Science

Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge

7.13 Problems

7.1 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,

I 0Æ(y)KÆ(y)° IÆ(y)K0Æ(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 S

dS ‘(r) = 4ºR2′(0)

(b)

Z S

dSz'(r) = 4º 3

R4 @’ @z

ØØØØ r=0

7.4 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 R3

x̂ + V0 R3

(y2 ° x2)ŷ ° V0 R

ẑ.

There is no other charge anywhere. Express your answer in terms of trigonometric functions of µ and ¡.

7.5 Green’s Formula

Let n̂ be the normal to an equipotential surface at a point P . If R1 and R2 are the principal radii of curvature of the surface at P . A formula due to George Green relates normal derivatives (@/@n ¥ n̂ ·r) of the potential ‘(r) (which satisfies Laplace’s equation) at the equipotential surface to the mean curvature of that equipotential surface ∑ = 12 (R

°1 1 + R

°1 2 ):

@2′ @n2

+ 2∑ @’ @n

= 0.

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 po- tential 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.

1 1

02 2

x

y d

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).

1 2

4 3

r

x

O

1 2

(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(¡) = ≤0

2ºR

∑ sin( 12Æ + ¡)

1° cos( 12Æ + ¡) +

sin( 12Æ° ¡) 1° cos( 12Æ° ¡)

∏ .

(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

C13 = ° ≤0 º

ln 2.

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

2 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 12V 00(z)Ω Ω̂ ° V 0(z)ẑ

for points near the symmetry axis where Ω ø p

|V 0(z)/V 000(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

Ω̈ = z̈Ω0 + ż2Ω00 = q

2m ΩV 00(z).

(c) Use Newton’s second law and an approximate form of conservation of energy (valid when vz is large) to derive the trajectory equation

d2Ω dz2

+ 1 2

V 0

V dΩ dz

+ Ω 4

V 00

V = 0.

(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,

d2R dz2

= ° 3 16

R(z)

∑ V 0(z) V (z)

∏2 .

(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.

2 !

1 ”

2 ”

0″ #

2 V$#

1 !

1 V$#

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Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge

7.11 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.

x

VV

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

+ ∫ @’ @∫

+ @2′ @¡2

= 0

where ∫ = tan 12µ .

1. Separate variables and show that

‘(µ, ¡) = 4V º

1X 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 ° 12z 2 ± 13z

3 ° 14z 4 + · · · to sum the series and show

that

‘ = 4V º

tan°1 Ω

2 tan 12µ tan 1 2Æ

tan2 12µ ° tan2 1 2Æ

cos ¡

æ .

7.12 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(kmR) = 0, show that the potential inside the tube is

‘(Ω, z) = 2V R

1X m=1

exp(°kmz) km

J0(kmΩ) J1(kmR)

.

7.13 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 sẑ 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 R1R2(V2 ° V1)

R2 °R1

∑ 1

R21 ° 3s

R32 °R31 cos µ

∏ .

(b) To lowest order in s, show that the force exerted on the inner shell is

F =

Z dS

æ2

2≤0 n̂ = ẑ2ºR21

ºZ 0

dµ sin µ æ2(µ) 2≤0

cos µ = ° Q 2

4º≤0

sẑ R32 °R31

.

(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.

4

0

V

(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 + Fxz + Gyz + Hx2 + Iy2 + Jz2.

(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

1 2 d

V 0

y

x

a

(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 + Ey2 + Fxy.

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 2a

d

(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 |y

S

|=R dyS

‘̄(yS) |r ° yS |3

|r| < R

gives the potential at any point r inside a sphere if we specify the potential ‘̄(yS) 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 |r ° yS |

= 1X

`=0

r`

R`+1 P`(r̂ · ŷS).

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.

0

0

V

V

0

R

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

1

2

(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)ẑ 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

R

R( )a ( )b

z

7.23 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.

0x! x a!

0″ “!

0” !

z

x

7.24 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)ẑ in the space z > 0.

(a) Corrugate the plate slightly so z(x) = b sin kx with kbø 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

ºZ 0

d≥ V (z + iΩ cos ≥).

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

0

d¡ cos m¡ cos n¡ = º±mn (m 6= 0)

D DR

y

x

1M

0M

7.29 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).

x

z d

d! 0M! 0M

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/R 2)(zx̂ + xẑ).

(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 0

dssJ0(∏s) = J1(∏) and 2

Z 1 0

ds s J0(xns)J0(xms) = J 2 1 (xn)±nm.

The real numbers xm satisfy J0(xm) = 0.

R

d

L V

R V

7.32 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.

0!”V! ”

#

$

x

z

(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

!2L

aV”

V#

c∞ 2009 Andrew Zangwill 287

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) =

1Z 0

dk A(k)f(k, z)J0(kΩ),

where J0 is the zero-order Bessel function and

A(k) = 2V

1° e°2kL sin(ka)

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

1Z 0

dk sin(ka)

k J0(kΩ) =

8<: º/2 0 ∑ Ω ∑ a sin°1(a/Ω) Ω ∏ a.

(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≤0E 2N + 1

[(N + 1)PN+1(cos µ) + (N ° 2)PN°1(cos µ)] æ

where æ0 is the induced charge density for ≤ = 0. Along the way, confirm and use the fact that the

normal to the surface is n̂ = r̂° ≤@Pn @µ

µ̂ + O(≤2). Hint: (2N + 1)PN (x)P1(x) = NPN°1(x) + (N +

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) = 1X

n=1

Æn ≥ r

R

¥n Pn(cos µ).

Let ẑ point along µ = 0 and insert a solid conducting sphere of radius R < a at the origin. Show that the force exerted on the sphere when it is connected to ground is in the z direction and

Fz = 4º≤0

1X n=1

(n + 1)ÆnÆn+1.

Hint: The Legendre polynomials satisfy (n + 1)Pn+1(x) + nPn°1(x) = (2n + 1)xPn(x).

c∞ 2009 Andrew Zangwill 288

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