A proposal for detection of absolute rotation using superconductors and large voltages
AA proposal for detection of absolute rotation usingsuperconductors and large voltages
E.M. Forgan, C.M. Muirhead, A.I.M. Rae & C.C. Speake
School of Physics & Astronomy, University of Birmingham, B15 2TT, U.K.
Abstract
We describe designs for practical detectors of absolute rotation, which relyon the creation of magnetic fields by charged objects that are rotating withrespect to an inertial frame. Our designs, motivated by an original suggestionby R.M. Brady, utilize the properties of superconductors, both to shield andconfine the magnetic fields, and also as the basis of a SQUID detector ofthe fields produced. We show that with commercially available SQUIDs, ourdesigns can have sufficient sensitivity and signal-to-noise ratio to measure thesidereal rate of rotation of the Earth. We consider three different designs:two of these can also be configured to provide a confirmation of the formthat Maxwell’s equations take in a rotating frame. We can also make adirect experimental test of whether low-frequency electromagnetic energyexperiences the same inertial rest-frame as matter.
Keywords:
Superconductivity, Non-inertial Frames, SQUIDs
1. Introduction
Consider a long normal metal cylinder carrying a uniform excess electri-cal charge around its curved surface. If we now rotate the cylinder about itslong axis, the charge will move with the metal, and constitute a circulatingcurrent resulting in a magnetic field parallel to the axis of the cylinder. If wecan measure this field and know the distribution of charges, we can use it todeduce the rate of rotation. The experimental demonstration that physicalmovement of electrostatic charge has the same magnetic effect as an elec-tric current was first conclusively demonstrated by Rowland & Hutchinson’s
Email address:
[email protected] (E.M. Forgan)
Preprint submitted to Physics Letters A December 4, 2020 a r X i v : . [ phy s i c s . i n s - d e t ] N ov ioneering experiments [1] in the 19th century, and were important in thedevelopment of understanding of electromagnetism. Here we use the sameprinciple for a different purpose. If we aligned our cylinder with the Earth’saxis then, with sufficient sensitivity, we could use it to measure the rate ofrotation of the Earth relative to the inertial frame, which is presumably therest of the Universe (Mach’s principle), without observing the stars. The ideaof using superconductors and high voltages was tried by R.M. Brady in the1980’s with the technology of the time using rotation rates ∼ radians/sec.His first design[2] gave signals that appear to be spurious, as they do not re-verse on reversing the direction of rotation. The results from his later designwere never published in full: Ref. [3] is a brief account only available online,showing results limited by considerable drift and other complicating factors,but apparently of the magnitude expected from his and our predictions. Wedescribe here three different designs, intended to avoid unwanted signals,and give a theoretical treatment of Brady’s design in the Appendix. We findthat it is realistic to construct an apparatus that can accurately measurethe Earth’s sidereal rotation rate with a reasonable noise-integration time,so long as external magnetic fields are very low and remaining flux lines arewell-pinned in the superconducting parts. Two forms of the apparatus mayalso be used to check the form of Maxwell’s equations in rotating framesby experiments using rotations with a shorter period. The behaviour of thesuperconducting parts may at first appear counter-intuitive so we introducethe design concepts in a heuristic way
2. Heuristic description of the design
In Figs. 1 & 2 we show the evolution of a conceptual design. 1(a) showsthe cross section of a positively charged long hollow cylinder of normal metalor insulator, rotating in a clockwise direction, thus constituting a circulatingsurface current density, J Amps per unit length. As in a solenoid, this givesrise to a magnetic field B i , pointing downwards inside the cylinder. In prac-tice, we could make the cylinder the earthed plate of a cylindrical capacitor,with another normal metal cylinder outside, as shown in Fig. 1(b). Thecharges on this co-rotating outer cylinder make an equal and opposite con-tribution to the magnetic field, which changes direction but not magnitudeand it now appears as B o in the gap between the two cylinders and is zeroinside the middle cylinder. The charges on the opposite plates of a capacitorare equal and opposite, so the currents J o & J are equal and opposite in this2ase. Figure 1: Evolution of a conceptual design. The colours on the surfaces of the componentsrepresent the relevant surface charges (red positive; blue negative). Normal metal is lightgrey, superconductor is dark grey. Surface current densities are J ’s, magnetic fields are B ’s and cross-sectional areas between the cylinders are A ’s (a) Isolated positively chargedrotating normal metal cylinder; (b) with oppositely charged counter-electrode; (c) outercylinder made of superconductor; (d) inner and outer cylinder of superconductor: this lastchange removes the magnetic field we intended to measure. Recognising that this field is very small, the outer cylinder is then madeof a superconductor to shield the interior from external magnetic fields, as inFig. 1(c). Ignoring for now an effect called the ‘London moment’, which weconsider later, the total flux inside a superconducting cylinder is quantized,and taking it as zero before the voltage was applied to the capacitor, itremains zero afterwards. Therefore, a current density J o is induced on theinside of the outer cylinder, which gives rise to the field B o which maintainszero total flux inside that cylinder. J o is a dissipation-free current density,which may be partly due to rotating charges and partly supercurrent: its total value is the only relevant quantity and is determined by flux quantization.In this case of a superconducting outer cylinder, J o is not equal and oppositeto J ; these two current densities give rise to a magnetic field B i in the inner3 igure 2: Schematic of an all-superconducting design. This represents the central cross-section of long concentric charged cylinders with the inner one having a narrow slit alongits length. The slit forces the charge on it to rotate, again giving a surface current density J ; additional supercurrents flow in the surfaces of the cylinders to maintain the magneticfield conditions required by superconductivity. area A i , plus an oppositely directed field B o in the outer area A o , and zerototal flux.A normal metal inner cylinder will create Johnson noise in the magneticfield inside it, so one would like to make this superconducting too, as in Fig.1(d). However, quantization means that the flux in the inner cylinder mustremain zero as the capacitor is charged, and this means that the total currentflowing around the inner cylinder, and the field inside it must be zero. Fluxquantization in the outer cylinder ensures that the field in the outer area isalso zero, so this modification has removed the effect we wish to observe! Away of avoiding this is shown in Fig. 2.Here we have added a slit along the length of the superconducting innercylinder. Electrostatics ensures that the charge remains on the outside of theinner cylinder and therefore must rotate with it. Flux quantization in theouter cylinder and zero magnetic field in the bulk of both superconductorsare maintained by current densities J i and J o . It will be noted that J i passesthrough the slit and flows on the inner surface of the inner cylinder, givingrise to the magnetic field B i inside it. This is the field we intend to measure:4t is in an earthed, electric-field-free region. Our calculations will indicatethat with ± ∼ kV applied to an apparatus of radius ∼ is rotating with and insidea superconductor , the London moment, is invisible, as the rotation affectsthe superconducting electrons in the SQUID in the same way as the bulksuperconductor around it [3, 5]. Hence, when the whole apparatus is inthe rotating frame, we pick up only the effects of the rotating electrostaticcharges. This was realized by Brady and is the basis of his work to de-velop an electromagnetic method of measuring rotation. This considerationis important because the field associated with the London moment can becomparable to that due to rotating charges if the SQUID is not co-rotating.We note that the London moment is a response due to the inertial mass ofthe electrons in the bulk uncharged superconductor, so does not change whenthe voltage applied between the cylinders is reversed. The London momentof a cylindrical superconducting shell has been considered [6]. So long as theshell is thick compared with the microscopic superconducting coherence andmagnetic penetration depths, and not close to T c , then the bulk expression [4]for the London moment applies.(ii) At first sight, it may appear that the fields just inside and outside theslit in Fig. 2 cannot be different because there is no current flowing acrossthe slit between them – apparently breaking Amp`ere’s theorem. However,near the slit, the electric field lines are no longer radial. As the split cylinder5otates, the change in the displacement of the electric field d D /dt , measured- like the current - in the non-rotating frame, gives rise to a tangential dis-placement current near the slit; this can be shown to ‘complete the circuit’.
3. Basic Theory
For simplicity, and to illustrate how the magnetic fields depend on theform of the apparatus sketched in Fig. 2, we carry out the calculations for thefields at the central cross-section of long cylinders, with the magnetic fieldsgiven by the expressions for the internal fields of long solenoids. Also, thecurrents due to charges will be carried within the electrostatic penetrationdepth of a metal surface and the supercurrents within the (longer, but stillmicroscopic) magnetic penetration depth; we ignore this minor effect andtake all currents to flow ‘in the surface’.A charged cylinder of radius r and length (cid:96) , carries a surface chargedensity σ = ε E where E is the radial electric field at the surface. (Thisstill applies in the presence of a dielectric, where σ is the sum of the mobileand polarisation charges, which rotate together in the apparatus. This pointwas also made in Ref. [3].) The total charge carried by the cylinder is Q =2 πr(cid:96)ε E . If the cylinder is rotating about its axis at angular frequency ω ,the circulating current per unit length, J is given by: J = ωQ π(cid:96) = ωrε E . (1)This is the current density represented by J in Figs. 1 & 2 and used inthe theory below. We adopt the sign convention that clockwise currents anddownward-directed magnetic fields are positive. We first calculate the casesin Fig 1(b) and (c), before turning to Fig 2.In 1(b), the magnitudes of the charges on the normal metal outer andinner cylinders are equal and opposite and rotate together, so J o = − J , and: B o = − µ J . (2)Substituting for J , we obtain an expression for B o in terms of the electricfield E in the cylindrical capacitor. B o = − µ ωrε E = − ωrE/c . (3)We note the term c in the denominator, indicating that the magnetic fieldis small. 6e now treat the situation in Fig. 1(c) with an outer superconductingcylinder and an inner normal cylinder. Flux quantization gives an unchangedzero total magnetic flux inside the outer shield when the capacitor is charged: B o A o + B i A i = 0 . (4)The Meissner effect ensures that there will be zero field in the bulk of thesuperconducting outer cylinder and there is a magnetic field B o just insideit. Hence the net current density J o on its inside surface satisfies: µ J o = B o . (5)The current density J carried around by the charge on the inner cylindergives the difference between inner and outer fields: B i = B o + µ J . (6)From Eqns. 6 & 4: B i = µ J − B i . ( A i /A o ) . (7)Hence: B i = µ J . A o / [ A i + A o ] . (8)Now we consider the intended design shown in Fig. 2, with a superconductingsplit inner cylinder. Eqns. 4 & 5 still apply. The field B o in the outer area A o does not penetrate the bulk of the superconducting surfaces bounding it,so we have: µ ( J − J i ) = − µ J = − B o . (9)Similarly, the field B i in the inner area A i is related to the surface currentdensity on the inside of the inner cylinder: µ J i = B i . (10)From Eqns. 10 & 4: µ J i = − B o . ( A o /A i ) . (11)Hence, using Eqn. 9: µ J + B o . ( A o /A i ) = − B o . (12)Thus: B o = − µ J . A i / [ A i + A o ] . (13)7lux quantization (Eqn. 4) relates B i to B o , giving finally: B i = µ J . A o / [ A i + A o ] . (14)This is exactly the same expression as Eqn. 8, the normal inner cylindercase. The reason is that the current density J i goes both clockwise andanticlockwise around the inner cylinder, and so does not alter the value ofthe field inside the inner cylinder relative to the normal case. However,as mentioned above, with a normal metal inner cylinder we have Johnsonnoise, whereas with superconducting parts we can employ further strategiesdescribed below to increase the sensitivity. These are necessary, as the fieldsproduced are very small; for instance, with ∼ B i due to the Earth’srotation is calculated to be ∼ − T. Nevertheless, with a suitable design,fields of this magnitude can be detected with a SQUID magnetometer.In the following sections we consider three different designs to measurethe rotation effect and analyze their sensitivity. In the case of the cylindricaldesigns, we assume ‘long’ cylinders to obtain simple results. In practice,the outer cylinder would be longer than the inner cylinder with caps whichmostly close the ends to provide a magnetic shield for the contents. In thegap due to the difference in length of outer and inner cylinders, the lines offlux due to B o would cross the ends of the inner cylinder and would becomethe equal return flux B i . The ends of the inner cylinder would be partiallyclosed in such a way as to allow axial magnetic flux to enter, but not radialelectric field. In the Appendix, we also give an approximate treatment ofBrady’s second design[3] and compare its sensitivity with our designs.
4. Central pickup coil design
We first define symbols for the relevant dimensions of the design repre-sented schematically in Fig. 3. The outside radius of the inner cylinder is r o ,with a gap of t across which the voltage is applied. The inside radius of theinner cylinder is r i . We will find in a practical system, that the gap betweenouter and inner cylinders t (cid:28) r o , so that Eqn. 14 may be written: B i = µ J . r o t/ (cid:0) r i + 2 r o t (cid:1) . (15)At first sight, we would be inclined to make r i small in order to enhance B i .As noted above, the fields are so small that a SQUID detector is required;8 igure 3: Schematic view of central pick-up design and SQUID detection system this detects flux rather than field, so the criteria are different. The detectionsystem is a pickup coil inside r i connected to the SQUID input coil, whichfeeds flux into the SQUID. In practice, the SQUID input coil would be in ascreened region and arranged to make a negligible contribution to the fieldin the cylinders.The pickup coil of radius r p has n closely-coupled turns, giving an induc-tance L p . It responds to the flux turns Φ p = nπr p B i to which it is exposed.This drives a current I S into the SQUID input coil of inductance L S , whichis tightly coupled to the SQUID. The SQUID and its integrated input coil isa readily available commercial item and is chosen to have a small equivalentinput current noise ( S I ) . . In order to obtain maximum signal to noise inour proposed experiment we therefore need to maximise the current I S intothe SQUID. Standard flux-transformer theory gives us: I s = Φ p / ( L S + L p ) . (16)If we treat the n turns of wire on the pickup coil as close to each other, so thatthe coupling coefficient between turns ≈ unity and each turn has inductance9 ∼ µ r p , then L p = n L and n = ( L p /L ) . = ( L p /µ r p ) . , (17)Φ p varies as n but L p varies as n , so I S has a maximum when L p = L S so I S = nB i πr p / L p = nB i πr p / L S , (18)and with n now set to ( L S /L ) . , we can also write: I S = B i πr p / L S L ) . . (19)We will ignore any screening of I S by the surrounding superconductor. Thiswill be true if r p is not too close to r i . We let r p = γr i , with γ ∼ .
8, so thatthe area of the pickup coil is not too large a fraction of that of the hole.Using Eqns. 15 & 18, we find I S = ( µ /L S ) / πtr o ( γr i ) / ( r i + 2 tr o ) × J , (20)This is maximised when r i = (6 tr o ) / , (21)giving: I S = 1 . µ /L S ) / ( tr o ) / γ / × J . (22)Using Eqn. 1, we can also re-express Eqn. 22 in terms of the applied voltage V = Et and the angular frequency of rotation ω : I S = 1 . µ L S ) − / r / o t − / γ / × V ω/c . (23)This emphasises that small t increases the signal, but in practice there willbe a lower limit on t for given V , set by the breakdown electric field in thegap, which can be increased by filling the gap with a dielectric. We havealready noted that the presence of dielectric does not affect our derivation.For comparison with other designs it is useful to express the results interms of L , the inductance of a single turn of the pickup coil, the flux Φ picked up by a single turn, and the total flux Φ S in the SQUID input coil. L ≈ µ r p = µ γr i = µ γ (6 tr ) . , (24)Φ = πr p B i = γ πtr o × µ J = γ πr o V ω/c , (25)10 S = n × Φ / Ls = Φ / L S L ) . , (26)and: Φ S = L S I S = n × Φ / L S /L ) . Φ / . (27)In section 7, we consider the role of Φ , L S and L in determining thesignal to noise achievable with this and the two other designs described insections 5 & 6.
5. Solenoidal pickup design
In the previous section, we considered a simple pickup coil. This would besuitable for in-principle tests for shorter rotation periods ∼
10 seconds, butwe can greatly increase the sensitivity by having a pickup coil more stronglycoupled to the inner split cylinder and occupying essentially all its area. Thesetup is shown in Fig. 4. As before, we carry out our calculations for thefields due to long cylinders, ignoring end effects. As in the situation describedin Fig. 2, the field B o in the outer area A o is related to J , J i & J o by Eqn. 9.The field B g in the gap between solenoid and split cylinder is given by: B g = µ J i . (28)The field in the central area is due to the sum of the induced current density J i on the inner surface of the split cylinder, plus the current per unit axiallength due to the SQUID input current. (The latter is the sum of currentsflowing on the inner and outer surfaces of the turns of the solenoid.). Wetherefore have: B i = µ ( J i + nI S /(cid:96) ) . (29)Flux quantization in the SQUID input circuit gives: L S I S + nB i A i = 0 . (30)Combining this with Eqn. 29 we find: nµ J i = − n µ I S /(cid:96) − L S I S /A i . (31)Flux quantization in the outer cylinder gives: B o A o + B g A g + B i A i = 0 , (32)11 igure 4: Schematic diagram with a solenoidal pickup coil, which is represented in crosssection by the inner grey circle. It is a superconducting cylinder of length (cid:96) cut into asingle-layer solenoid of n turns, with very small gaps between the turns. The ends of thesolenoid are connected to a SQUID pickup coil, which takes a current I S , so the currentper unit length of the cylinder is nI S /(cid:96) . The solenoid is fairly closely-fitting inside thesplit cylinder of inside radius r i , with a gap of width g and area A g = 2 πr i g , kept small toreduce the flux in this region. The outer radius of the split cylinder is r o with a gap t tothe outer can, giving A o = 2 πr o t . Note that main purpose of the split cylinder is to shieldthe SQUID circuit completely from strong electric fields, and also from charging currentsand their magnetic fields when the voltage is changed. This is achieved in practice bymaking the slit very narrow. and using Eqns. 9, 28 & 30: nµ ( J i − J ) A o + nµ J i A g − L S I S = 0 . (33)We then substitute for J i , using Eqn. 31 to give a relationship between theSQUID current and the current density J due to rotation: n µ A o (cid:96) I S + n µ A g (cid:96) I S + nµ A o J + L S I S ( A o + A g ) A i + L S I S = 0 . (34)Writing A tot = A o + A g + A i , we have after some manipulation: I S = − (cid:18) A i A tot (cid:19) nµ A o ( L S + n L ) J . (35)The minus sign on the RHS merely indicates that I S is in the opposite direc-tion to J . The second term on the denominator is the effective inductanceof the n -turn solenoid, which is L eff = n × L , the inductance for a singleturn, which is given by: L = (cid:18) A i A tot (cid:19) µ ( A o + A g ) (cid:96) . (36)12he numerator in Eqn. 35 varies as n , but the denominator contains n , sowe may vary n to maximise | I S | for a given value of L S . This gives: n = L S /L . (37)This gives the SQUID current as: | I S | = (cid:18) A i A tot (cid:19) nµ A o L S J = (cid:18) A i A tot (cid:19) µ πr o t L S L ) . J . (38)There is a further optimisation to perform: as in the previous section, wetake A i (cid:29) ( A o + A g ) so that the prefactor A i /A tot ≈
1. However, the value of( L ) . in the denominator also depends on this factor and on r i . Plotting | I S | versus r i results in a broad peak, at a value of r i that can only be obtainednumerically. However setting r i ≈ (4 tr o ) / gives a sensitivity within 1% ofoptimum, and still gives the prefactor ≈
1. We shall take it as unity forsimplicity. We may now write various results in terms of the dimensions ofthe apparatus: L = µ π ( r o t + r i g ) (cid:96) , (39)and to get a simple expression, we set g = t , and write for the mean radius r m = ( r i + r o ) / L = µ πr m t(cid:96) . (40)We note that the value of L is not proportional to the area A i inside thesolenoid, but instead depends on the much smaller area of the gaps outsideits windings. This is a consequence of the shielding effects of the surroundingsuperconductors.For a given applied voltage V , J ∝ E = V /t , so the t in the numerator ofEqn. 38 disappears. We may also incorporate the expressions for J and L to give: | I S | ≈ πµ r ( L S L ) . × ε V ω = (cid:18) π(cid:96)r o tr m µ L S (cid:19) . × V ω/c . (41)Another way of writing this is in terms of the flux transferred to the SQUIDinput inductance:Φ S = L S | I S | ≈ (cid:18) L S L (cid:19) . πr V ω/c = n × πr V ω/c = n × Φ / , (42)13here Φ = 2 πr V ω/c is the flux picked up by a single turn.In section 7, we shall consider what sensitivity this and other designs cangive, and the role of the values of the flux and the inductances.
6. Disk design
Figure 5: Schematic drawing of ‘disk’ design; (a) View along axis; (b) View in cross section.The n -turn pickup coil, split washer and outer shield are all superconducting. In the cylindrical design of section 5, all the turns of the pickup coil sharethe same flux, and the inductance varies as the square of the number ofturns. We now describe an alternative design, which consists of a number ofidentical cells which are magnetically isolated from each other, so that thetotal inductance of the pickup coils varies linearly with the number of cells,which are distributed along the rotation axis. A single cell is represented in14 igure 6: Detail of one side of the washer, showing the directions of surface currentdensities, magnetic fields and symbols for dimensions
Fig. 5, with detail given in Fig. 6. This design is equivalent to the previ-ous solenoid design, where the outer cylinder, which provided shielding andmaintained flux quantization is now represented by the outer box. Inside, theslotted cylinder is folded into a C-cross-section slotted washer, which shieldsthe ‘pancake’ SQUID pickup coil. The coil has n turns carrying a current I S within a radial width w . The spacings g & t and of the slot, and the axiallength of the disk, would all be relatively much smaller than represented inthe schematic figures.The algebraic analysis given below was verified usingfinite-element analysis modelling, to choose the dimensions finally adopted.The rotating charge on the outer surfaces of the washer gives rise to acurrent density J on its top and bottom surfaces. The current in the SQUIDpickup coil flows primarily in its top and bottom surfaces, giving rise to asurface current density nI S / w . This induces a current density J i on theinner surfaces of the washer, which returns via the slot as a surface currentdensity J (cid:48) i on the outer surfaces. Most of this current flows on the surfacesof width w & W , so continuity of current gives: J (cid:48) i = J i w/W . (43)15ll these currents give rise to magnetic fields B o & B i , outside and inside thewasher body. As before, the charged outer superconducting shield carries asurface current density between B o in the gap and zero field in the bulk ofthe shield.In this design, B o & B i are radial fields, so conservation of flux impliesthat in the regions of constant height t & g they should vary as 1 /r . Hencethe surface current densities in the adjacent superconductors must also varysimilarly with r . However for the purposes of analysis, we make the approx-imation of calculating the values that currents, fields and fluxes take at themean radius from the axis, r m . At this radius, B o = µ ( J (cid:48) i − J ) = µ ( J i w/W − J ) , (44)and the total flux Φ out passing through the axial hole due to B o is given by:Φ out = 2 πr m tB o = 2 πr m tµ ( J i w/W − J ) . (45)The field around the pickup coil obeys B i = µ J i = µ nI S / w . (46)The total flux Φ in going through the pickup coil due to this current densityis: Φ in = 2 πr m gB i = 2 πr m gµ J i . (47)Flux quantization in the SQUID input circuit gives: L s I S + n (Φ out + Φ in ) = 0 , (48)i.e.: L s I S + n [2 πr m tµ ( nI s / W − J ) + 2 πr m gµ nI S / w ] = 0 . (49)Thus: I S (cid:2) L s + n πµ r m ( t/W + g/w ) (cid:3) = n πr m tµ J . (50)The second term on the LHS is the effective screened inductance L c of theSQUID pickup coil in a single cell. Thus we may write the following expres-sion for I S in terms of the flux Φ c contributed by a single cell to the SQUIDinput: I S = Φ c ( L S + L c ) . (51)16ere: L c = n µ πr m ( t/W + g/w ) , (52)and Φ c = n πr m tµ J = n × πr m V ω/c , (53)while the flux in the SQUID input inductance may be written:Φ S = L S I S = L S ( L S + L c ) Φ c . (54)In the following section, we shall consider how the signal to noise may beoptimised for a set of N cells of this design and compare it with other designs.
7. Optimisation of sensitivity and calculation of integration timesto acquire a given accuracy
We first consider designs, such as those described in sections 4 & 5, inwhich we have a pickup loop with an effective inductance having the value L for one turn. With n closely-coupled turns it would have an inductance L n , and the n turns would pick up a flux n Φ . In this case, the current I S flowing in the SQUID input coil of inductance L S is given by: I S = n Φ ( L S + L n ) . (55)A commercial SQUID (e.g. Ref. [7]) has a spectral density of flux noise whichmay be represented as the current noise in its input coil ( S I ) / Amp.Hz − / .We wish to maximise the signal to noise ratio I S / ( S I ) / . Now SQUID man-ufacturers generally provide a range of models with different values of in-ductance of the input coil, each of which has a different mutual inductancecoupling it to the same design of SQUID. In these circumstances, ( S I ) / isnot a constant independent of L S . Rather, for ideal coupling of the inputcoil to the SQUID, the quantity L S S I is constant as it is equal to S E , theenergy sensitivity of the SQUID. The coupling coefficient for modern SQUIDdesigns is close to unity, so for simplicity we shall write L S S I as 2 S E . Thishas the units J/Hz - the same as Planck’s constant. Using S E we may write: I S ( S I ) / = I S (2 S E /L S ) / = (cid:18) L S S E (cid:19) / n Φ ( L S + L n ) . (56)17 Optimizing this with respect to n , we find L S = L n , giving: I S ( S I ) / = (cid:18) L S S E (cid:19) / (cid:18) L S L (cid:19) / Φ L S = Φ S E L ) . . (57)We see that the signal to noise is independent of L S , and we may choose aSQUID from a range with a good energy resolution and adjust n to matchthe apparatus to the input inductance of that SQUID. Also, we find thatthe signal to noise is maximised when the inductance of a single turn of thepickup coil is as small as possible for a given flux coupling Φ to a single turn.If we compare these quantities for the small pickup coil versus the solenoidalpickup coil, we see that the former has both a smaller Φ and a larger L , sowill be less sensitive. Its only advantage is that it allows SQUID and chargedcylinders to be rotated independently, allowing experiments on the effects ofrelative rotation.The inverse of the expression in Eqn. 57 is the fractional error in therotational angular frequency ω . The time necessary to obtain unity signal tonoise in the rotation frequency is therefore given by: τ ∼ S I I S = 8 S E L Φ , (58)and for noise ∼
1% of the signal we require a 10 × longer measuring time.We shall later use this equation to estimate the integration times for varioussetups and rotation rates.We now consider the optimisation of a design, such as the disk designdescribed in section 6, in which we have a set of N cells, each with a pickuploop of effective inductance L c picking up a flux Φ c . These are added in seriescausing a current I S to flow in the SQUID input coil of inductance L S : I S = N Φ c ( L S + N L c ) . (59)In this case, the signal to noise ratio is given by: I S ( S I ) / = I S (2 S E /L S ) / = (cid:18) L S S E (cid:19) / N Φ c ( L S + N L c ) . (60)To increase the sensitivity, the number N of cells is chosen to be aslarge as possible, subject to experimental constraints. We can optimize with18espect to L S , choosing a SQUID input coil best matched to the apparatus.Alternatively, for a given L S , we can optimise the number of turns in thepickup coil in each cell, which gives the same sensitivity. In either case, wehave L S = N L c , giving: I S ( S I ) / = (cid:18) N L c S E (cid:19) / N Φ c N L c = Φ c S E L c /N ) . . (61)Comparing this with Eqn. 57, we see that the design with the higher sensi-tivity depends on how Φ c compares with Φ , and L c /N with L . To knowthe value of integration time for a measurement, which involves a choice ofactual dimensions and commercially available SQUIDs, we carry out calcu-lations which are reported in the following section.Here, we have analysed designs motivated by Brady’s original sugges-tion [2], using coaxial cylinders and disks. His later design [3] consisted oftwo oppositely-charged flexible superconducting sheets separated by insulat-ing material, wound some 50 times around a central superconducting cylinderto form a ‘Swiss roll’, with the ends of each sheet connected by supercon-ducting joints. The current in one of these joints ran through the input coilof a SQUID. In the Appendix, we supply an approximate analysis of thisdesign, which gives a similar sensitivity to our best designs. However, wehave not considered it as a candidate as it presents some notable practicaldifficulties, as recorded by Brady [3]. These are that the SQUID input circuitis exposed to high electric fields, making it sensitive to charging and leakagecurrents. Also, the sharp ends of the thin sheets making the turns of the spi-ral give a tendency to electrical breakdown, which can destroy the SQUID.The spiral has a very low inductance - smaller than that of the connectingleads. This reduces its sensitivity relative to our designs, which have totalinductances considerably larger than the connecting leads. Finally, the thinsuperconducting sheets are not conducive to mechanical stability, which isnecessary to avoid signals due to background flux. It has also been argued [8]that because the electric field transforms into a magnetic field in the rotatingframe, there is a strong cancellation of the signal in Brady’s design.
8. Discussion
There has been some controversy concerning what magnetic field dueto a rotating charged cylinder would be observed when rotating with the19ylinder. One might think that when rotating with the cylinder, there isno circulating current and therefore no magnetic field. However, just asnew ‘fictitious forces’ – Coriolis and centrifugal – appear in mechanics ina rotating frame, one might expect that there can be ‘fictitious currents’in the electromagnetic case. There is also the question of which frame theobserver is in; these matters are discussed in Ref. [8]. An experiment totest this with constant voltage applied can be performed by a back & forthrotation while observing the resulting oscillation in magnetic field strength.Note that such measurements at constant voltage would be measuring theeffects of relative motion with respect to the lab frame rather than absoluterotation. The earth’s rotation would only cause a very small and constantoffset of magnetic field value. The measurements could be done (using thelower sensitivity pickup-coil setup - or the higher sensitivity solenoid pickupif carefully engineered to ensure free rotation, while retaining a small gap) inthree ways:(i) with the whole apparatus rotated together as envisaged for detectingthe absolute rotation rate of the Earth.(ii) with the SQUID and its pickup coil on the same axis as the cylinders,but at rest while the cylinders are rotated - or(iii) vice versa - with the SQUID rotating and the cylinders at rest.In the latter two cases, at zero applied voltage the SQUID would detectthe field due to the London moment either of the surrounding rotating super-conductor or of its own rotation. The London field is given by B L = 2 ωm/e ,where m and e are the free electron mass and charge[4]. When the voltageis applied, the SQUID would detect the sum of B L and B i . This experi-ment would provide a test that the B field observed in a rotating frame isthe same as would be calculated if the rotating apparatus were consideredfrom an inertial frame. By operating at constant voltage and using backand forth rotation, we avoid effects of charging currents and complicationsin making electrical connections, when the two parts of the apparatus arerotating differently.In Table I, we show under what conditions the measurements describedabove could be performed. In all cases we propose that either the appliedvoltage or the rotation direction is reversed, and the resultant change inSQUID output is phase sensitively detected. This avoids zero errors, but thedetection frequency, and hence the SQUID noise, is limited by the frequencyat which these reversals can be made. For short rotation periods, one coulduse the small pickup coil, which could comparatively easily be rotated inde-20onditions ⇓ Size ⇒ Small Scaled Small Small SmallMethod of Oscn. of ⇒ ± V ± V ± ω ± ω ± V measurement @ freq. ⇒ ⇒ ⇓ Noise ⇒
10% 1% 1% 1% 1%Solenoidal Int. τ ⇒ (cid:96) /cm ⇒
15 72 15 15 15design r o /cm ⇒ n ⇒
67 68 67 67 67Stacked Int. τ ⇒ (cid:96) /cm ⇒
15 57 15 15 15design r o /cm ⇒ n ⇒ N ⇒
19 62 19 19 19Single disk Int. τ ⇒
146 hr 0.7 s 700 s 70 s r o = 3 . n ⇒
16 16 16 16Small Int. τ ⇒
317 hr 1 hr 1.5 s 1500 s 150 spickup (cid:96) /cm ⇒
15 290 15 15 15design r o /cm ⇒ n ⇒
17 7.9 17 17 17
Table 1: Table of calculations of the performance of the three designs. All calculationswere carried out for a voltage of 1000 V applied across a gap t of 0.2 mm. Where applicable,the gap g was also set to 0.2 mm. All other dimensions are those given below or scaledup from them. To calculate the integration times required, we took the specifications of aparticular Magnicon ® [7] SQUID, which has an input inductance of 1 . µ H and a noiseat 0.1 Hz of ( S I ) . = 1 .
48 pA / √ Hz. For frequencies in this region, S I rises as 1 /f , so itmatters at what frequency the signal is phase-sensitively detected. For the back-and-forthrotations, which are to measure the effects of rotations relative to the lab frame, we haveassumed the total rotation is half a turn before reversing, so that the frequency at whichthe rotation rate is oscillated is equal to the rotation rate. For the small pickup coil design,we take r o = 3 . r i ∼ .
65 cm and set γ = 0 .
8. For the smallsolenoidal pickup coil design, we take r o = 3 . r i ∼ r m = 3 .
05 cm, W = 0 . w = 0 . N ∼
19 cells occupy approximately the same outer radius and length as the solenoidalpickup design. We also calculate the performance of a single cell. , the flux per single turn of pickup.For the stacked disk design, the larger value of L , the inductance per singleturn, is counteracted by the √ N increase in sensitivity for N cells in series.For high accuracy measurement of the Earth’s rotation with either ofthese designs, a scaled up apparatus is required. To maximise sensitivity,only the radial extent of the components and the axial length are increased.All gaps and (in the disk design) the thickness of the disks are left unscaled,which keeps the electric field and inductance values constant. We find thatthe strongest scaling dependence is in Φ , which increases as the square of theradius. The value of L is independent of scale for both designs. However, thedisk design has a √ N increase in sensitivity, and the inductance of N diskscan be adjusted to match the available SQUID input inductance. The otherway of increasing the sensitivity is to increase the applied voltage V , and thegap in proportion, so that one stays at a fixed fraction of the breakdown fieldof the dielectric. This increases both Φ and L , but since the latter appearsas a square root (for instance in Eqn. 57) this increases the sensitivity as √ V .We note that the SQUID parameters used in the Table correspond toan energy sensitivity S E ≈ h . With a measurement frequency of 0.1Hz, we are a factor 30 below the frequency of 3 Hz, at which 1 /f noisebecomes important for this SQUID. Thus at high frequencies, it has whitenoise with S E ≈ h . Since the integration time is proportional to S E ,it is advantageous to have both white noise and 1 /f noise at the measuringfrequency as close as possible to the quantum limit ∼ h .It should be noted that to carry out any of these experiments is not with-out experimental difficulties. At a rotation speed of 1 Hz, we can generate aflux of ∼ . , which is much larger than the noise level of modern SQUIDs,which is measured in µ Φ / √ Hz. However the flux due to the Earth’s fieldpassing through the SQUID pickup coil would be ∼ Φ , so a very lowfield environment and constancy of the residual flux is essential for success.The size of the signal depends on the geometry of the apparatus and themagnitude of the charging voltage, so it is unlikely that we would be ableto provide an absolute calibration for measuring the Earth’s rotation rateto high accuracy. Instead, to measure the Earth’s rotation rate, one would22ount the cylinder parallel to the Earth’s axis and servo the cylinder rotationto counter that of the Earth and hence reduce the signal to zero. To checkthat one obtains zero signal at zero rotation rate, the apparatus could beoperated at 45 º latitude, so that one can rotate the axis of the cylinder from (cid:107) to ⊥ to the Earth’s axis. (In both (cid:107) & ⊥ cases, the cylinder axis wouldbe at 45 ◦ to local vertical.) One can also check that the signal is zero at zeroapplied voltage.The operation of our apparatus can provide a check on the sometimesdisputed expressions for electromagnetism in rotating frames (In a frame co-rotating with the cylinders there is no current flowing, but there should stillbe the same magnetic field[8]). We may compare our apparatus with fibre-optic and ring-laser gyroscopes [9, 10], which have to be ‘tickled’ to avoidlocking. When this is done, they have been used successfully to measurethe Earth’s rotation rate to a rather greater accuracy than we expect toachieve - e.g. Ref. [10]. However, they test Mach’s principle using opticaltechniques while our apparatus is using very low frequency electromagnetism,and rotating- or vibrating-mass gyroscopes [11] are testing Mach’s principleapplied to mechanics.
9. Conclusions
We have described the principles of operation for a detector of absoluterotation, which relies on the properties of superconductors when charged tohigh voltages. It can also act as a test/confirmation of the laws of electro-magnetism at low frequencies in a rotating frame, and these measurementsmay be performed in a small-scale version of the apparatus. We have demon-strated that if the size of the apparatus is scaled up by a factor ∼
10. Acknowledgements
We thank Antonello Ortolan, Giovanni Carugno & Giuseppe Ruoso of theUniversity of Padua for discussions and suggestions. The designs describedin this paper are being implemented in a project at INFN Legnaro, Padua,Italy. 23his research did not receive any specific grant from funding agencies inthe public, commercial, or not-for-profit sectors.
Appendix A. The Brady spiral
In Ref. [3], Brady considers a multi-turn double spiral consisting of twosuperconducting sheets separated by insulating layers. The sheets have thick-nesses greater than the penetration depth so that there is zero magnetic fieldin their bulk, and the supercurrents flowing on each side of a sheet do soindependently. The spiral is constructed by rolling the two sheets arounda brass cylinder coated with a superconducting lead film, to form a “Swissroll” as represented in Fig. A.7. The inner and outer ends of the outer sheetare connected with a superconducting wire and a SQUID superconductinginput coil is connected between the inner and outer ends of the other sheetso that each sheet forms part of a superconducting loop. The double spiralis surrounded by an earthed superconducting can and the central cylinder isalso earthed. A high voltage is applied to the outer spiral, while the innerspiral (which is connected to the SQUID input coil) is held at earth poten-tial. The potential difference induces opposite charge densities on oppositelyfacing surfaces of the spirals, and also the inner surface of the outer can. Thewhole apparatus is rotated about the spiral axis, and the rotating chargesconstitute surface-current densities, as in the case of the rotating cylindersdiscussed in the body of the paper.In Brady’s account of his work[3], a detailed derivation of its sensitivity isnot given. However, he gives the dimensions and a calculated value, L eff =11 nH, for the effective inductance of his spiral. In our derivation below,we obtain a formula for L eff , and using his dimensions, we obtain the samevalue. His approach of calculating the LI energy in the pickup circuit andcomparing it with the energy sensitivity of the SQUID to calculate integrationtimes is equivalent to ours. We do not know exactly how he treated the effectof stray inductances, but subject to that, we believe that our treatment ofBrady’s design is at least qualitatively similar to his.We illustrate the principles of such a device, showing the double spiral incross section in Fig. A.8. This apparatus can only be treated approximately,because the superconducting connections are made at one axial end of thespirals, so the induced current densities are non-uniform near the connections.However, for most turns of a multi-turn spiral, the currents are uniform. Wenote that, as the separation between the layers is much shorter than their24 igure A.7: Schematic representation of the superconducting sheets and connections ofthe Brady Spiral [3] For clarity, the central cylinder and the outer can are not included.A cross section is shown in Fig. A.8 for the purpose of analysis. total length, the magnetic fields in the gaps for these turns will be parallelto the axis of the spiral. However, there will be tangential components atthe inner and outer ends of the spirals where the induced currents are non-uniform. The magnetic fields in the gaps are screened from the bulk ofthe superconducting sheets by surface currents and hence have the values µ ( J − J ) and µ ( J − J ). These fields point out of the page inside the chargedred surfaces and into the page inside the charged blue surfaces. Assumingno trapped flux, the total flux within the outer can must be zero so we have µ [ N ( J − J ) A − J A ] − µ N ( J − J ) A = 0 , (A.1)where we have generalised to a double spiral of N turns ( N = 2 in theFigures) and made the simplifying assumptions that all the relevant areasare equal to A and all the turns have similar radii, so that the value of therotation current J is the same in each turn. In the case of large N , it followsthat J ≈ J , and for simplicity we shall take them as equal. The large N approximation is in any case necessary, since this will minimise the additionaleffects on the flux from the non-uniform current densities at the ends of thespirals. Representing the axial length of the spirals as (cid:96) , we see that thecurrents in the two external circuits are equal (and opposite) with the value25 igure A.8: The cross section of a 2-turn double spiral. (In practice, the spiral wouldhave many turns.) The red and blue spirals represent positively and negatively chargedsuperconducting sheets respectively and the blue circle represents a closed superconduct-ing cylinder, which also carries charge. The opposite ends of each sheet are connectedby superconducting wires as shown. The arrows labelled J represent the current densitiesresulting from the rotation; those labelled J and J represent induced supercurrent densi-ties. The total current flowing through the blue link and the SQUID is I s . In the interestsof clarity, the spirals are drawn with quite large spacings between them. In practice, thespirals were actually wound quite tightly[3]. I S = ( J + J ) (cid:96) = 2 J (cid:96) . Since both red and blue circuits are superconducting,the flux inside each must remain constant when the apparatus is charged andset into rotation. This will be ensured by a complicated pattern of inducedsupercurrent flows very near the axial ends of the spirals next to the externalcircuits. These currents could only be calculated by a detailed numericalsimulation, which is beyond the scope of this paper. A representation of thecurrents in one sheet is shown in Fig. A.9.To analyze this situation, we use the condition that the total flux inthe shorted circuit must remain zero. This means that the superconductingsheets will respond in such a way that the flux coming up inside the redspiral will not cross the top of this sheet and contribute to flux in thatcircuit. Instead, this flux will cross the blue sheet and go back down thegap inside it (See Fig. A.9). In making a detailed calculation of the effectsof flux quantization, we note that the effective inductance of the spirals isvery small, so that we must include an inductance L w due to the connecting26 igure A.9: The blue spiral unwrapped: this shows the rotation current density J , whichis nearly uniform, and the induced supercurrent densities J & J , which together form thecurrent I S passing through the SQUID input coil. The total flux in this superconductingcircuit is zero, so L S I S plus the contribution of the stray inductance L w I S is equal andopposite to that passing through the circuit area A loop due to the effects of J , J & J .The major contribution to this flux is that passing up one side of the sheet and down theother side. This flux only extends a distance ∼ t above the end of the spiral. This pictureis only approximate, as there may be contributions to the flux from more distant turns ofthe spiral. These cannot be calculated in general, because their contribution depends onthe distance of the upper part of A loop from the top of the spiral. These contributions willbe small if the distance is small compared with the radius of the spiral but large comparedwith t . wires (which includes inductance due to the high supercurrent densities atthe ends of the spirals). If we ignored this stray inductance, the flux crossingthe red spiral would be zero. In practice, a small amount crosses the redcircuit and flux quantization ensures that this flux balances L w I S due to thestray inductance of that circuit.The remainder of the flux inside the red circuit crosses the blue circuit,and is balanced by the flux in the stray inductance and the SQUID inputcircuit. Flux quantization in the blue circuit gives: µ ( J − J ) N A − L w I S = I S ( L S + L w ) . (A.2)27sing J (cid:96) = I S /
2, we obtain I S = µ N AJL S + 2 L w + L eff , (A.3)where L eff = µ N A/ (cid:96) is the effective inductance of the spiral as viewedfrom the SQUID connections. This result may also be written: I S = 2 J (cid:96)L eff L S + 2 L w + L eff . (A.4)In the absence of the stray inductance, the best signal to noise would beobtained, as usual, by setting L S = L eff , but in this case we have to set L S = L tot = (2 L w + L eff ). This results in the optimised value for I S beingreduced to: I S = J (cid:96) × L eff L tot = J (cid:96) × L eff L S . (A.5)The electrostatic interaction between charges on the sheets ensures that thecharges rotate with the spirals, so the surface-current density, J , due torotation is the charge per turn multiplied by the rotation rate. Given theabove simplifying assumptions, we have J = ε Er m ω . (A.6)where E is the electric field produced by the applied voltage. We note that N A is proportional to the total cross-sectional area of the Swiss roll, so forgiven E (usually the limiting factor, to avoid breakdown in the insulator), I s is independent of N for a given apparatus dimensions. However, thegap t between the turns is necessarily smaller at large N , so the requiredvoltage V decreases, which is an experimental advantage, although balancedby the fact that thinner turns are less dimensionally stable. We also notethat the number of turns does not alter the effective inductance of a spiral.This inductance is smaller than the stray inductance, which reduces thesensitivity. The input inductances of modern SQUIDs can be as low as L tot ,so the additional complication of a flux transformer is not required.An alternative expression for the optimised I S may be written in termsof the spacing t between turns and the applied voltage V : I s = µ N πr m t × ε ( V /t ) r m ωL S = πN r m L S × V ω/c . (A.7)28here we have written A = 2 πr m t . This can be used in comparing sensitivi-ties.Throughout this discussion, we have assumed that the number of spiralturns, N , is large, which is certainly true in Brady’s set-up where N = 54.The more general case would require numerical simulation as the inducedsupercurrents and fields are non-uniform. We have not attempted this, butbelieve that all corrections to our treatment are ∝ /N . We have also usedthe simplification that all turns of the spiral have essentially the same radius;we expect that using a mean radius r m will be a good approximation. As-suming this, there is an optimum value for the inner radius r i of the spiral forgiven outer radius r o . Reducing r i increases N , but reduces r m in Eqn. A.7.Numerical optimisation gives the optimum r i ≈ . r o , so r m ≈ . r o , whichis very similar to the optimum for our solenoid design.We now discuss the sensitivity of the spiral design. For comparison withour designs, we take a spiral with the same external dimensions, gaps andapplied voltage as our solenoidal design with outside diameter 3.5 cm. Hencewe take the width of the insulating gaps (and also the thickness of the su-perconducting lead sheets) as 0.2 mm. This gives 0.8 mm as the pitch ofthe double spiral. Fitting the spiral on an r i = 1 . N = 28 turns within an outer radiusof r o = 3 . r m = 2 . ∼ µ r o . This gives the effective inductance of the spiral L eff = 3 . L w = 85 . L tot ∼
89 nH.This is much less than L S = 1 . µ H, the input inductance of the SQUID weconsidered in the main text. The easiest way to deal with this is to assumethat another model from the same range has L S = L tot ∼
90 nH and thesame value of S E : this would give the same signal to noise as if an ideal fluxtransformer were used to match L tot with L S = 1 . µ H. Taking all this intoaccount, we find that the ratio between signal to noise for the spiral and thatfor our solenoidal pickup model is:( I S / ( S I ) . ) spiral ( I S / ( S I ) . ) solenoidal = 2 r m r o (cid:20) L eff L tot (cid:21) . N . ∼ . . (A.8)Thus we find that the multi-turn spiral has in principle a similar sensitivityto our solenoidal pickup design. However, the practical problems found byBrady with this design, which are described in the main text, indicate thatthe designs described in this paper are preferable.29 eferences [1] H.A. Rowland & C.T. Hutchinson, On the electromagnetic effect ofconvection-currents, Phil. Mag , 445 (1889).[2] R.M. Brady, A superconducting gyroscope with no moving parts, IEEETrans. Magn. Mag- ∼ rmb4/[4] N. F. Brickman, Rotating Superconductors, Phys. Rev. , 460 (1969).[5] J.C. Satterthwaite & E.T. Gawlinski, Concerning superconducting in-ertial guidance gyroscopes inside superconducting magnetic shields, J.Appl. Phys. , 5829 (1997).[6] J.Berger, Nonlinearity of the field induced by a rotating superconductingshell, Phys. Rev. B , 31. DOI 10.3390/uni-verse6020031[9] C. Ciminelli, F. Dell’Olio, C.E. Campanella, and M.N. Armenise, Pho-tonic technologies for angular velocity sensing, Adv. Opt. Photon. ,370-404 (2010)[10] N Beverini et al. High-Accuracy Ring Laser Gyroscopes: Earth RotationRate and Relativistic Effects, J. Phys.: Conf. Ser. , 012061 (2016).[11] D.M. Rozelle, The Hemispherical Resonator Gyro: From Wineglass tothe Planets, Adv. Astronaut. Sci.134