PHYSICAL REVIEW A VOLUME 33,NUMBER 1 JANUARY 1986 Comments Comments are short papers which comment on papers of other authors previously published in the Physical Review.Each Comment should state clearly to which paper it refers and must be accompanied by a brief abstract.The same publication schedule as for regular articles is followed,and page proofs are sent to authors. Improved result for the accuracy of Coulomb's law:A review of the Williams,Faller,and Hill experiment Lewis P.Fulcher Department of Physics and Astronomy,Bowling Green State University, Bowling Green,Ohio 43403-0224 (Received 28 January 1985) About 15 years ago Williams,Faller,and Hill (WFH)carried out a modern version of the classical null experiment which tests Coulomb's law.Their high-precision measurements of the voltage furnish the most stringent upper limit for the parameter 8,which measures a (possible)deviation from the inverse-square form of Coulomb's law.We show that the experiment of WFH is actually about three times more sensitive to the parameter 8 than they supposed by carrying out a careful analysis of the geometrical factor involved in the interpretation of their experiment.The new upper limit for 8 is(1.0+1.2)x10-16. I.INTRODUCTION down by about 14 orders of magnitude.Interest in experi- ments that improve the accuracy with which Coulomb's law Experimental tests of the accuracy of Coulomb's law have is known to be true will undoubtedly continue since any enjoyed a long and interesting history,as summarized in positive result for 6 would have profound consequences for Table I.Most of the experiments have used the principle the structure of Maxwell's theory of electricity and magne- established in the classic experiment of Cavendish,where a tism and the theories based on this. search for a charge or potential difference inside a charged, The smallest value for the upper limit has been obtained closed conductor is carried out.The experimenters have by Williams,Faller,and Hill2 (WFH)and it is the result often interpreted their results as a means of setting an upper usually quoted in the textbooks.3 In order to obtain an esti- limit for a parameter 8,which is introduced as a (possible) mate of the sensitivity of their experiment to the parameter violation of Coulomb's law in the scale invariant form 8,they used the geometrical factor, r-2+.During the 200 yr period described by Table I the sensitivities of the experiments have increased so much that M(a,b)- a+b 4a2 (1) the upper limit for the absolute value of 8 has been pushed a-b 2-b2 TABLE I.Summary of experimental tests of Coulomb's law Upper limit Experimenter Apparatus or for the (date) geometry parameter 8 Robison (1769) Gravitational torque on a pivot arm 0.06 Cavendish (1773) Two concentric metal spheres 0.02 Coulomb (1785) Torsion balance 0.04 Maxwell (1873) Two concentric spheres 21600 Plimpton and Lawton (1938) Two concentric spheres 2×10-9 Cochran (1967) Concentric cubical conductors 9.2×10-12 Bartlett,Goldhagen, and Phillips (1970) Five concentric spheres 1.3×10-13 Williams,Faller, and Hill (1971) Five concentric icosahedrons (2.7±3.1)×10-16 33 759 c1986 The American Physical Society
PHYSICAL REVIE%' A VOLUME 33, NUMBER 1 Comments JANUARY 1986 proofs are sent to authors. Improved result for the accuracy of Coulomb's law: A review of the Williams, Faller, and Hill experiment Lewis P. Fulcher Department of Physics and Astronomy, Bowling Green State University, Bowling Green, Ohio 43403-0224 {Received 28 January 1985) About 15 years ago illiams, Faller, and Hill (%'FH) carried out a modern version of the classical null experiment which tests Coulomb's law. Their high-precision measurements of the voltage furnish the most stringent upper limit for the parameter 8, which measures a (possible) deviation from the inverse-square form of Coulomb's law. %e show that the experiment of ~FH is actually about three times more sensitive to the parameter 5 than they supposed by carrying out a careful analysis of the geometrical factor involved in the interpretation of their experiment. The new upper limit for 5 is (1.0+1,2) x10 I. INTRODUCTION Experimental tests of the accuracy of Coulomb's law have enjoyed a long and interesting history, as summarized in Table I. Most of the experiments have used the principle established in the classic experiment of Cavendish, where a search for a charge or potential difference inside a charged, closed conductor is carried out. The experimenters have often interpreted their results as a means of setting an upper limit for a parameter 8, which is introduced as a (possible) violation of Coulomb's law in the scale invariant form r +~. During the 200 yr period described by Table I the sensitivities of the experiments have increased so much that the upper limit for the absolute value of 5 has been pushed down by about 14 orders of magnitude. Interest in experiments that improve the accuracy with which Coulomb's law is known to be true will undoubtedly continue since any positive result for 8 would have profound consequences for the structure of Maxwell's theory of electricity and magnetism and the theories based on this. ' The smallest value for the upper limit has been obtained by Williams, Faller, and Hill (WFH) and it is the result usually quoted in the textbooks. 3 In order to obtain an estimate of the sensitivity of their experiment to the parameter 5, they used the geometrical factor, t M(a, b) =——ln 1 a a+ P — 4a2 ln 2 b a —b a2 —P2 TABLE I. Summary of experimental tests of Coulomb's law. Experimenter (date) Apparatus or geometry Upper limit for the parameter 8 Robison (1769) Cavendish {1773) Coulomb (1785) Maxwell (1&73) Plimpton and Lawton (1938} Cochran (1967) Bartlett, Goldhagen, and Phillips (1970) Williams, Faller, and Hill (1971) Gravitational torque on a pivot arm Two concentric metal spheres Torsion balance Two concentric spheres Two concentric spheres Concentric cubical conductors Five concentric spheres Five concentric icosahedrons 0.06 0.02 0.04 1 21 600 2x 10—9 9.2x 10 1.3 x 10 {2.7 +3.1)x 10-l6 33 759 1986 The American Physical Society
760 COMMENTS 33 which was derived by Maxwell for two concentric spheres dius a containing a uniformly distributed charge (Ref.6), of radii a and b.Since the apparatus of WFH consisted of five concentric conductors,it is apparent that a more refined calculation should be done in order to obtain an accurate v)-2ar是-r+a)1+8-r-al1+1. Q (2) result for the sensitivity of the experiment.In particular,it where the use of absolute value sign extends the validity of is important that the contributions of all sources of the Eq.(2)to the entire range of r.Since both the BGP and (possible)anomalous internal fields be included.(Maxwell WFH experiments were done with alternating currents,the considered only a single charged source.)Indeed,as we outermost two spheres are assumed to have equal but oppo- show below,the appropriate generalization of Eq.(1)is the site charges at all times during the experiment.Thus,using same as that derived by Bartlett,Goldhagen,and Phillips superposition,we calculate the potential inside a pair of (BGP),whose experiment was also based on five concentric spheres of radii r4 (charge and r3 (charge -with conductors.The correct geometrical factor is about three r4>r3,which is given by times larger than that calculated by WFH.This factor of 3 leads to an improvement in sensitivity to the parameter 8 by V(r)= 24r-6n+r)1*-(4-)1+8] 0 the same amount. 2r-2+r)1+6-(r,-r)1+]. (3) II.THE GEOMETRICAL FACTOR The measurement of the induced voltage signal is carried Our derivation for the geometrical factor for an arrange- out interior to both of the charged spheres.Thus,we re- ment of four concentric spheres begins with the exact ex- quire the expression for the potential difference between a pression for the potential at distance r from a sphere of ra- pair of pick points interior to both spheres,that is ）-n-24n-两+，1+--+1-2是-两+n1*0-(-)1*a 2ni-两l+w4-1+2n&-+n-n-w*y. Q (4) In order to obtain a simplified expression,it is necessary to (6.4+7.3)x 10-13 V peak to peak,which is of course con- expand Eq.(4)in powers of 8 using x!+8=x(1+8Inx). sistent with zero.WFH used Eq.(1)to take into account After judiciously combining the various logarithmic terms, the geometrical factors.Substituting b=1.6a there yields we obtain a first-order expression for the potential differ- M=0.232 and leads to their value listed in Table I. ence.This may be written as a ratio, To apply the considerations of Sec.II to the concentric V(r2)-V(r)- icosahedrons requires some geometrical interpretation.It V(r3)-V(r4) 84-Mr3,r2)-M(r,川 seems reasonable that one could choose an effective radius r4-r3 to incorporate the differences between the spheres and the _8[Mr4,r2)-M(r4,i川， icosahedrons.For the time being we follow the considera- (5) r4-r3 tions of WFH,who were confronted with a similar problem in analyzing their experiment to obtain an upper limit for where we have used the lowest-order expression for the the photon's rest mass.?Their value for the ratio of the ef- capacitance of the two external spheres to eliminate the un- fective radius to the length L of the triangular sides depend- known charge O.The geometrical factor M in Eq.(5)is the ed upon how they determined the effective radius.3 If the Maxwellian factor defined in Eq.(1)above.From the form icosahedron is to have the same surface area as the sphere, of Eq.(5)it is apparent that a single Maxwellian factor is then not adequate for an accurate calculation of sensitivity of the R=0.83L four conductor experiments,but that the correct expression (6) involves a combination of four such factors.Our result for If instead the radius of the inscribed sphere is to be used, the potential ratio agrees with that calculated by BGP in then Ref.5. R=0.76L (7) III.RESULTS WFH suggested that uncertainties in the definition of the effective radius might lead to a difference of 10-20%in the The experiments of BGP and WFH used five concentric determination of an upper limit to the photon rest mass. conductors.However,the middle conductor was not an ac- Scale invariance makes the problem of geometrical interpre- tive element in either experiment;its function was to screen tation much simpler for the calculation of the sensitivity to out stray electric fields that leaked into the interior of the 8.Since all of the factors that appear in Eq.(5)depend on charged outer pair.Thus,the four-conductor analysis of the the ratio of the radii,one obtains the same result for the previous section is sufficient.WFH maintained a potential sensitivity using either Eqs.(6)or (7)or any other linear difference of 10 kV between their outermost pair of spheres relationship between the effective radius and L that is the at a frequency of 4 MHz.Using a phase-sensitive detector same for all conductors. they searched for a potential difference between the inner- The lengths of the sides of the four icosahedrons are most pair of spheres.Their result for the induced signal is 23.37,37,and 50 in.,respectively.Using the surface
33 which was derived by Maxwell4 for two concentric spheres of radii a and b. Since the apparatus of %FH consisted of five concentric conductors, it is apparent that a more refined calculation should be done in order to obtain an accurate result for the sensitivity of the experiment. In particular, it is important that the contributions of all sources of the (possible) anomalous internal fields be included. (Maxwell considered only a single charged source. ) Indeed, as we show below, the appropriate generalization of Eq. (I) is the same as that derived by Bartlett, Goldhagen, and Phillips5 (BGP), whose experiment was also based on five concentric conductors. The correct geometrical factor is about three times larger than that calculated by %'FH. This factor of 3 leads to an improvement in sensitivity to the parameter 5 by the same amount. dius a containing a uniformly distributed charge (Ref. 6) Q, V(.) = [(r+g)1+5 (r g(I+8] 2ar(1 —g') where the use of absolute value sign extends the validity of Eq. (2) to the entire range of r Si.nce both the BGP and WFH experiments were done with alternating currents, the outermost two spheres are assumed to have equal but opposite charges at all times during the experiment. Thus, using superposition, we calculate the potential inside a pair of spheres of radii r4 (charge Q) and rq (charge —Q) with ~4 Q f3 which is given by V(r) = [(rq+ r)'+ —(r4 r)'+ ]- 2r4r(1 —8 ) II. THE GEOMETRICAL FACTOR Q [(& + &)1+5 (& &)1+8] 2rgr(1 —g') (3) Our derivation for the geometrical factor for an arrangement of four concentric spheres begins with the exact expression for the potential at distance r from a sphere of raThe measurement of the induced voltage signal is carried out interior to both of the charged spheres. Thus, we require the expression for the potential difference between a pair of pick points interior to both spheres, that is V(r2) —V(r~) = [(r4+ r2)'+ —(r4 r2)'—+ 2r4r2(1 —82) ) — 2r4r~ (1—52) [(r4+ r~)'+ —(r4 rt)'+ ]- [(rs+r2)'+ —(rz —r2)'+ ]+ 2 [(rg+r~)'+ —(r3 rt) ] 2r)r2(1 —5') 2rgrt(1 —g') (4) In order to obtain a simplified expression, it is necessary to expand Eq. (4) in powers of 8 using x'+a x(1+8lnx). After judiciously combining the various logarithmic terms, we obtain a first-order expression for the potential difference. This may be written as a ratio, V(r2) —V(r~) 5r4 [M(r3,f2) —M(r), r~) ] V r3 V r4 r4 —rg (5) ~here we have used the lowest-order expression for the capacitance of the two external spheres to eliminate the unknown charge Q. The geometrical factor M in Eq. (5) is the Maxwellian factor defined in Eq. (1) above. From the form of Eq. (5) it is apparent that a single Maxwellian factor is not adequate for an accurate calculation of sensitivity of the four conductor experiments, but that the correct expression involves a combination of four such factors. Our result for the potential ratio agrees with that calculated by BGP in Ref. 5. III. RESULTS The experiments of BGP and %FH used five concentric conductors. However, the middle conductor was not an active element in either experiment; its function was to screen out stray electric fields that leaked into the interior of the charged outer pair. Thus, the four-conductor analysis of the previous section is sufficient. ~FH maintained a potential difference of 10 kV between their outermost pair of spheres at a frequency of 4 MHz. Using a phase-sensitive detector they searched for a potential difference between the innermost pair of spheres. Their result for the induced signal is (6.4 k7.3) X 10 ' U peak to peak, which is of course consistent with zero. WFH used Eq. (1) to take into account the geometrical factors. Substituting b =1.6a there yields ~M ~ =0.232 and leads to their value listed in Table I. To apply the considerations of Sec. II to the concentric icosahedrons requires some geometrical interpretation. It seems reasonable that one could choose an effective radius to incorporate the differences between the spheres and the icosahedrons. For the time being we follow the considerations of %'FH, who were confronted with a similar problem in analyzing their experiment to obtain an upper limit for the photon's rest mass. ' Their value for the ratio of the effective radius to the length L of the triangular sides depended upon how they determined the effective radius. If the icosahedron is to have the same surface area as the sphere, then R =0.83L If instead the radius of the inscribed sphere is to be used, then R =0.76L %'FH suggested that uncertainties in the definition of the effective radius might lead to a difference of 10—20% in the determination of an upper limit to the photon rest mass. Scale invariance makes the problem of geometrical interpretation much simpler for the calculation of the sensitivity to 5 Since all of the fac.tors that appear in Eq. (5) depend on the ratio of the radii, one obtains the same result for the sensitivity using either Eqs. (6) or (7) or any other linear relationship between the effective radius and L that is the same for all conductors. The lengths of the sides of the four icosahedrons are 23~, 37, 37~, and 50 in. , respectively. Using the surface
心 COMMENTS 761 area criterion to define the effective radii leads to the values The new value of 8 determined from Eq.(8)is r1=49.54cm,r2=78.00cm,r3=78.53cm,andr4=105.41 cm.The geometrical factor F,which includes all of the ex- 8=(1.0±1.2)×10-16 (9) pressions on the right-hand side of Eq.(5)except the com- mon factor 8 is thus given by F-0.629.And the sensi- which is a factor of about 2 times smaller than that pub- tivity of the experiment of WFH to 8 can be expressed lished by WFH.The experiment of WFH is about 2+times more sensitive to the parameter 6 than they thought.Equa- V(r2)-V(r) =|F8=0.6298. (8) tion (9)represents the most stringent upper limit for delta V(r3)-V(r4) to date. IAn alternative interpretation of Coulomb's law experiments is as a 4J.C.Maxwell,A Treatise on Electricity and Magnetism (Dover.New means of setting an upper limit for the photon rest mass.It is York,1954),Vol.I. possible to reconcile such an interpretation with the special theory 5D.F.Bartlett,P.E.Goldhagen,and E.A.Phillips,Phys.Rev.D 2. of relativity as pointed out by A.S.Goldhaber and N.M.Nieto, 483(1970). Rev.Mod.Phys.43,277(1971). 6L.P.Fulcher and M.A.Telljohann,Am.J.Phys.44,366 (1976). 2E.R.Williams,J.E.Faller,and H.A.Hill,Phys.Rev.Lett.26, 7Incidentally,the analysis of WFH for the sensitivity of their experi- 721(1971). ment to a possible photon rest mass is correct. 3See,for example,J.D.Jackson,Classical Electrodynamics (Wiley, 8E.R.Williams,Ph.D.thesis,Wesleyan University,1970 (unpub- New York,1975);or R.Resnick and D.Halliday,Physics,Part / lished). (Wiley,New York,1978)
33 area criterion to define the effective radii leads to the values r~ =49.54 cm, r2=78.00 cm, r3 78 ~ 53 cm and r4 105.41 cm. The geometrical factor I, which includes all of the expressions on the right-hand side of Eq. (5) except the common factor 8 is thus given by ~F~ =0.629. And the sensitivity of the experiment of WFH to 5 can be expressed = /F/ /8I=O. 6298 . V r3 —V r4 The new value of 8 determined from Eq. (8) is 8= (1.0+1.2) x10 which is a factor of about 2~ times smaller than that published by %FH. The experiment of %FH is about 2~ times more sensitive to the parameter 5 than they thought. Equation (9) represents the most stringent upper limit for delta to date. 'An alternative interpretation of Coulomb's law experiments is as a means of setting an upper limit for the photon rest mass. It is possible to reconcile such an interpretation with the special theory of relativity as pointed out by A, S. Goldhaber and N. M. Nieto, Rev. Mod. Phys. 43, 277 (1971). 2E. R. Williams, J. E. Faller, and H. A. Hill, Phys. Rev. Lett. 26, 721 (1971). 3See, for example, J, D. Jackson, Classica/ E/ectrodynamics (iley, New York, 1975); or R. Resnick and D. Halliday, Physics, Part II (Wiley, New York, 1978). J. C. Maxwell, A Treatise on E/ectricity and Magnetism (Dover, New York, 1954), Vol. I. 5D. F. Bartlett, P. E. Goldhagen, and E. A. Phillips, Phys. Rev. D 2, 483 (1970). 6L. P. Fulcher and M. A. Telljohann, Am. J. Phys. 44, 366 (1976). 7Incidentally, the analysis of WFH for the sensitivity of their experiment to a possible photon rest mass is correct. E. R. Williams, Ph.D. thesis, %esleyan University, 1970 (unpublished)
