G Spavieri et al that is,the charge has the dimensions of length times the square sensitivities comparable with charge detection capabilities of root of a force.The corresponding unit was called the Franklin the cryogenic single-electron transistors. (or simply u.e.s)and was such that a force of I dyne would Coulomb's Law is also at the heart of pedagogical act between two unit charges separated by a distance of Icm physics.It is measured and tested by students in laboratory (1 Franklin 1 u.e.s 1cm3/2 g12s-1). classes virtually every day.There are several types of In the rationalized Gaussian system,k was also precision apparatus that are commercially available for student dimensionless and had a numerical value of k =1/4.The experiments on Coulomb's Law.In fact,a Coulomb null added complexity of incorporating a numerical constant into experiment that enables physics students to obtain rigorous the statement of Coulomb's Law is amply compensated by the upper bounds on the photon mass using an apparatus that disappearance of the 4 in the integral relations(as in Gauss operates with subnanovolt signals has been devised by Law).In this system,the unit of the electric charge was derived Crandall [46].It is inexpensive to assemble and use,and the using arguments similar to those of the first case,but the unit experimental arrangement can be adapted for use at several was seldom used. college levels.Of course,interest in the pedagogic aspects of The accepted standard,of course,is the International Coulomb's Law goes well beyond the obvious applications System or SI,within which the unit of electric charge is the of it to electrostatics.By introducing the superposition coulomb [C].The constant k now takes the value (4xeo) principle and the theory of special relativity,students are taught where eo.which has the defined value eo =8.854....x how generalizations of Coulomb's Law lead to Maxwell's 10-12Fm-,is the 'electric constant',i.e.the permittivity of equations [47]. free space or the dielectric constant in vacuo.In this case. In this review,we have presented a description of the k is neither a pure number nor a dimensionless ratio,instead ongoing development of empirical studies of the forces having the dimensions of [eo] between charges,beginning with the very early work by In spite of the ubiquity of the SI,use of the Gaussian people like Cavendish and others that established the nature system persists among many theoretical physicists because of the inverse square law,and then proceeding to the modern it offers a convenient form of k and velocities appear in experiments that search for deviations from exactness of the dimensionless form (instead of v the quotient v/c is used, inverse square law.Both Newton's Law of gravity and where c is the speed of light in vacuo).Of course,the SI is Coulomb's Law possess the same spatial dependence,1/r2. used uniquely for any measurements and all modern electrical and it is fair to say that these statements of the inverse square instruments and devices are calibrated accordingly. law have served as critical markers in guiding much of the development of theoretical physics,in parallel with the long history of improvement of the experimental situation.With 8.Modern interpretations and conclusions the development of modern physics,Newton's Law is now understood to be the weak-field limit of Einstein's general In considering the various tests of Coulomb's Law,or in relativity,while Coulomb's Law,as one statement of the applications of it to diverse physical phenomena,it is important electromagnetic interaction,has been absorbed into the unified to remember that this fundamental tenet of electrostatics is an theory of electro-weak interactions idealization.Therefore,care must be taken when applying it However,according to general relativity,matter and when exploring its limitations,as pointed out by Saranin introduces curvature into space-time and this is what [21].Even so,it finds use in all fields of science.For example, modifies the law of gravitation as originally formulated by it is employed in meteorology to test models of thunderstorm Newton.The result is that the theory is nonlinear in the charge distributions,as done by Stolzenburg and Marshall [42] sense that the principle of superposition of effects is not They considered the charge distribution in a stratiform region valid.On the contrary,Coulomb's Law is not altered in the of a mesoscale convective system and determined that the description of physical phenomena provided by modern unified general vertical charge structure of that region is horizontally theories.In the unified theory,the interaction (attraction extensive over at least 100 km.Charge distributions were or repulsion)between charges is no longer described in modelled in a three-dimensional domain using an approach that terms of forces but rather in terms of an exchange of virtual correctly calculates the electric field due to all the point charges particles (photons)that yields the same effect as the classical and their corresponding images on the basis of Coulomb's electrostatic force.If the photon mass is zero,Coulomb's Law Law.Atmospheric physics,planetary physics,astrophysics remains a fundamental law of electrodynamics,where linearity and plasma physics are vast domains of application for efforts and the principle of superposition of effects are valid.That is, of this type,and this hints at the range of scales over which it generally speaking,even unified field theories would predict, is routinely applied. for the observable interaction between stationary charges,the Coulomb's torsion balance was in essence the first same result predicted by Coulomb's Law,always supposing precision mechanical detector of charge,i.e.the first high- that the photon mass is zero.All this can introduce the thought precision electrometer.The modern analogues of such that Coulomb's Law is thus more fundamental than Newton's instruments are semiconductor-based field-effect devices,the Law and that electromagnetic interactions are a more primary most sensitive of which are cryogenically cooled transistors kind of fundamental interaction at the basis of any attempt of that function at the single-electrons level [43,44].Also, unification of the forces and interactions of nature. recently,a working nanometre-scale mechanical electrometer Finally,in discussing the physical implications of was constructed by Cleland and Roukes [45],who state that, Coulomb's Law,it is tempting to consider the remarkably high in principle,devices such as theirs should ultimately reach degree of precision with which the inverse square law holds, S168 Metrologia,41 (2004)S159-S170G Spavieri et al that is, the charge has the dimensions of length times the square root of a force. The corresponding unit was called the Franklin (or simply u.e.s) and was such that a force of 1 dyne would act between two unit charges separated by a distance of 1 cm (1 Franklin = 1 u.e.s = 1 cm3/2 g1/2 s−1). In the rationalized Gaussian system, k was also dimensionless and had a numerical value of k = 1/4π. The added complexity of incorporating a numerical constant into the statement of Coulomb’s Law is amply compensated by the disappearance of the 4π in the integral relations (as in Gauss’ Law). In this system, the unit of the electric charge was derived using arguments similar to those of the first case, but the unit was seldom used. The accepted standard, of course, is the International System or SI, within which the unit of electric charge is the coulomb [C]. The constant k now takes the value (4π0)−1 where 0, which has the defined value 0 = 8.854.... × 10−12 F m−1, is the ‘electric constant’, i.e. the permittivity of free space or the dielectric constant in vacuo. In this case, k is neither a pure number nor a dimensionless ratio, instead having the dimensions of [0] −1. In spite of the ubiquity of the SI, use of the Gaussian system persists among many theoretical physicists because it offers a convenient form of k and velocities appear in dimensionless form (instead of v the quotient v/c is used, where c is the speed of light in vacuo). Of course, the SI is used uniquely for any measurements and all modern electrical instruments and devices are calibrated accordingly. 8. Modern interpretations and conclusions In considering the various tests of Coulomb’s Law, or in applications of it to diverse physical phenomena, it is important to remember that this fundamental tenet of electrostatics is an idealization. Therefore, care must be taken when applying it and when exploring its limitations, as pointed out by Saranin [21]. Even so, it finds use in all fields of science. For example, it is employed in meteorology to test models of thunderstorm charge distributions, as done by Stolzenburg and Marshall [42]. They considered the charge distribution in a stratiform region of a mesoscale convective system and determined that the general vertical charge structure of that region is horizontally extensive over at least 100 km. Charge distributions were modelled in a three-dimensional domain using an approach that correctly calculates the electric field due to all the point charges and their corresponding images on the basis of Coulomb’s Law. Atmospheric physics, planetary physics, astrophysics, and plasma physics are vast domains of application for efforts of this type, and this hints at the range of scales over which it is routinely applied. Coulomb’s torsion balance was in essence the first precision mechanical detector of charge, i.e. the first high￾precision electrometer. The modern analogues of such instruments are semiconductor-based field-effect devices, the most sensitive of which are cryogenically cooled transistors that function at the single-electrons level [43, 44]. Also, recently, a working nanometre-scale mechanical electrometer was constructed by Cleland and Roukes [45], who state that, in principle, devices such as theirs should ultimately reach sensitivities comparable with charge detection capabilities of the cryogenic single-electron transistors. Coulomb’s Law is also at the heart of pedagogical physics. It is measured and tested by students in laboratory classes virtually every day. There are several types of precision apparatus that are commercially available for student experiments on Coulomb’s Law. In fact, a Coulomb null experiment that enables physics students to obtain rigorous upper bounds on the photon mass using an apparatus that operates with subnanovolt signals has been devised by Crandall [46]. It is inexpensive to assemble and use, and the experimental arrangement can be adapted for use at several college levels. Of course, interest in the pedagogic aspects of Coulomb’s Law goes well beyond the obvious applications of it to electrostatics. By introducing the superposition principle and the theory of special relativity, students are taught how generalizations of Coulomb’s Law lead to Maxwell’s equations [47]. In this review, we have presented a description of the ongoing development of empirical studies of the forces between charges, beginning with the very early work by people like Cavendish and others that established the nature of the inverse square law, and then proceeding to the modern experiments that search for deviations from exactness of the inverse square law. Both Newton’s Law of gravity and Coulomb’s Law possess the same spatial dependence, 1/r2, and it is fair to say that these statements of the inverse square law have served as critical markers in guiding much of the development of theoretical physics, in parallel with the long history of improvement of the experimental situation. With the development of modern physics, Newton’s Law is now understood to be the weak-field limit of Einstein’s general relativity, while Coulomb’s Law, as one statement of the electromagnetic interaction, has been absorbed into the unified theory of electro-weak interactions. However, according to general relativity, matter introduces curvature into space–time and this is what modifies the law of gravitation as originally formulated by Newton. The result is that the theory is nonlinear in the sense that the principle of superposition of effects is not valid. On the contrary, Coulomb’s Law is not altered in the description of physical phenomena provided by modern unified theories. In the unified theory, the interaction (attraction or repulsion) between charges is no longer described in terms of forces but rather in terms of an exchange of virtual particles (photons) that yields the same effect as the classical electrostatic force. If the photon mass is zero, Coulomb’s Law remains a fundamental law of electrodynamics, where linearity and the principle of superposition of effects are valid. That is, generally speaking, even unified field theories would predict, for the observable interaction between stationary charges, the same result predicted by Coulomb’s Law, always supposing that the photon mass is zero. All this can introduce the thought that Coulomb’s Law is thus more fundamental than Newton’s Law and that electromagnetic interactions are a more primary kind of fundamental interaction at the basis of any attempt of unification of the forces and interactions of nature. Finally, in discussing the physical implications of Coulomb’s Law, it is tempting to consider the remarkably high degree of precision with which the inverse square law holds, S168 Metrologia, 41 (2004) S159–S170