G Spavieri et al As an alternative to equations (7)and (8),de Broglie [9] The voltage across the inductor of capacity C is then considered a simple generalization of Maxwell's equations given [15]by involving a small non-zero rest mass of the photon.In this case,two charges will repel each other by a Yukawa force V(r)-V(a)= -(a2-r2).(14) C 6 derived from the potential Save for the standard termq/C(which is zero when there U(r)= e-ur e-rlic 一= (9) is no charge on the inner shell),the term dependent on u in r equation(14)coincides with that of equation (10). where u=myc/h=c is the inverse Compton wavelength of the photon.In the limit ua <1,U(r)=1/r-u+u2r 4.Direct tests of Coulomb's Law and equation(6)yields After the development of the phase-sensitive detectors such V(r)-V(a) V(a) 6u2a2-r为 as lock-in amplifiers,new and more sensitive attempts to test (10) Coulomb's Law were made such as the ones by Plimpton and Lawton [16].Cochran and Franken [17],and Bartlett Although U(r)contains the term u,what is tested and Phillips [18].In this section,we consider Maxwell's experimentally is the result of equation (10).Since the other derivation (equations(6)and(7))applied to the simple case tests of Coulomb's Law are explicitly sensitive to u and not to of a conducting sphere containing a smaller concentric sphere u,the quadratic dependence of V(r)-V(a)on u makes a test The potential of the outer sphere is raised to a value V and based on this approach comparable to other tests.Thus,the the potential difference between them is measured.The actual potential difference V(r)-V(a)is not zero if Coulomb's Law shape of these conductors should not be relevant because the is invalid or,equivalently,if the photon rest mass is non-zero. electric field inside a cavity of any shape vanishes unless For direct tests of Coulomb's Law that consist of measuring Coulomb's Law is violated.Thus,Cochran and Franken [17] the static potential difference of charged concentric shells,one could use conducting rectangular boxes in their experiment may use either equation(7)or equation(10).However,one can and set a limit of s≤0.9×l0-l also test Coulomb's Law by determining u with independent, The experiments of both Cavendish and Maxwell required indirect methods.In general,these would rely on either finding connecting the inner sphere to an electrometer.The accuracy possible variations due to the presence of the Yukawa potential of the experiment was thus limited by fluctuations in the (9)or on the standard fields of massless electrodynamics,such contact potentials while measuring the inner sphere's voltage as,e.g.,measurements at either large distances or long times, Another problem was that of spontaneous ionization between where the percentage effect would be much higher.Typical the spheres.These problems were overcome by Plimpton and of these approaches are those that involve the magnetic field Lawton [16]by using alternating potentials.They developed a of the Earth.For example,one might consider (a)satellite quasi-static method and charged the outer sphere with a slowly verification that the magnetic field of the Earth falls off as alternating current.The potential difference between the inner 1/r3 out to distances at which the solar wind is appreciable and outer spheres was detected with a resonant frequency [10],(b)observation of the propagation of hydromagnetic electrometer.It consisted of an undamped galvanometer with waves through the magnetosphere [11],(c)application of the amplifier.placed within the globes,and with the input resistor Schrodinger external field method [12],or other methods such of the amplifier forming a permanent link connecting them as those described below.The three approaches outlined above so as to measure any variable potential difference.No effect should all give roughly the same limit,u10-11 cm-1. was observed when a harmonically alternating high potential In the high-frequency (direct)null test of Coulomb's Law V (>3000V),from a condenser generator operating at the described below,it is convenient to start from a relativistically low resonance frequency of the galvanometer,was applied invariant linear generalization of Maxwell's equations,namely to the outer globe.The sensitivity was such that a voltage the Proca equations [13],which allow for a finite rest mass of of 10-6V was easily observable above the small level of the photon.Proca's equations for a particle of spin I and mass background noise.With this technique they succeeded in my are [14] reducing Maxwell's limit to s≥2×l0-9. 4π +2)Ay= (11) Another of the classic 'null experiments'that tests the exactness of the electrostatic inverse square law was performed and Gauss'Law becomes by Bartlett et al [19].In this experiment,the outer shell of a spherical capacitor was raised to a potential V with respect to 7·E=4rp-up (12) a distant ground and the potential difference V(r)-V(a)of Equation (12)may be applied to two concentric, equations (7)and (10)induced between the inner and outer conducting,spherical shells of radii r and a (a >r)with an shells was measured.Five concentric spheres were used and a inductor across (i.e.in parallel with)this spherical capacitor. potential difference of 40kV at 2500 Hz was imposed between If a potential Voe is applied to the outer shell,the resulting the two outer spheres.A lock-in detector with a sensitivity electric field is [15] of about 0.2 nV measured the potential difference between the inner two spheres.Any deviation in Coulomb's law shouldlead E(r)=(gr-2u2Voeir)i, (13) to a non-null result for V(r)-V(a)proportional to s as shown by equation (7).The result obtained by these authors was where g is the total charge on the inner shell. s<I x 10-13.A comparable result was found even when S162 Metrologia,41 (2004)S159-S170G Spavieri et al As an alternative to equations (7) and (8), de Broglie [9] considered a simple generalization of Maxwell’s equations involving a small non-zero rest mass of the photon. In this case, two charges will repel each other by a Yukawa force derived from the potential U (r) = e−µr r = e−r/λC r , (9) where µ = mγ c/h¯ = λ−1 C is the inverse Compton wavelength of the photon. In the limit µa  1, U (r) = 1/r − µ + 1 2µ2r and equation (6) yields V (r) − V (a) V (a) = −1 6 µ2 (a2 − r2 ). (10) Although U (r) contains the term µ, what is tested experimentally is the result of equation (10). Since the other tests of Coulomb’s Law are explicitly sensitive to µ2 and not to µ, the quadratic dependence of V (r)−V (a) on µ makes a test based on this approach comparable to other tests. Thus, the potential difference V (r)−V (a) is not zero if Coulomb’s Law is invalid or, equivalently, if the photon rest mass is non-zero. For direct tests of Coulomb’s Law that consist of measuring the static potential difference of charged concentric shells, one may use either equation (7) or equation (10). However, one can also test Coulomb’s Law by determining µ with independent, indirect methods. In general, these would rely on either finding possible variations due to the presence of the Yukawa potential (9) or on the standard fields of massless electrodynamics, such as, e.g., measurements at either large distances or long times, where the percentage effect would be much higher. Typical of these approaches are those that involve the magnetic field of the Earth. For example, one might consider (a) satellite verification that the magnetic field of the Earth falls off as 1/r3 out to distances at which the solar wind is appreciable [10], (b) observation of the propagation of hydromagnetic waves through the magnetosphere [11], (c) application of the Schrodinger external field method [12], or other methods such ¨ as those described below. The three approaches outlined above should all give roughly the same limit, µ
10−11 cm−1. In the high-frequency (direct) null test of Coulomb’s Law described below, it is convenient to start from a relativistically invariant linear generalization of Maxwell’s equations, namely the Proca equations [13], which allow for a finite rest mass of the photon. Proca’s equations for a particle of spin 1 and mass mγ are [14] ( + µ2 )Aν = 4π c Jν (11) and Gauss’ Law becomes ∇ · E = 4πρ − µ2 ϕ. (12) Equation (12) may be applied to two concentric, conducting, spherical shells of radii r and a (a>r) with an inductor across (i.e. in parallel with) this spherical capacitor. If a potential V0eiωt is applied to the outer shell, the resulting electric field is [15] E(r) =  qr−2 − 1 3µ2 V0eiωtr  r,ˆ (13) where q is the total charge on the inner shell. The voltage across the inductor of capacity C is then given [15] by V (r) − V (a) =
a r E · dl = q C − µ2 V0eiωt 6 (a2− r2 ). (14) Save for the standard term q/C (which is zero when there is no charge on the inner shell), the term dependent on µ in equation (14) coincides with that of equation (10). 4. Direct tests of Coulomb’s Law After the development of the phase-sensitive detectors such as lock-in amplifiers, new and more sensitive attempts to test Coulomb’s Law were made such as the ones by Plimpton and Lawton [16], Cochran and Franken [17], and Bartlett and Phillips [18]. In this section, we consider Maxwell’s derivation (equations (6) and (7)) applied to the simple case of a conducting sphere containing a smaller concentric sphere. The potential of the outer sphere is raised to a value V and the potential difference between them is measured. The actual shape of these conductors should not be relevant because the electric field inside a cavity of any shape vanishes unless Coulomb’s Law is violated. Thus, Cochran and Franken [17] could use conducting rectangular boxes in their experiment and set a limit of ε
0.9 × 10−11. The experiments of both Cavendish and Maxwell required connecting the inner sphere to an electrometer. The accuracy of the experiment was thus limited by fluctuations in the contact potentials while measuring the inner sphere’s voltage. Another problem was that of spontaneous ionization between the spheres. These problems were overcome by Plimpton and Lawton [16] by using alternating potentials. They developed a quasi-static method and charged the outer sphere with a slowly alternating current. The potential difference between the inner and outer spheres was detected with a resonant frequency electrometer. It consisted of an undamped galvanometer with amplifier, placed within the globes, and with the input resistor of the amplifier forming a permanent link connecting them, so as to measure any variable potential difference. No effect was observed when a harmonically alternating high potential V (>3000 V), from a condenser generator operating at the low resonance frequency of the galvanometer, was applied to the outer globe. The sensitivity was such that a voltage of 10−6 V was easily observable above the small level of background noise. With this technique they succeeded in reducing Maxwell’s limit to ε 2 × 10−9. Another of the classic ‘null experiments’ that tests the exactness of the electrostatic inverse square law was performed by Bartlett et al [19]. In this experiment, the outer shell of a spherical capacitor was raised to a potential V with respect to a distant ground and the potential difference V (r) − V (a) of equations (7) and (10) induced between the inner and outer shells was measured. Five concentric spheres were used and a potential difference of 40 kV at 2500 Hz was imposed between the two outer spheres. A lock-in detector with a sensitivity of about 0.2 nV measured the potential difference between the inner two spheres. Any deviation in Coulomb’s law should lead to a non-null result for V (r)−V (a) proportional to ε as shown by equation (7). The result obtained by these authors was ε
1 × 10−13. A comparable result was found even when S162 Metrologia, 41 (2004) S159–S170