292 REVIEWS OF MODERN PHYSICS.JULY 1971 w is as big as possible to minimize thermal noise com- corrected to make V obey proper boundary conditions, pared to the signal.We choose arbitrarily 2asheu=1 m, is reduced by a factor o(kd),where d is the k a=x=/4.This gives f(x)0.16 (almost ) distance between plates of the capacitors,and between cos k a=2-1/2,and either capacitor and the grounded shield.As a result, (r)0.12(ur)2Q/a, sensitivity to4 is reduced by a factor≈kd≈l0s a≈2X108Hz.(6.15) 10-,meaning that even if the postulated detection effi- One may ask if some other geometry would have ciency could be achieved,it would still lead to a result no superior sensitivity to the spherical arrangement used better than that of Bartlett et al.(1970). in most previous experiments.To deal with this,we Our theorem,Eq.(6.19),also refutes the suggestion prove the following: of Franken and Ampulski (1971)that a "table-top" Theorem:If field generating apparatus and field apparatus (of dimension D)could be used to detect a detectors are confined to a region with maximum shift of 6(@2)=(uc)2 in the resonance frequency in dimensions of length D,then effects of a finite photon an LC circuit because o2 should really be given by mass u are of order o[(uD)2]or smaller,regardless of 02=1/LC(uc)2.The problems inherent in this field oscillation frequency. suggestion are analyzed by Goldhaber and Nieto, (1971),but the crucial point is that the natural Proof:We recall from Sec.II the expression for the resonant frequency wg=(LC)-1/cannot be shifted field in momentum space by more than a fraction of order (uD)2,since the Faa(k)=-(4m/c)i[1/(p2-k2)](koJa(k)-kgJa(k)), felds acting on charges and currents in the circuit are only changed by this amount.Therefore,in particu- (6.16) lar,a resonance at w=uc will occur for wguc,not and rewrite this as wg=0.The photon mass effect on the resonant fre- 产aa()=[1-u2/(2-2)门Fso(k),(6.17)） quency is practically negligible.The same conclusion has also been reached by Kroll (1971),Park and where Fa8(0)()is the field produced with u=0.In Williams (1971),Meyer (1971),and Boulware (1971). coordinate space,with fixed frequency w,we get Armed with these results we may assert confidently Fa8(w,I)=FaB(0)(w,I)+6FaB(@,I), that the spherical geometry is superior to any other for Coulomb's law tests.The reason is alluded to by 8Faa(w,r)=-(u2/4x)∫dr'exp(i|k||r-r'l） Bartlett et al.(1970),and Bartlett and Phillips (1969) and was doubtless familiar to earlier workers.In con- XFa8o(w,r)|r-r-1,(6.18)） ventional (=0)electrodynamics,an oscillating As long as the integral converges well it provides an spherically symmetric potential cannot occur within excellent means of estimating corrections due to finite radius r<R,unless there is an oscillating charge density photon mass.In particular,if charges and currents are within that region.In performing an experiment to confined to a region of dimensions <D in all directions, measure such a potential,it is always hard to exclude and observations are made in that region,then we have stray inductance effects with plane parallel geometry, but only an actual charge leakage could produce a F=(uD)Fa() (6.19 spurious effect in the spherical geometry.So,since,as where Fas)is a typical value of F)in the region. we have just seen,the expected effect of u0 is of This result is not altered by the fact that radiation similar size for apparatus of similar dimensions,the may be emitted from the region,since it is easy to spherical arrangement is preferable because it will have verify that the integral over the radiation fields is of a comparable signal but a smaller synchronous back- the same order (even smaller if k D<1). ground voltage than any other geometry. What are the ultimate limits on the Cavendish This theorem contradicts the conclusion of Gertsen- method?Let us consider possible improvements on the shtein and Solovei (1969)that a limit on 1000 times experiment of Williams et al.(1971).First of all,one can smaller than the geomagnetic limit could be obtained by reduce the Johnson noise voltage (Johnson,1928;Ny- a proposed electrical experiment with a geometry of quist,1928)by increasing the time of observation. parallel planes.A capacitor is driven with an oscillating Williams et al.would be able to obtain a limit compara- voltage.A second capacitor is shielded from the first ble to the geomagnetic value (and five times better than by a grounded conducting sheet parallel (geometrically) their present result)by running their experiment for a to both capacitors.The voltage across the second year-about the maximum time one could practically capacitor is interpreted as due to the passage of a consider (Faller,private communication).26 Secondly, longitudinal electric wave through the conducting shield.They correctly deduce what in our notation 26 In connection with lowering noise voltage,we should note would be the statement that the potential across the the possibility of using low temperatures to make a super- conducting apparatus.R.Y.Chiao has pointed out to us that second capacitor is=(u2/k2)V.However,they assume voltage sensitivity of better than 10-1 V'can be achieved,about that V is the potential that would exist in the absence a thousand times better than the sensitivity of Williams etal. of a conducting shield.When this assumption is However,cooling such a large apparatus to liquid helium tem- peratures is a formidable task