Problems
Application 5.2 Coulomb Blockade
5.7 Real Conductors
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Next, define
1 2 = e2
0
∂n0
∂μ (5.88)
and substitute (5.87) into (5.85) to get
∇2ϕ(r)−ϕ(r)
2 = −qδ(r). (5.89)
It is not difficult to solve (5.89) if we focus on points away from the origin, where (5.89) simplifies to
∇2ϕ(r)−ϕ(r)
2 =0. (5.90)
A point charge is spherically symmetric. Therefore,ϕ(r)=ϕ(r) and a brief calculation confirms that (5.90) simplifies to
d2(rϕ) dr2 −rϕ
2 =0. (5.91)
The solution to (5.91) which satisfies the physical boundary conditions ϕ(r→ ∞)=0 and ϕ(r→0)=q 4π 0ris
ϕ(r)= q
4π 0rexp(−r ). (5.92)
It is not difficult to check using (5.85) that the electron density implied by (5.92) satisfies
d3r n(r)= −q. (5.93)
We conclude from (5.93) that the mobile negative charge attracted toward the origin exactly compen- sates the positive charge of the point impurity. The decaying exponential in (5.92) makes it clear that the electric field is essentially zero at any distance greater than aboutfrom the point charge.13For this reason,is called thescreening length.
For a sample of finite size, Gauss’ law provides a boundary condition at the conductor surface that can be used with (5.90) to confirm our assertion that the positively charged layer in Figure 5.16 extends a distanceinto the conductor. Similarly, if we replace the point charge by a uniform, static, external electric field, one finds from (5.90) that the electric field penetrates a distanceinto the bulk of the conductor. All of this shows that, for a sample with a characteristic sizeD, the ratio Dmeasures how nearly “perfectly” the conductor responds to electrostatic perturbations.
5.7.1 Debye-H¨uckel and Thomas-Fermi Screening
The qualitative physics of the screening layer lies in the proportionality between ∂n0 ∂μand the compressibility quoted in footnote 12. By definition, a perfect conductor has zero screening length.
Therefore, according to (5.88), this system is infinitely compressible in the sense that it costs no energy to squeeze all the screening charge into an infinitesimally thin surface layer. By contrast, the screening charge in a classical thermal plasma (like the cytoplasm of a cell or a doped semiconductor) has finite compressibility because the charges gain configurational entropy by spreading out in space.
The screening electrons in a metal similarly resist compression because, as the quantum mechanical particle-in-a-box problem teaches us, the kinetic energy of the electrons increases as the volume of the box decreases. For these reasons, the infinitesimally thin surface distributions of a perfect conductor broaden into the screening layers illustrated in Figure 5.16.
13 Compare the exponential in (5.92) with the exponential in (2.63).
Sources, References, and Additional Reading 151
Quantitatively, we need the functionn0(μ) to computefrom (5.88). For our “classical” examples, it is a standard result of Boltzmann statistics that the particle density depends on chemical potential as
n0(μ)= (2π mkT)3/2 h3 exp
μ kT
. (5.94)
The associated screening length calculated from (5.88) is DH =
)0kT
e2n . (5.95)
The subscript “DH” in (5.95) honors P. Debye and E. H¨uckel, who identified this quantity in their analysis of screening in electrolytes. Typically,DH∼10−100 ˚A in such solutions and also in doped semiconductors. For a metal, Fermi statistics atT =0 gives the dependence of the particle density on chemical potential as
n0(μ)= 8π 3
2mμ h2
3/2
. (5.96)
Using (5.96) in (5.88) gives the screening length, TF=
)π aB
4kF
, (5.97)
whereμ=−h2k2F/2mandaB=4π 0−h2/me2. Here, the subscript “TF” remembers L.H. Thomas and E. Fermi because the approximations of this section were used by them in a statistical theory of atomic structure. In a good metal,TF∼1 ˚A.
Sources, References, and Additional Reading
The Cavendish quotation at the beginning of the chapter is taken from Section 98 of “An attempt to explain some of the principal phenomena of electricity by means of an elastic fluid”,Philosophical Transactions61, 584 (1771).
The article is reprinted (with a penetrating commentary) in
J.C. Maxwell,The Electrical Researches of the Honourable Henry Cavendish(Frank Cass, London, 1967).
Section 5.1 Textbooks that discuss the physics of conductors without an immediate plunge into boundary value potential theory include
V.C.A. Ferraro,Electromagnetic Theory(Athlone Press, London, 1954).
W.T. Scott,The Physics of Electricity and Magnetism,2ndedition (Wiley, New York, 1966).
R.K. Wangsness,Electromagnetic Fields, 2ndedition (Wiley, New York, 1986).
In the course of his study of thermal fluctuations, Einstein considered the effect of a heat bath on the charge carried by the surface of a conductor. He proposed to use amplification by successive steps of electrostatic induction to observe the tiny changes in charge predicted by his theory. The story is told in detail in
D. Segers and J. Uyttenhove, “Einstein’s ‘little machine’ as an example of charging by induction”,American Journal of Physics74, 670 (2006).
Section 5.2 Application 5.1 is taken from the surprisingly readable
G. Green,An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828). Facsimile edition by Wez¨ata-Melins Aktieborg, G¨oteborg, Sweden (1958), Section 12.
Techniques used to findσ(rS)for a conducting disk different from the one used in Application 5.1 include (i) solving Laplace’s equation in ellipsoidal coordinates; (ii) solving an integral equation; and (iii) a purely geometric approach. These methods are discussed, respectively, in
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152 CONDUCTING MATTER: ELECTROSTATIC INDUCTION AND ITS CONSEQUENCES
J.A. Stratton,Electromagnetic Theory(McGraw-Hill, New York, 1941).
L. Egyes,The Classical Electromagnetic Field(Dover, New York, 1972).
A.M. Portis,Electromagnetic Fields: Sources and Media(Wiley, New York, 1978).
Section 5.3 A pedagogical discussion of electrostatic shielding which complements the one given in the text is S.H. Yoon, “Shielding by perfect conductors: An alternative approach”,American Journal of Physics71, 930 (2003).
Section 5.4 The approximate self-capacitance formula (5.41) and the exact formulae graphed in Figure 5.6 are discussed, respectively, in
Y.L. Chow and M.M. Yovanovich, “The shape factor for the capacitance of a conductor”,Journal of Applied Physics53, 8470 (1982).
L.D. Landau and E.M. Lifshitz,The Electrodynamics of Continuous Media(Pergamon, Oxford, 1960).
Example 5.4 is a simplified version of a problem analyzed in
K.K. Likharev, N.S. Bakhvalov, G.S. Kazacha, and S.I. Serdyukova, “Single-electron tunnel junction array”,IEEE Transactions on Magnetics25, 1436 (1989).
The boxed material after Example 5.4 comes from
A. Marcus, “The theory of the triode as a three-body problem in electrostatics”,The American Physics Teacher 7, 196 (1939).
The capacitance matrix is important to the design of very large scale integrated (VLSI) circuits. A typical example is Z.-Q. Ning, P.M. Dewilde, and F.L. Neerhoff, “Capacitance coefficients for VLSI multilevel metallization lines”, IEEE Transactions on Electron Devices34, 644 (1987).
Section 5.5 Entry points to learn about Coulomb blockade and quantum dots are
H. Grabert and M.H. Devoret,Coulomb Blockade Phenomena in Nanostructures, NATO ASI Series B, vol- ume 294 (Plenum, New York, 1992).
M.W. Keller, A.L. Eichenberger, J.M. Martinis, and N.M. Zimmerman, “A capacitance standard based on counting electrons”,Science285, 1707 (1999).
Section 5.6 Our “thermodynamic” approach to the forces exerted on conductors was inspired by the masterful but characteristically terse discussion in Section 5 ofLandau and Lifshitz(see Section 5.4 above).
Examples 5.5 and 5.6 were adapted, respectively, from
O.D. Jefimenko,Electricity and Magnetism(Appleton-Century-Crofts, New York, 1966).
W.M. Saslow,Electricity, Magnetism, and Light(Academic, Amsterdam, 2002).
Section 5.7 An excellent introduction to the screening response of “real” conductors (both classical and quan- tum) to static and non-static perturbations is
J.-N. Chazalviel,Coulomb Screening by Mobile Charges(Birkh¨auser, Boston, 1999).
Problems
5.1 A Conductor with a Cavity A solid conductor has a vacuum cavity of arbitrary shape scooped out of its interior. Use Earnshaw’s theorem to prove thatE=0 inside the cavity.
5.2 Two Spherical Capacitors A spherical conducting shell with radiusbis concentric with and encloses a conducting ball with radiusa. Compute the capacitanceC=Q/ϕwhen
(a) the shell is grounded and the ball has chargeQ.
(b) the ball is grounded and the shell has chargeQ.
5.3 Concentric Cylindrical Shells A capacitor is formed from three very long, concentric, conducting, cylindrical shells with radiia < b < c. Find the capacitance per unit length of this structure if a fine wire connects the inner and outer shells andλbis the uniform charge per unit length on the middle cylinder.
Problems 153 5.4 A Charged Sheet between Grounded Planes Two infinite conducting planes are held at zero potential at z= −dandz=d. An infinite sheet with uniform charge per unit areaσ is interposed between them at an arbitrary point.
(a) Find the charge density induced on each grounded plane and the potential at the position of the sheet of charge.
(b) Find the force per unit area which acts on the sheet of charge.
0 z d d
0 0
5.5 The Charge Distribution Induced on a Neutral Sphere A point chargeqlies a distancer > Rfrom the center of an uncharged, conducting sphere of radiusR. Express the induced surface charge density in the form
σ(θ)= ∞
=1
σP(cosθ)
whereθis the polar angle measured from a positivez-axis which points from the sphere center to the point charge.
(a) Show that the total electrostatic energy is UE = 1
0
∞
=0
σ
2+1 R3σ
2 4π
2+1+q R+2 r+1
.
(b) Use Thomson’s theorem to findσ(θ).
5.6 Charge Transfer between Conducting Spheres A metal ball with radiusR1 has chargeQ. A second metal ball with radiusR2 has zero charge. Now connect the balls together using a fine conducting wire.
Assume that the balls are separated by a distanceRwhich is large enough that the charge distribution on each ball remains uniform. Show that the ball with radiusR1possesses a final charge
Q1= QR1
R1+R2
1+ (R1−R2)R2
(R1+R2)R
.
5.7 Concentric Spherical Shells Three concentric spherical metallic shells with radiic > b > ahave charges ec,eb, andea, respectively. Find the change in potential of the outermost shell when the innermost shell is grounded.
5.8 Don’t Believe Everything You Read in Journals A research paper published in the journalApplied Physics Lettersdescribes experiments performed with three identical spherical conductors suspended from above by insulating wires so a (fictitious) horizontal plane passes through the center of all three spheres. It was reported that a large voltageV applied to one sphere induced equal and opposite rotation in the two isolated spheres (see top view below). The authors suggested that the isolated spheres were set into motion by electrostatic torque. Show, to the contrary, that this torque is zero.
V
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5.9 A Dipole in a Cavity A point electric dipole with momentpis placed at the center of a hollow spherical