Table Of ContentCasimir Force in Non-Planar Geometric
Configurations
Sung Nae Cho
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Physics
Tetsuro Mizutani, Chair
John R. Ficenec
Harry W. Gibson
A. L. Ritter
Uwe C. Tauber
April 26, 2004
Blacksburg, Virginia
Keywords: Casimir Effect, Casimir Force, Dynamical Casimir Force, Quantum
Electrodynamics (QED), Vacuum Energy
Copyright (cid:176)c 2004, Sung Nae Cho
Casimir Force in Non-Planar Geometric Configurations
Sung Nae Cho
(ABSTRACT)
The Casimir force for charge-neutral, perfect conductors of non-planar geometric configurations have been investi-
gated. Theconfigurationswere: (1)theplate-hemisphere,(2)thehemisphere-hemisphereand(3)thesphericalshell.
TheresultingCasimirforcesforthesephysicalarrangementshavebeenfoundtobeattractive. TherepulsiveCasimir
forcefoundbyBoyerforasphericalshellisaspecialcaserequiringstringentmaterialpropertyofthesphere,aswell
as the specific boundary conditions for the wave modes inside and outside of the sphere. The necessary criteria in
detectingBoyer’srepulsiveCasimirforceforaspherearediscussedattheendofthisthesis.
Acknowledgments
I would like to thank Professor M. Di Ventra for suggesting this thesis topic. The continuing support and encour-
agement from Professor J. Ficenec and Mrs. C. Thomas are gracefully acknowledged. Thanks are due to Professor
T. Mizutani for fruitful discussions which have affected certain aspects of this investigation. Finally, I express my
gratitudeforthefinancialsupportoftheDepartmentofPhysicsofVirginiaPolytechnicInstituteandStateUniversity.
Contents
Abstract ii
Acknowledgments iii
ListofFigures vi
1. Introduction 1
1.1. Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. CasimirEffect 5
2.1. QuantizationofFreeMaxwellField . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Casimir-PolderInteraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. CasimirForceCalculationBetweenTwoNeutralConductingParallelPlates . . . . . . . . . . . . . . 11
2.3.1. Euler-MaclaurinSummationApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2. VacuumPressureApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3. TheSourceTheoryApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. ReflectionDynamics 18
3.1. ReflectionPointsontheSurfaceofaResonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2. SelectedConfigurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1. HollowSphericalShell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2. Hemisphere-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.3. Plate-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3. DynamicalCasimirForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1. FormalismofZero-PointEnergyanditsForce . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2. EquationsofMotionfortheDrivenParallelPlates . . . . . . . . . . . . . . . . . . . . . . . 31
4. ResultsandOutlook 34
4.1. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.1. HollowSphericalShell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2. Hemisphere-HemisphereandPlate-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2. InterpretationoftheResult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3. SuggestionsontheDetectionofRepulsiveCasimirForceforaSphere . . . . . . . . . . . . . . . . . 41
4.4. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.1. Sonoluminescense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.2. CasimirOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
AppendicesonDerivationDetails 44
A. ReflectionPointsontheSurfaceofaResonator 45
B. MappingBetweenSets(r,θ,φ)and(r0,θ0,φ0) 72
iv
Contents
C. SelectedConfigurations 74
C.1. HollowSphericalShell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
C.2. Hemisphere-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
C.3. Plate-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
D. DynamicalCasimirForce 91
D.1. FormalismofZero-PointEnergyanditsForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D.2. EquationsofMotionfortheDrivenParallelPlates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
E. ExtendedListofReferences 102
Bibliography 106
v
List of Figures
2.1. Twointeractingmoleculesthroughinduceddipoleinteractions. . . . . . . . . . . . . . . . . . . . . 8
2.2. Across-sectionalviewoftwoinfiniteparallelconductingplatesseparatedbyagapdistanceofz =d.
Thelowestfirsttwowavemodesareshown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3. A cross-sectional view of two infinite parallel conducting plates. The plates are separated by a gap
distanceofz =d.Also,thethreeregionshavedifferentdielectricconstantsε (ω). . . . . . . . . . . 17
i
3.1. Theplaneofincidenceviewofplate-hemisphereconfiguration. Thewavest(cid:176)hataresupp(cid:176)ortedthrough
internalreflectionsinthehemispherecavitymustsatisfytherelationλ≤2(cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176). . . . . . . 19
2 1
3.2. Thethicklineshownhererepresentstheintersectionbetweenhemispheresurfaceandtheplaneofinci-
(cid:176) (cid:176)
dence.Theunitvectornormaltotheincidentplaneisgivenbynˆ0 =−(cid:176)(cid:176)n~0 (cid:176)(cid:176)−1(cid:80)3 † k0 r0 eˆ.
p,1 p,1 i=1 ijk 1,j 0,k i
21
3.3. Thesurfaceofthehemisphere-hemisphereconfigurationcanbedescribedrelativetothesystemorigin
throughR~,orrelativetothehemispherecentersthroughR~0. . . . . . . . . . . . . . . . . . . . . . . 22
3.4. Inside the cavity, an incident wave k~0 on first impact point R~0 induces a series of reflections that
i i
propagatethroughouttheentireinnercavity. Similarly,awavek~0i incidentonthe(cid:176)imp(cid:176)actpointR~0i+
aRˆ0 ,whereaisthethicknessofthesphere,inducesreflectedwaveofmagnitude(cid:176)(cid:176)k~0 (cid:176)(cid:176).Theresultant
i i
wavedirectionintheexternalregionisalongR~0 andtheresultantwavedirectionintheresonatoris
i
along−R~0 duetothefactthereisexactlyanotherwavevectortravelinginoppositedirectioninboth
i
regions. Inbothcases,thereflectedandincidentwaveshaveequalmagnitudeduetothefactthatthe
sphereisassumedtobeaperfectconductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5. Thedashedlinevectorsrepresentthesituationwhereonlysingleinternalreflectionoccurs. Thedark
linevectorsrepresentthesituationwheremultipleinternalreflectionsoccur. . . . . . . . . . . . . . . 26
3.6. Theorientationofadiskisgiventhroughthesurfaceunitnormalnˆ0 .Thediskisspannedbythetwo
p
unitvectorsθˆ0 andφˆ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
p p
3.7. Theplate-hemisphereconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8. The intersection between oscillating plate, hemisphere and the plane of incidence whose normal is
(cid:176) (cid:176)
nˆ0 =−(cid:176)(cid:176)n~0 (cid:176)(cid:176)−1(cid:80)3 † k0 r0 eˆ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
p,1 p,1 i=1 ijk 1,j 0,k i
3.9. Becausetherearemorevacuum-fieldmodesintheexternalregions,thetwocharge-neutralconducting
platesareacceleratedinwardtillthetwofinallystick. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.10.Aonedimensionaldrivenparallelplatesconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1. Boyer’s configuration is such that a sphere is the only matter in the entire universe. His universe
extendstotheinfinity,hencetherearenoboundaries. Thesenseofvacuum-fieldenergyflowisalong
theradialvectorrˆ,whichisdefinedwithrespecttothespherecenter. . . . . . . . . . . . . . . . . . 34
4.2. Manufacturedsphere,inwhichtwohemispheresarebroughttogether,resultsinsmallnon-spherically
symmetric vacuum-field radiation inside the cavity due to the configuration change. For the hemi-
spheres made of Boyer’s material, these fields in the resonator will eventually get absorbed by the
conductorresultinginheatingofthehemispheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3. Theprocessinwhichaconfigurationchangefromhemisphere-hemispheretosphereinducingvirtual
photoninthedirectionotherthanrˆisshown. Thevirtualphotonhereisreferredtoasthequantaof
energyassociatedwiththezero-pointradiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vi
ListofFigures
4.4. A realistic laboratory has boundaries, e.g., walls. These boundaries have effect similar to the field
modesbetweentwoparallelplates. In3D,theeffectsaresimilartothatofacubicallaboratory,etc. . 36
4.5. Theschematicofspheremanufacturingprocessinarealisticlaboratory. . . . . . . . . . . . . . . . . 36
4.6. The vacuum-field wave vectors k~0 and k~0 impart a net momentum of the magnitude kp~ k =
(cid:176) (cid:176) i,b i,f net
(cid:126)(cid:176)(cid:176)k~0 −k~0 (cid:176)(cid:176)/2ondifferentialpatchofanareadAonaconductingsphericalsurface. . . . . . . 37
i,b i,f
4.7. To deflect away as much possible the vacuum-field radiation emanating from the laboratory bound-
aries,thewalls,floorandceilingareconstructedwithsomeoptimalcurvaturetobedetermined. The
apparatusisthenplacedwithinthe“ApparatusRegion.” . . . . . . . . . . . . . . . . . . . . . . . . 41
4.8. The original bubble shape shown in dotted lines and the deformed bubble in solid line under strong
acousticfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.9. The vacuum-field radiation energy flows are shown for closed and unclosed hemispheres. For the
hemispheresmadeofBoyer’smaterial,thenon-radialwavewouldbeabsorbedbythehemispheres. . 43
A.1. Asimplereflectionofincomingwavek~0 fromthesurfacedefinedbyalocalnormaln~0. . . . . . . . 47
i (cid:176) (cid:176)
A.2. Parallelplanescharacterizedbyanormalnˆ0 =−(cid:176)(cid:176)n~0 (cid:176)(cid:176)−1(cid:80)3 † k0 r0 eˆ. . . . . . . . . . 52
p,1 p,1 i=1 ijk 1,j 0,k i
A.3. ThetwoimmediateneighboringreflectionpointsR~0 andR~0 areconnectedthroughtheangleψ .
1 2 1,2
Similarly, the two distant neighbor reflection points R~0 and R~0 are connected through the angle
i i+2
Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ψi,i+1,ψi+1,i+2
vii
1. Introduction
The introduction is divided into three parts: (1) physics, (2) applications, and (3) developments. A brief outline of
the physics behind the Casimir effect is discussed in item (1). In the item (2), major impact of Casimir effect on
technologyandscienceisoutlined. Finally,theintroductionofthisthesisisconcludedwithabriefreviewofthepast
developments,followedbyabriefoutlineoftheorganizationofthisthesisanditscontributionstothephysics.
1.1. Physics
When two electrically neutral, conducting plates are placed parallel to each other, our understanding from classical
electrodynamicstellusthatnothingshouldhappenfortheseplates. Theplatesareassumedtobethatmadeofperfect
conductorsforsimplicity.In1948,H.B.G.CasimirandD.Polderfacedasimilarprobleminstudyingforcesbetween
polarizable neutral molecules in colloidal solutions. Colloidal solutions are viscous materials, such as paint, that
contain micron-sized particles in a liquid matrix. It had been thought that forces between such polarizable, neutral
molecules were governed by the van der Waals interaction. The van der Waals interaction is also referred to as
the Lennard-Jones interaction. It is a long range electrostatic interaction that acts to attract two nearby polarizable
molecules.CasimirandPolderfoundtotheirsurprisethatthereexistedanattractiveforcewhichcouldnotbeascribed
to the van der Waals theory. Their experimental result could not be correctly explained unless the retardation effect
wasincludedinthevanderWaals’theory. ThisretardedvanderWaalsinteractionorLienard-Wiechertdipole-dipole
interaction[1]isnowknownastheCasimir-Polderinteraction[2].Casimir,followingthisfirstwork,elaboratedonthe
Casimir-Polderinteractioninpredictingtheexistenceofanattractiveforcebetweentwoelectricallyneutral,parallel
platesofperfectconductorsseparatedbyasmallgap[3]. ThisalternativederivationoftheCasimirforceisintermsof
thedifferencebetweenthezero-pointenergyinvacuumandthezero-pointenergyinthepresenceofboundaries. This
force has been confirmed by experiments and the phenomenon is what is now known as the “Casimir Effect.” The
forceresponsiblefortheattractionoftwounchargedconductingplatesisaccordinglytermedthe“CasimirForce.” It
wasshownlaterthattheCasimirforcecouldbebothattractiveorrepulsivedependingonthegeometryandthematerial
propertyoftheconductors[4,5,6].
The Casimir effect is regarded as macroscopic manifestation of the retarded van der Waals interaction between
uncharged polarizable atoms. Microscopically, the Casimir effect is due to interactions between induced multipole
moments,wherethedipoletermisthemostdominantcontributorifitisnon-vanishing. Therefore,thedipoleinterac-
tionisexclusivelyreferredto,unlessotherwiseexplicitlystated,throughoutthethesis. Theinduceddipolemoments
can be qualitatively explained by quantum fluctuations in matter which leads to the energy imbalance 4E due to
charge-separation between virtual positive and negative charge contents that lasts for a time interval 4t consistent
with the Heisenberg uncertainty principle 4E4t ≥ h/4π, where h is the Planck constant. The fluctuations in the
induced dipoles then result in fluctuating zero-point electromagnetic fields in the space around conductors. It is the
presence of these fluctuating vacuum fields that lead to the phenomenon of the Casimir effect. However, the dipole
strengthisleftasafreeparameterinthecalculationsbecauseitcannotbereadilycalculated. Itsvaluemustbedeter-
minedfromexperiments.
Once this idea is accepted, one can then move forward to calculate the effective, temperature averaged, energy
due to the dipole-dipole interactions with the time retardation effect folded in. The energy between the dielectric
(or conducting) media is obtained from the allowed modes of electromagnetic waves determined by the Maxwell
equationstogetherwiththeboundaryconditions. TheCasimirforceisthenobtainedbytakingthenegativegradient
of the energy in space. This approach, as opposed to full atomistic treatment of the dielectrics (or conductors), is
justifiedaslong asthemost significantfield wavelengthsdeterminingthe interactionarelargewhencompared with
thespacingofthelatticepointsinthemedia. Theeffectofallthemultipledipolescatteringbyatomsinthedielectric
(orconducting)mediasimplyenforcesthemacroscopicreflectionlawsofelectromagneticwaves. Forinstance,inthe
caseofthetwoparallelplates,themostsignificantwavelengthsarethoseoftheorderoftheplategapdistance. When
thiswavelengthislargecomparedwiththeinteratomicdistances,themacroscopicelectromagnetictheorycanbeused
1
1. Introduction
withimpunity. But,tohandletheeffectivedipole-dipoleinteractionHamiltonian,theclassicalelectromagneticfields
havetobequantized. Thenthegeometricconfigurationcanintroducesignificantcomplications,whichisthesubject
matterthisstudyisgoingtoaddress.
Finally, itistobenoticedthattheCasimirforceontwouncharged, perfectlyconductingparallelplatesoriginally
calculatedbyH.B.G.Casimirwasdoneundertheassumptionofabsolutezerotemperature. Insuchcondition, the
occupationalnumbern forphotoniszero;andhence,therearenophotonsinvolvedinCasimir’scalculationforhis
s
parallelplates. However,theoccupationnumberconventionforphotonsreferstothosephotonswithelectromagnetic
energy in quantum of E = (cid:126)ω, where (cid:126) is the Planck constant divided by 2π and ω, the angular frequency.
photon
Thezero-pointquantumofenergy, E = (cid:126)ω/2,involvedinCasimireffectatabsolutezerotemperatureisalsoof
vac
electromagnetic origin in nature; however, we do not classify such quantum of energy as a photon. Therefore, this
quantumofelectromagneticenergy,E =(cid:126)ω/2,willbesimplydenoted“zero-pointenergy”throughoutthisthesis.
vac
Byconvention,thelowestenergystate,thevacuum,isalsoreferredtoasazero-point.
1.2. Applications
InordertoappreciatetheimportanceoftheCasimireffectfromindustry’spointofview, wefirstexaminethetheo-
reticalvaluefortheattractiveforcebetweentwounchargedconductingparallelplatesseparatedbyagapofdistance
d:F =−240−1π2d−4(cid:126)c,wherecisthespeedoflightinvacuumanddistheplategapdistance. Togetasenseof
C
themagnitudeofthisforce,twomirrorsofanareaof∼ 1cm2 separatedbyadistanceof∼ 1µmwouldexperience
an attractive Casimir force of roughly ∼ 10−7N, which is about the weight of a water droplet of half a millimeter
in diameter. Naturally, the scale of size plays a crucial role in the Casimir effect. At a gap separation in the ranges
of∼ 10nm,whichisroughlyaboutahundredtimesthetypicalsizeofanatom,theequivalentCasimirforcewould
beintherangeof1atmosphericpressure. TheCasimirforcehavebeenverifiedbyStevenLamoreaux[7]in1996to
within an experimental uncertainty of 5%. An independent verification of this force have been done recently by U.
MohideenandAnushreeRoy[8]in1998towithinanexperimentaluncertaintyof1%.
TheimportanceofCasimireffectismostsignificantfortheminiaturizationofmodernelectronics. Thetechnology
already in use that is affected by the Casimir effect is that of the microelectromechanical systems (MEMS). These
aredevicesfabricatedonthescaleofmicronsandsub-micronsizes. TheorderofthemagnitudeofCasimirforceat
suchasmalllengthscalecanbeenormous. ItcancausemechanicalmalfunctionsiftheCasimirforceisnotproperly
takenintoaccountinthedesign,e.g.,mechanicalpartsofastructurecouldsticktogether,etc[9]. TheCasimirforce
maysomedaybeputtogooduseinotherfieldswherenonlinearityisimportant. Suchpotentialapplicationsrequiring
nonlinearphenomenahavebeendemonstrated[10]. ThetechnologyofMEMSholdmanypromisingapplicationsin
scienceandengineering. WiththeMEMSsoontobereplacedbythenextgenerationofitskind,thenanoelectrome-
chanicalsystemsorNEMS,understandingthephenomenonoftheCasimireffectbecomeevenmorecrucial.
Aside from the technology and engineering applications, the Casimir effect plays a crucial role in accurate force
measurements at nanometer and micrometer scales [11]. As an example, if one wants to measure the gravitational
forceatadistanceofatomicscale, notonlythesubtractionofthedominantCoulombforcehastobedone, butalso
theCasimirforce,assumingthatthereisnoeffectduetostrongandweakinteractions.
Mostrecently, anewCasimir-likequantumphenomenonhavebeenpredictedbyFeigel[12]. Thecontributionof
vacuumfluctuationstothemotionofdielectricliquidsincrossedelectricandmagneticfieldscouldgeneratevelocities
of∼50nm/s.UnliketheordinaryCasimireffectwhereitscontributionissolelyduetolowfrequencyvacuummodes,
thenewCasimir-likephenomenonpredictedrecently byFeigelisdue tothecontributionof highfrequencyvacuum
modes. Ifthisphenomenonisverified,itcouldbeusedinthefutureasaninvestigatingtoolforvacuumfluctuations.
Otherpossibleapplicationsofthisneweffectlieinfieldsofmicrofluidicsorprecisepositioningofmicro-objectssuch
ascoldatomsormolecules.
EverythingthatwassaidabovedealtwithonlyoneaspectoftheCasimireffect,theattractiveCasimirforce.Inspite
ofmanytechnicalchallengesinprecisionCasimirforcemeasurements[7,8],theattractiveCasimirforceisfairlywell
established. This aspect of the theory is not however what drives most of the researches in the field. The Casimir
effectalsopredictsarepulsiveforceandmanyresearchersinthefieldtodayarefocusingonthisphenomenonyetto
be confirmed experimentally. Theoretical calculations suggest that for certain geometric configurations, two neutral
conductors would exhibit repulsive behavior rather than being attractive. The classic result that started it all is that
of Boyer’s work on the Casimir force calculation for an uncharged spherical conducting shell [4]. For a spherical
conductor, the net electromagnetic radiation pressure, which constitute the Casimir force, has a positive sign, thus
2
1. Introduction
beingrepulsive. Thisconclusionseemstoviolatefundamentalprincipleofphysicsforthefieldsoutsideofthesphere
take on continuum in allowed modes, where as the fields inside the sphere can only assume discrete wave modes.
However,noonehasbeenabletoexperimentallyconfirmthisrepulsiveCasimirforce.
The phenomenon of Casimir effect is too broad, both in theory and in engineering applications, to be completely
summarized here. I hope this informal brief survey of the phenomenon could motivate people interested in this
remarkableareaofquantumphysics.
1.3. Developments
Casimir’s result of attractive force between two uncharged, parallel conducting plates is thought to be a remarkable
applicationofQED.Thisattractiveforcehavebeenconfirmedexperimentallytoagreatprecisionasmentionedearlier
[7, 8]. However, it must be emphasized that even these experiments are not done exactly in the same context as
Casimir’soriginalconfigurationduetotechnicaldifficultiesassociatedwithCasimir’sidealizedperfectlyflatsurfaces.
Casimir’sattractiveforceresultbetweentwoparallelplateshasbeenunanimouslythoughttobeobvious. Itsorigin
can also be attributed to the differences in vacuum-field energies between those inside and outside of the resonator.
However, in 1968, T. H. Boyer, then at Harvard working on his thesis on Casimir effect for an uncharged spherical
shell, had come to a conclusion that the Casimir force was repulsive for his configuration, which was contrary to
popularbelief. HisresultisthewellknownrepulsiveCasimirforcepredictionforanunchargedsphericalshellofa
perfectconductor[4].
ThesurprisingresultofBoyer’sworkhasmotivatedmanyphysicists,bothintheoryandexperiment,tosearchforits
evidence. Onthetheoreticalside,peoplehavetrieddifferentconfigurations,suchascylinders,cube,etc.,andfound
many more configurations that can give a repulsive Casimir force [5, 13, 14]. Completely different methodologies
weredevelopedinstrivingtocorrectlyexplaintheCasimireffect. Forexample,the“SourceTheory”wasemployed
bySchwingerfortheexplanationoftheCasimireffect[14,15,16,17].Inspiteofthesuccessinfindingmanyboundary
geometriesthatgaverisetotherepulsiveCasimirforce,theexperimentalevidenceofarepulsiveCasimireffectisyet
to be found. The lack of experimental evidence of a repulsive Casimir force has triggered further examination of
Boyer’swork.
ThephysicsandthetechniquesemployedintheCasimirforcecalculationsarewellestablished. TheCasimirforce
calculationsinvolvesummingupoftheallowedmodesofwavesinthegivenresonator.Thisturnedouttobeoneofthe
difficulties in Casimir force calculations. For the Casimir’s original parallel plate configuration, the calculation was
particularlysimpleduetothefactthatzeroesofthesinusoidalmodesareprovidedbyasimplefunctionalrelationship,
kd = nπ,wherek isthewavenumber,distheplategapdistanceandnisapositiveinteger. Thistechniquecanbe
easilyextendedtootherboundarygeometriessuchassphere,cylinder,coneoracube,etc. Forasphere,thefunctional
relationthatdeterminestheallowedwavemodesintheresonatoriskr = α ,wherer istheradiusofthesphere;
o s,l o
andα ,thezeroesofthesphericalBesselfunctionsj .Inthenotationα denoteslthzeroofthesphericalBessel
s,l s s,l
function j . The same convention is applied to all other Bessel function solutions. The allowed wave modes of a
s
cylindricalresonatorisdeterminedbyasimplefunctionalrelationka =β ,wherea isthecylinderradiusandβ
o s,l o s,l
arenowthezeroesofcylindricalBesselfunctionsJ .
s
OneofthemajordifficultiesintheCasimirforcecalculationfornontrivialboundariessuchasthoseconsideredin
thisthesisisindefiningthefunctionalrelationthatdeterminestheallowedmodesinthegivenresonator. Forexample,
forthehemisphere-hemisphereboundaryconfiguration,theradiationoriginatingfromonehemispherewouldenterthe
otherandrunthroughacomplexseriesofreflectionsbeforeescapingthe(cid:176)hemispheric(cid:176)alcavity. Theal(cid:176)lowedvacuu(cid:176)m-
field modes in the resonator is then governed by a functional relation k(cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176) = nπ, where (cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176) is
2 1 2 1
thedistancebetweentwosuccessivereflectionpointsR~01 andR~02 ofthe(cid:176)resonator,a(cid:176)sisillustratedinFigure3.1. As
willbeshowninthesubsequentsections, theactualfunctionalformfor(cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176)isnotsimpleeventhoughthe
2 1
(cid:176) (cid:176)
physics behind (cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176) is particularly simple: the application of the law of reflections. The task of obtaining
2 1
(cid:176) (cid:176)
the functional relation k(cid:176)(cid:176)R~0 −R~0 (cid:176)(cid:176) = nπ for the hemisphere-hemisphere, the plate-hemisphere, and the sphere
2 1
configurationformedbybringingintwohemispherestogetheristothebestofmyknowledgemyoriginaldevelopment.
Itconstitutesthemajorpartofthisthesis.
ThisthesisisnotaboutquestioningthetheoreticaloriginoftheCasimireffect. Instead,itsemphasisisonapplying
theCasimireffectasalreadyknowntodeterminethesignofCasimirforcefortherealisticexperiments. Inspiteofa
3
