connected to ground is in the z direction and
Fz = 4º≤0
1
X
(n + 1)Æn Æn+1 .
n=1
Hint: The Legendre polynomials satisfy (n + 1)Pn+1 (x) + nPn°1 (x) = (2n + 1)xPn (x).
c 2009 Andrew Zangwill
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