The problem of two unequal electrodes within a shielding cavity in the presence of a dielectric partially filling the gap is treated by means of overrelaxation methods in the axisymmetric case. A discrepancy in the literature concerning the boundary equation between two dielectrics is resolved and the difference equation for the dielectric corner point is obtained by means of Gauss’ law. Some numerical results are described.
REFERENCES
1.
G.
Kirchhoff
, Gesammelte Abhandlungen
(Johann Ambrosius Barth
, Leipzig
, 1882
), pp. 101
–117
.2.
H. J.
Wintle
and S.
Kurylowicz
, “Edge corrections for strip and disc capacitors
,” IEEE Trans. Instrum. Meas.
IM-34
(1
), 41
–47
(1985
).3.
G. J.
Sloggett
, N. G.
Barton
, and S. J.
Spencer
, “Fringing fields in disc capacitors
,” J. Phys. A
19
, 2725
–2736
(1986
);G. J.
Sloggett
, N. G.
Barton
, and S. J.
Spencer
,also “Addendum to ‘Fringing fields in disc capacitors’
,” J. Phys. A
20
, 4061
–4062
(1987
).4.
G. W.
Parker
, “What is the capacitance of parallel plates?
” Comput. Phys.
5
, 534
–540
(1991
).5.
G. T.
Carlson
and B. L.
Illman
, “The circular disk parallel plate capacitor
,” Am. J. Phys.
62
, 1099
–1105
(1994
).6.
G. W.
Parker
, “Electric field outside a parallel plate capacitor
,” Am. J. Phys.
70
(5
), 502
–506
(2002
).7.
T. T.
Grove
, M. F.
Masters
, and R. E.
Miers
, “Determining dielectric constants using a parallel plate capacitor
,” Am. J. Phys.
73
(1
), 52
–56
(2005
).8.
T. V.
Rao
, “Capacity of the circular plate condenser: Analytical solutions for large gaps between the plates
,” J. Phys. A
38
, 10037
–10056
(2005
).9.
A. H.
Scott
and H. L.
Curtis
, “Edge correction in the determination of dielectric constant
,” J. Res. Natl. Bur. Stand.
22
, 747
–775
(1939
).10.
N.
Sneddon
, Mixed Boundary Value Problems in Potential Theory
(Wiley
, New York
, 1966
), p. 240
.11.
A.
Naini
and Mark
Green
, “Fringing fields in a parallel-plate capacitor
,” Am. J. Phys.
45
(9
), 877
–879
(1977
).12.
F.
Thompson
and J. K.
Gagon
, “Fringe capacitance of a parallel-plate capacitor
,” Phys. Educ.
17
, 80
–82
(1982
).13.
S.
Burt
, N.
Finney
, and J.
Young
, “Fringe field of parallel plate capacitor
,” AAPT 129th National Meeting
, Sacramento, CA, 31 Jul–4 August 2004
; paper available from Professor Younes Ataiiyan, Santa Rosa Junior College, Department of Engineering and Physics, 1501 Mendoncino Ave., Santa Rosa, CA 95401.14.
D. F.
Bartlett
, P. E.
Goldhagen
, and E. A.
Phillips
, “Experimental test of Coulomb’s law
,” Phys. Rev. D
2
, 483
–487
(1970
).15.
J. D.
Jackson
, “A curious and useful theorem in two-dimensional electrostatics
,” Am. J. Phys.
67
(2
), 107
–115
(1998
).16.
H.
Houtman
, F. W.
Jones
, and C. J.
Kost
, “Laplace and Poisson equation solution by RE-LAX3D
,” Comput. Phys.
8
(4
), 469
–479
(1994
).17.
F.
Pinto
, “Analytical and experimental investigation on a vibrating annular membrane attached to a central free, rigid core
,” J. Sound Vib.
291
, 1278
–1287
(2006
).18.
J.
Conway
and A. P.
Anderson
, “Electromagnetic techniques in hyperthermia
,” Clin. Phys. Physiol. Meas.
7
(4
), 287
–318
(1986
).19.
L. D.
Landau
and E. M.
Lifshitz
, Electrodynamics of Continuous Media
(Butterworth-Heinemann
, Oxford
, 1998
), Sec. 7.20.
M.
DiStasio
and W. C.
McHarris
, “Electrostatic problems? Relax!
” Am. J. Phys.
47
(5
), 440
–444
(1979
).21.
E. M.
Purcell
, Electricity and Magnetism
(McGraw-Hill
, Boston
, 1985
); Prob. 3.30 and Chap. 10.22.
W. H.
Press
, S. A.
Teukolski
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes in Fortran
(Cambridge U. P.
, Cambridge
, 1986
), pp. 854
–860
.23.
J. C.
Strikwerda
, Finite Difference Schemes and Partial Differential Equations
(SIAM
, Philadelphia
, 2004
), pp. 325
–361
.24.
W. M.
MacDonald
, “Discretization and truncation errors in a numerical solution of Laplace’s equation
,” Am. J. Phys.
62
(2
), 169
–173
(1994
).25.
J. A.
Harrison
, “A computer study of uniform field electrodes
,” Br. J. Appl. Phys.
18
, 1617
–1627
(1967
).26.
C.
Weber
, “Numerical solutions of Laplace’s and Poisson’s equations and the calculation of electron trajectories and electron beams
,” in Focusing of Charged Particles
, edited by A.
Septier
(Academic
, New York
, 1967
), Vol. 1
, pp. 45
–162
.27.
D. F.
Bartlett
and T. R.
Corle
, “The circular parallel plate capacitor: A numerical solution for the potential
,” J. Phys. A
18
, 1337
–1342
(1985
).28.
J. A.
Seeger
, “Solution of Laplace’s equation in a multidielectric region
,” Proc. IEEE
56
, 1393
–1394
(1968
). There is an important Errata in Ref. 28; in line 3 of Eq. (3) there should be a factor of 4 in front of the ratio of dielectric constants, and in line 1 of Eq. (4) and should be interchanged.29.
J. E.
Boers
, “Digital computer solution of Laplace’s equation including dielectric surfaces
,” Mathematics Notes, SC-RR-69–446, Research Report, Sandia Laboratories, December 1969
.30.
T. T.
Crow
, “Solutions to Laplace’s equation using spreadsheets on a personal computer
,” Am. J. Phys.
55
(9
), 817
–823
(1987
).31.
M. V. K.
Chari
and S. J.
Salon
, Numerical Methods in Electromagnetism
(Academic
, San Diego
, 2000
), pp. 109
–111
.32.
J. C.
Maxwell
, A Treatise on Electricity and Magnetism
(Dover
, New York
, 1954
), Art. 201, pp. 308
–309
.33.
R. A.
Kromhout
and W. G.
Moulton
, “Effect of fringing field on capacitance and measurement of
,” Am. J. Phys.
24
(9
), 631
–632
(1956
).34.
N. N.
Lebedev
, “The electric field at the edge of a plane condenser containing a dielectric
,” Sov. Phys. Tech. Phys.
3
, 1234
–1243
(1958
).35.
I. M.
Minkov
, “Electrostatic field of a condenser with dielectric insert
,” Sov. Phys. Tech. Phys.
5
, 1143
–1146
(1961
).36.
K.
Asano
, A. W.
Dipert
, and C. D.
Hendricks
, “A note on the uniform field approximation when guard rings are used
,” Am. J. Phys.
38
(12
), 1452
–1454
(1970
).37.
D. A.
Hastings
, “Computational methods for electrical potential and other field problems
,” Am. J. Phys.
43
(6
), 518
–524
(1975
).38.
F. X.
Hart
, “Validating spreadsheet solutions to Laplace’s equation
,” Am. J. Phys.
57
(11
), 1027
–1034
(1989
).39.
P. W.
Gash
, “Improved numerical solutions of Laplace’s equation
,” Am. J. Phys.
59
(6
), 509
–515
(1991
).40.
C. F.
Gauss
, Brief an Gerling, Werke, Vol. 9 (1823
), translated by G. Forsythe in “Gauss to Gerling on Relaxation
,” Math. Tables Aids Comput.
5
, 255
–258
(1951
).41.
D. M.
Young
, Iterative Solutions of Large Linear Systems
(Dover
, Mineola, NY
, 1971
), Chap. 6.42.
R. S.
Varga
, Matrix Iterative Analysis
(Springer
, Berlin
, 2000
), pp. 301
–304
and Chap. 9.43.
R. V.
Southwell
, Relaxation Methods in Engineering Science
(Clarendon
, Oxford
, 1940
), pp. 233
–243
.44.
R. V.
Southwell
, Relaxation Methods in Theoretical Physics
(Clarendon
, Oxford
, 1946
), pp. 20
–24
and 89
–92
.45.
G. E.
Forsythe
and W. R.
Wasow
, Finite-Difference Methods for Partial Differential Equations
(Dover
, Mineola, NY
, 2004
), pp. 241
–283
.46.
R.
Barrett
, M.
Berry
, T. F.
Chan
, J.
Demmel
, J.
Donato
, J.
Dongarra
, J. V.
Eijkhout
, R.
Pozo
, C.
Romine
, and H. V.
der Vorst
, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
(SIAM
, Philadelphia
, 1994
), pp. 12
–13
.47.
R.
Haberman
, Applied Partial Differential Equations
(Pearson Prentice Hall
, Upper Saddle River, NJ
, 2004
), pp. 260
–266
.48.
J. W.
Thomas
, Numerical Partial Differential Equations, Conservation Laws and Elliptic Equations
(Springer
, New York
, 1999
), Chap. 10, especially Sec. 10.11.49.
G. D.
Smith
, Numerical Solution of Partial Differential Equations
(Clarendon
, Oxford
, 1985
).50.
J. P.
McKelvey
, “Electrostatic fields in inhomogeneous dielectrics
,” Am. J. Phys.
56
(8
), 713
–718
(1988
).51.
P.
Lorraine
, D. P.
Corson
, and F.
Lorraine
, Electromagnetic Fields and Waves
(Freeman
, New York
, 1988
).© 2007 American Association of Physics Teachers.
2007
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