We investigate the restoring force on a dielectric slab being pulled from within the volume of a parallel plate capacitor connected to a battery. Using a conformal mapping to treat the fringing electric field exactly, we numerically obtain an expected Hooke's Law restoring force for small displacements, and a diminishing force for a displacement up to half the length of the dielectric.
REFERENCES
1.
S.
Margulies
, “Force on a dielectric slab inserted into a parallel-plate capacitor
,” Am. J. Phys.
52
(6
), 515
–518
(1984
). The author presents a physical analysis showing that the edge effects are responsible for the restoring force. A detailed calculation of the effect is not included.2.
R.
Serway
and J. W.
Jewett
,Jr. , Physics for Scientists and Engineers with Modern Physics
, 7th ed. (Brooks Cole
, USA
, 2008
), chapter 26, problem 56, page 750
.The usual incorrect result is given in
Ralph V.
McGrew
, Instructor's Solutions Manual for Serway and Jewett's Physics for scientists and Engineers with modern Physics
, 7th ed., Vol. 2
(Brooks
, Cole, USA
, 2008
), pp. 93
–94
.3.
Eric R.
Dietz
, “Force on a dielectric slab: Fringing field approach
,” Am. J. Phys.
72
(12
), 700
–701
(2004
) has calculated the force for the single case that the dielectric is precisely half removed. The symmetry of the arrangement allows for the result to be obtained exactly. The method illustrates the central role of the fringing field.4.
J. C.
Maxwell
, A Treatise on Electricity and Magnetism
, 3rd ed., Vol. 1 (reprinted in the USA by Oxford U.P. in the Oxford Classic Series
, 1998
), Chap. XII, pp. 309
–310
.W.
Rogowski
, “Rogowski extended Maxwell's sketch of the method
,” Archiv fur Electrotechnik.
12
, 1
–8
(1923
).5.
See supplementary material at http://dx.doi.org/10.1119/1.4875058 for the executable Mathematica notebook (as well as a .pdf version) entitled “Conformal Capacitor”. The user is warned that the calculations takes about half an hour to run on a Dell Latitude E6510 with an Intel i7 processor.
6.
G. L.
Pollack
and D. R.
Stump
, Electromagnetism
(Addison-Wesley
, USA
, 2002
), pp. 163
–171
. The mapping is example 9.7.
Michael P.
Bradley
, “Edge Effects in the Parallel Plate Capacitor: The Maxwell Transformation and the Rogowski Profile,” 2004 lecture, University of Saskatchewan. This lecture is available, with permission from the author, at the supplementary material link in Ref. 5.8.
R. V.
Churchill
, Complex Variables and Applications
, 2nd ed. (McGraw-Hill
, USA
, 1960
), pp. 174
–217
. The transformation considered here is example 18 on page 217.9.
J. C.
Maxwell
, ibid, appendix PLATES, Fig. XII.10.
This particular result appears either not to have been noticed previously, or if noticed, not to have been exploited, probably for lack of the necessary computational resources.
11.
A Mathematica Module is the equivalent of a subroutine in many computer languages.
12.
For the convenience of the interested reader, the points plotted here are: (0,0), (0.005, −0.00366405), (0.015, −0.01099192), (0.025, −0.01831908), (0.035, −0.02564507), (0.045, −0.03296942), (0.055, −0.04029165), (0.065, −0.04761131), (0.075, −0.05492793), (0.085, −0.06224104), and (0.095, −0.06955018). Complete numerical results are given in Ref. 5 above.
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