Physical implications of Coulomb's Law the frequency was reduced to 250 Hz and the detector was measurement of interaction forces between two macroscopic synchronized with the charging current rather than with the charged bodies. charge itself In view of the high levels of precision achieved in several The best result obtained so far through developments of these tests,it is interesting to consider what possible of the original Cavendish technique is still that from 1971 competing effects gravity might introduce into them.The by Williams et al [15],who improved an earlier experiment result f(a,r)in equation(5)was derived strictly from classical [20].They used five concentric metallic shells in the form electrodynamics.In it.a uniform charge distribution is of icosahedra rather than spheres in order to reduce the errors assumed and the effect of gravity is neglected.As noted due to charge dispersion.A high voltage and frequency signal by Plimpton and Lawton [16].if electrons have weight was applied to the external shells and a very sensitive detector meg the electron density on the conducting sphere must be checked for any trace of a signal related to variable charging asymmetrical,being greater at the bottom where the electrons of the internal shell.The detector worked by amplifying the are pulled by the force of gravity.For the experiment of signal of the internal shell and comparing it with an identical Plimpton and Lawton this effect is insignificant as it leads to a reference signal,progressively out of phase at arhythm of 360 maximum potential difference over the globe of 10-10 V,which per half hour.Any signal from the detector would indicate a is far less than the minimum detectable voltage of 10-6V. violation of Coulomb's Law.In order to avoid introducing Thus,while such a gravitational effect should be negligible unrelated fields,the reference signal and the detector output in the relatively low sensitivity experiment of Plimpton and signal were transmitted by means of optical fibres.The outer Lawton,it could conceivably be important in experimental shell,of about 1.5m diameter,was charged 10kV peak-to-peak tests of higher sensitivity.According to this model,the overall with a 4 MHz sinusoidal voltage.Centred inside this charged effect of gravity is to produce a distortion in what should conducting shell is a smaller conducting shell.Any deviation otherwise be a uniform charge distribution.Of course,the from Coulomb's Law is detected by measuring the line integral more general problem is to account for effects of a non- of the electric field between these two shells with a detection uniform charge distribution regardless of the origin of the sensitivity of about 10-12 V peak-to-peak. non-uniformity.In equation (5),the null result comes from The null result of this experiment expressed in the form of the assumption that Coulomb's Law is valid and that the the photon rest mass squared (equation (14)or equation (10)) charge is distributed uniformly on the sphere.However, is u2=2.3 x 10-19 cm-2.Expressed as a deviation from Shaw [22]objected to the assumption that the charge will Coulomb's Law in the form of equation (7),their result is distribute itself uniformly over a conducting spherical shell, =6 x 10-16,extending the validity of Coulomb's Law by even in the absence of any gravitational effect.In conventional two orders of magnitude beyond the findings of Bartlett et al. electrostatics,the uniform charge distribution for Coulomb and Yukawa potentials follows from the symmetry of the problem and the uniqueness of the solution.If these potentials are not 5.Limits due to the effects of gravity valid there is no guarantee of a uniform charge distribution and thus irregularities in the spherical surface would bias the We have mentioned above that null experiments that test the concept that the inner potential does not depend on the shape validity of Coulomb's Law are typically more precise than of the outer sphere.However,considering that any violation those that attempt to directly measure the interaction force of Coulomb's Law is very small,departures from the expected between charges.One of the problems arising when making uniformity should give [19]only second-order corrections to direct measurements of the force between two macroscopic equations (7)and (8). charged bodies,as done when using a torsion balance,is that the charges are distributed over conducting surfaces of 6.Indirect tests of Coulomb's Law finite size.In the ideal case,Coulomb's Law describes the interaction between two point charges separated by a precisely In addition to the tests discussed in the previous sections,there known distance.In any practical arrangement,even the have also been a number of indirect experimental verifications charge on a microscopically small conducting ball cannot be of Coulomb's Law,and these will be discussed briefly in what considered to be truly point-like-as if placed at the centre- follows. but rather distributed over the ball's surface.If the charged ball is interacting with another charged ball,the distribution 6.1.Geomagnetic and astronomical tests on the surface is no longer uniform and has to be determined using the method of images.Saranin [21]has studied in detail A consequence of Coulomb's Law is that the magnetic field the departures from Coulomb's Law that can occur when two produced by a dipole goes as 1/r3 at distances from its conducting spheres interact electrostatically with each other. centre for which the dipole approximation is valid.For the By computing forces on them as a function of their separation, magnetic field of a planet,this distance is equivalent to about he found that at small distances a switch from repulsion to two planetary radii (at least).If the photon rest mass is not attraction occurs in the general case of arbitrarily but similarly zero-which is equivalent to a violation of Coulomb's Law- charged spheres.The only exception-and in it they always a Yukawa factor e-r/c is introduced in the 1/r terms for the repel each other-is the case in which the charges on the electrostatic and magnetostatic potentials.In this case,the spheres are related as the squares of their radii.The results of magnetic field produced by a dipole no longer goes as 1/r3 Saranin help corroborate the idea that,even in principle,null but contains corrections related to the Compton wavelength experiments can be more precise than tests based on the direct Ac =u=h/myc where my is the photon mass. Me1 ologia,41(2004)s159-S170 S163Physical implications of Coulomb’s Law the frequency was reduced to 250 Hz and the detector was synchronized with the charging current rather than with the charge itself. The best result obtained so far through developments of the original Cavendish technique is still that from 1971 by Williams et al [15], who improved an earlier experiment [20]. They used five concentric metallic shells in the form of icosahedra rather than spheres in order to reduce the errors due to charge dispersion. A high voltage and frequency signal was applied to the external shells and a very sensitive detector checked for any trace of a signal related to variable charging of the internal shell. The detector worked by amplifying the signal of the internal shell and comparing it with an identical reference signal, progressively out of phase at a rhythm of 360˚ per half hour. Any signal from the detector would indicate a violation of Coulomb’s Law. In order to avoid introducing unrelated fields, the reference signal and the detector output signal were transmitted by means of optical fibres. The outer shell, of about 1.5 m diameter, was charged 10 kV peak-to-peak with a 4 MHz sinusoidal voltage. Centred inside this charged conducting shell is a smaller conducting shell. Any deviation from Coulomb’s Law is detected by measuring the line integral of the electric field between these two shells with a detection sensitivity of about 10−12 V peak-to-peak. The null result of this experiment expressed in the form of the photon rest mass squared (equation (14) or equation (10)) is µ2 = 2.3 × 10−19 cm−2. Expressed as a deviation from Coulomb’s Law in the form of equation (7), their result is ε = 6 × 10−16, extending the validity of Coulomb’s Law by two orders of magnitude beyond the findings of Bartlett et al. 5. Limits due to the effects of gravity We have mentioned above that null experiments that test the validity of Coulomb’s Law are typically more precise than those that attempt to directly measure the interaction force between charges. One of the problems arising when making direct measurements of the force between two macroscopic charged bodies, as done when using a torsion balance, is that the charges are distributed over conducting surfaces of finite size. In the ideal case, Coulomb’s Law describes the interaction between two point charges separated by a precisely known distance. In any practical arrangement, even the charge on a microscopically small conducting ball cannot be considered to be truly point-like—as if placed at the centre— but rather distributed over the ball’s surface. If the charged ball is interacting with another charged ball, the distribution on the surface is no longer uniform and has to be determined using the method of images. Saranin [21] has studied in detail the departures from Coulomb’s Law that can occur when two conducting spheres interact electrostatically with each other. By computing forces on them as a function of their separation, he found that at small distances a switch from repulsion to attraction occurs in the general case of arbitrarily but similarly charged spheres. The only exception—and in it they always repel each other—is the case in which the charges on the spheres are related as the squares of their radii. The results of Saranin help corroborate the idea that, even in principle, null experiments can be more precise than tests based on the direct measurement of interaction forces between two macroscopic charged bodies. In view of the high levels of precision achieved in several of these tests, it is interesting to consider what possible competing effects gravity might introduce into them. The result f (a, r)in equation (5) was derived strictly from classical electrodynamics. In it, a uniform charge distribution is assumed and the effect of gravity is neglected. As noted by Plimpton and Lawton [16], if electrons have weight meg the electron density on the conducting sphere must be asymmetrical, being greater at the bottom where the electrons are pulled by the force of gravity. For the experiment of Plimpton and Lawton this effect is insignificant as it leads to a maximum potential difference over the globe of 10−10 V, which is far less than the minimum detectable voltage of 10−6 V. Thus, while such a gravitational effect should be negligible in the relatively low sensitivity experiment of Plimpton and Lawton, it could conceivably be important in experimental tests of higher sensitivity. According to this model, the overall effect of gravity is to produce a distortion in what should otherwise be a uniform charge distribution. Of course, the more general problem is to account for effects of a non￾uniform charge distribution regardless of the origin of the non-uniformity. In equation (5), the null result comes from the assumption that Coulomb’s Law is valid and that the charge is distributed uniformly on the sphere. However, Shaw [22] objected to the assumption that the charge will distribute itself uniformly over a conducting spherical shell, even in the absence of any gravitational effect. In conventional electrostatics, the uniform charge distribution for Coulomb and Yukawa potentials follows from the symmetry of the problem and the uniqueness of the solution. If these potentials are not valid there is no guarantee of a uniform charge distribution and thus irregularities in the spherical surface would bias the concept that the inner potential does not depend on the shape of the outer sphere. However, considering that any violation of Coulomb’s Law is very small, departures from the expected uniformity should give [19] only second-order corrections to equations (7) and (8). 6. Indirect tests of Coulomb’s Law In addition to the tests discussed in the previous sections, there have also been a number of indirect experimental verifications of Coulomb’s Law, and these will be discussed briefly in what follows. 6.1. Geomagnetic and astronomical tests A consequence of Coulomb’s Law is that the magnetic field produced by a dipole goes as 1/r3 at distances from its centre for which the dipole approximation is valid. For the magnetic field of a planet, this distance is equivalent to about two planetary radii (at least). If the photon rest mass is not zero—which is equivalent to a violation of Coulomb’s Law— a Yukawa factor e−r/λC is introduced in the 1/r terms for the electrostatic and magnetostatic potentials. In this case, the magnetic field produced by a dipole no longer goes as 1/r3 but contains corrections related to the Compton wavelength λC = µ−1 = h/m ¯ γ c where mγ is the photon mass. Metrologia, 41 (2004) S159–S170 S163