INSTITUTE OF PHYSICS PUBLISHING METROLOGIA Metrologia 41 (2004)S159-S170 P1:S0026-1394(04)80313-X Physical implications of Coulomb's Law G Spavieri,G T Gillies2 and M Rodriguez Centro de Astrofisica Teorica.Facultad de Ciencias,Universidad de Los Andes,Merida, 5101 Venezuela 2 Department of Mechanical and Aerospace Engineering,University of Virginia, PO Box 400746,Charlottesville,VA 22904.USA 3 Departamento de Fisica,FACYT,Universidad de Carabobo,Valencia,2001 Venezuela E-mail:spavieri@ula.ve and gtg@virginia.edu Received 3 March 2004 Published 16 September 2004 Online at stacks.iop.org/Met/41/S159 doi:10.1088/0026-1394/41/5/S06 Abstract We examine the theoretical and experimental foundations of Coulomb's Law and review the various roles it plays not only in electromagnetism and electrodynamics,but also in quantum mechanics,cosmology,and thermodynamics.The many implications of Coulomb's Law draw attention to its fundamental importance within virtually all branches of physics and make this elementary yet profound law one of the most useful of all scientific tools. 1.Introductory historical outlook philosopher who had broad scientific interests in physics, electricity,magnetism,and optics,in addition to chemistry.He Few investigations in physics have enjoyed as sustained an was a politically involved Unitarian preacher and a sympathizer interest as have tests of Coulomb's Law.As has been with the French Revolution,and these aspects of his life forced the case with most of the fundamental laws of physics,it him to move to America with his family in 1794.Priestley was discovered and elucidated through observations of basic is credited with the discovery of oxygen in 1774,which he phenomena.In his research,Coulomb was interested in the produced by focusing sunlight on mercuric oxide.During his mutual interactions between electric charges,a topic that had studies of this 'dephlogisticated air',he noticed that it made been studied previously by Priestley [1],and in fact even him light-headed and that it had a similar effect on animals. earlier,in 1755,during the experimental work of Franklin [2]. The background studies underpinning Coulomb's Law Franklin (1706-1790)was an American printer,writer. began when Franklin took a small sphere made of cork politician,diplomat,and scientist.He is credited with the and placed it inside a charged metallic cup (see figure 1) invention of such practical everyday items as bifocaleyeglasses and observed that it did not move,suggesting that there and a free-standing,wood-burning heater called the "Franklin was no interaction between it and the cup.After Franklin stove'.His principal connection to electrical experimentation communicated his finding to Priestley,the Englishman came via his investigations of the properties of Leyden Jars explained the phenomenon in 1767,providing a line of He is also commonly credited with giving the names'positive reasoning analogous to that used by Newton [3]to formulate and 'negative'to the two opposite species of electrical charge, and enunciate the law of universal gravitation. although his assignment convention was eventually reversed. Underlying Newtonian gravity was the observation that Also living in America at the time was Priestley the gravitational field inside a spherical shell of homogeneous (1733-1804),an English chemist and amateur natural material is null if the field is inversely proportional to the square of the distance r.i.e.if its intensity goes asr-2 4In1752.he flewakite attachedtoasilk string inathunderstorm,and showed By approximating Franklin's cup as a spherical shell,Priestley that a metal key tied to the thread would charge a Leyden jar.(Incidentally. deduced that the observed phenomenon should be physically the next two people who attempted the experiment were killed in the effort.) His experiments with Leyden jars showed that they discharged more easily if on Different Kinds of Airs(1774-1777)and in the three-volume Experiments near a pointed surface.He thus suggested the use of lightning rods. and Observations Relating to Various Branches of Natural Philosophie 5The objects of his chemical studies included'fixed air(carbon dioxide). (1779-1786).By dissolving fixed air in water,he invented carbonated 'nitrous air'(nitric oxide),'marine acid air'(hydrogen chloride),'alkaline air water.He also noticed that the explosion of inflammable air with common (ammonia),'vitriolic air'(sulfur dioxide),'phlogisticated nitrous air'(nitrous air produced dew.Lavoisier repeated this experiment and took credit for oxide,laughing gas).and 'dephlogisticated air'(oxygen).His chemical it.Priestley believed in the phlogiston theory,and was convinced that his writings were published in the three-volume Experiments and Observations discovery of oxygen proved it to be correct. 0026-1394/04/050159+12$30.00 2004 BIPM and IOP Publishing Ltd Printed in the UK S159
INSTITUTE OF PHYSICS PUBLISHING METROLOGIA Metrologia 41 (2004) S159–S170 PII: S0026-1394(04)80313-X Physical implications of Coulomb’s Law G Spavieri1, G T Gillies2 and M Rodriguez3 1 Centro de Astrof´ısica Teorica, Facultad de Ciencias, Universidad de Los Andes, M ´ erida, ´ 5101 Venezuela 2 Department of Mechanical and Aerospace Engineering, University of Virginia, PO Box 400746, Charlottesville, VA 22904, USA 3 Departamento de F´ısica, FACYT, Universidad de Carabobo, Valencia, 2001 Venezuela E-mail: spavieri@ula.ve and gtg@virginia.edu Received 3 March 2004 Published 16 September 2004 Online at stacks.iop.org/Met/41/S159 doi:10.1088/0026-1394/41/5/S06 Abstract We examine the theoretical and experimental foundations of Coulomb’s Law and review the various roles it plays not only in electromagnetism and electrodynamics, but also in quantum mechanics, cosmology, and thermodynamics. The many implications of Coulomb’s Law draw attention to its fundamental importance within virtually all branches of physics and make this elementary yet profound law one of the most useful of all scientific tools. 1. Introductory historical outlook Few investigations in physics have enjoyed as sustained an interest as have tests of Coulomb’s Law. As has been the case with most of the fundamental laws of physics, it was discovered and elucidated through observations of basic phenomena. In his research, Coulomb was interested in the mutual interactions between electric charges, a topic that had been studied previously by Priestley [1], and in fact even earlier, in 1755, during the experimental work of Franklin [2]. Franklin (1706–1790) was an American printer, writer, politician, diplomat, and scientist. He is credited with the invention of such practical everyday items as bifocal eyeglasses and a free-standing, wood-burning heater called the ‘Franklin stove’. His principal connection to electrical experimentation came via his investigations of the properties of Leyden Jars4. He is also commonly credited with giving the names ‘positive’ and ‘negative’ to the two opposite species of electrical charge, although his assignment convention was eventually reversed. Also living in America at the time was Priestley (1733–1804)5, an English chemist and amateur natural 4 In 1752, he flew a kite attached to a silk string in a thunderstorm, and showed that a metal key tied to the thread would charge a Leyden jar. (Incidentally, the next two people who attempted the experiment were killed in the effort.) His experiments with Leyden jars showed that they discharged more easily if near a pointed surface. He thus suggested the use of lightning rods. 5 The objects of his chemical studies included ‘fixed air’ (carbon dioxide), ‘nitrous air’ (nitric oxide), ‘marine acid air’ (hydrogen chloride), ‘alkaline air’ (ammonia), ‘vitriolic air’ (sulfur dioxide), ‘phlogisticated nitrous air’ (nitrous oxide, laughing gas), and ‘dephlogisticated air’ (oxygen). His chemical writings were published in the three-volume Experiments and Observations philosopher who had broad scientific interests in physics, electricity, magnetism, and optics, in addition to chemistry. He was a politically involved Unitarian preacher and a sympathizer with the French Revolution, and these aspects of his life forced him to move to America with his family in 1794. Priestley is credited with the discovery of oxygen in 1774, which he produced by focusing sunlight on mercuric oxide. During his studies of this ‘dephlogisticated air’, he noticed that it made him light-headed and that it had a similar effect on animals. The background studies underpinning Coulomb’s Law began when Franklin took a small sphere made of cork and placed it inside a charged metallic cup (see figure 1) and observed that it did not move, suggesting that there was no interaction between it and the cup. After Franklin communicated his finding to Priestley, the Englishman explained the phenomenon in 1767, providing a line of reasoning analogous to that used by Newton [3] to formulate and enunciate the law of universal gravitation. Underlying Newtonian gravity was the observation that the gravitational field inside a spherical shell of homogeneous material is null if the field is inversely proportional to the square of the distance r, i.e. if its intensity goes as r−2. By approximating Franklin’s cup as a spherical shell, Priestley deduced that the observed phenomenon should be physically on Different Kinds of Airs (1774–1777) and in the three-volume Experiments and Observations Relating to Various Branches of Natural Philosophie (1779–1786). By dissolving fixed air in water, he invented carbonated water. He also noticed that the explosion of inflammable air with common air produced dew. Lavoisier repeated this experiment and took credit for it. Priestley believed in the phlogiston theory, and was convinced that his discovery of oxygen proved it to be correct. 0026-1394/04/050159+12$30.00 © 2004 BIPM and IOP Publishing Ltd Printed in the UK S159
G Spavieri et al by the force of gravity acting on them.By knowing their weight,and by repeating the measurements at different distances,it was possible to calculate the size of the electrical force,evaluate its dependence on distance,and thus verify the metallic cup exactness of the hypothesized 1/r2 law. From his observations.Robison deduced that the law must have the functional form F x- (1) Cork where g represents a measure of the precision to which the 1/r2 behaviour is verified.He found an upper limit of 0.06 for s and thus could state that for masses in electrical repulsion to each other,the force went asr-206.However. Figure 1.Franklin placed a small sphere made of cork inside a for electrical attraction his limit was weaker,stated asr metallic cup,and when this was charged he observed that the sphere where c<2,but still essentially confirming the expected did not move,suggesting that there was no interaction between it r-2 dependence.Unfortunately,Robison did not publish and the cup.Priestley explained the phenomenon in 1767,providing his results until 1801,and by then Coulomb [7]had already a line of reasoning analogous to that used by Newton for the law of universal gravitation,implying that the field has an inverse square presented his.The parameter s appearing in equation(1)and dependence on distance.In the subsequent tests,instead of the cork related to the precision of the 1/r2 behaviour,is not very sphere inside a metallic cup,experimentalists considered a metallic useful from a theoretical point of view but is retained here shell enclosed within an outer charged shell or several because of its common historical use.Subsequent theoretical concentric shells. developments and improved understanding of the foundations for high precision tests of Coulomb's Law have led to the use analogous to the gravitational case and he thus concluded that of the quantityu=myc/=cwhich,as considered below, the electric force,like the gravitational force,must depend on is well motivated theoretically and represents the inverse distance as r-2.The lack of an observed ponderable force Compton wavelength of a photon of mass m.Coulomb's on the cork sphere inside the charged spherical shell was thus Law is violated if u0,i.e.if the photon mass is not zero. evidence of an inverse square law behaviour in the electric Before discussing Coulomb's experiment,we note that force. Cavendish,in addition to his celebrated measurement of the Franklin's work also served as inspiration for the efforts mean density of the Earth,also carried out an early experiment of Aepinus (1724-1802)[4],a German physicist.He on the physics of the electrical force.Inspired by the same idea made experimental and theoretical contributions to the study that motivated his predecessors,he too considered a metallic of electricity and in 1759 proposed in a theoretical essay, spherical shell enclosed within an outer shell consisting of written in Latin,the existence of two types of electric charges two hemispheres that could be opened or closed.In the closed (positive and negative)and a 1/r2 behaviour of the electric position,the two hemispheres were connected electrically to an force.His conjectures were made in analogy to what Newton electrostatic machine and charged while in ohmic contact with had proposed in order to explain Kepler's Law,the free the inner sphere.The hemispheres were then disconnected fall of bodies near the Earth's surface,and the outcome from the inner sphere and opened,and it was verified that they of Cavendish's laboratory experiment that investigated the remained charged.At this point,an electrometer was used to gravitational attraction between lead spheres [5]. check that the inner sphere was still uncharged,thus confirming All of these early qualitative phenomenological and the 1/r2 law but with an uncertainty that was smaller than that theoretical studies paved the way for the eventual quantitative of Robison (less than 1/60 of the charge moved to the inner verification of the basic law describing the electrical force.In shell over the thin wire interconnecting the two spheres).With fact,the essay of Aepinus was read by Robison(1739-1805) reference to equation (1),Cavendish obtained ss 0.03.An [6].an English physician who in 1769 carried out experimental improved version of the experiment was later performed by tests of the inverse square law and used the results to surmise Maxwell [8],who increased the precision of the test and found that it was indeed correct.His determination was made that the exponent of r in Coulomb's Law could differ from 2 somewhat before that of Coulomb,but history has given the by no more than s≥5×10-5. name of this interaction to the latter. We now turn to the famous experiment of Coulomb of 1788.Charles Augustin de Coulomb (1736-1806)was a 2.Early experimental verifications of Coulomb's French physicist and a pioneer in electrical theory.He was Law born in Angouleme.He served as a military engineer for France in the West Indies,but retired to Blois at the time of Robison's experiment was very straightforward.He measured the French Revolution to continue his research on magnetism, the repulsive force between two charged masses,equilibrated friction,and electricity.In 1777,he invented the torsion 6 He studied at Jena and Rostock and taught mathematics at Rostock from balance for the purpose of measuring the force of magnetic 1747 to 1755.After a brief stay in Berlin he went to St Petersburg as professor and electrical attractions.With this device,Coulomb was of physics and academician,remaining there until 1798 and rising to a high able to formulate the principle,now known as Coulomb's position as courtier to Catherine the Great. Law,governing the interaction between electric charges.In S160 Metrologia,41 (2004)S159-S170
G Spavieri et al Figure 1. Franklin placed a small sphere made of cork inside a metallic cup, and when this was charged he observed that the sphere did not move, suggesting that there was no interaction between it and the cup. Priestley explained the phenomenon in 1767, providing a line of reasoning analogous to that used by Newton for the law of universal gravitation, implying that the field has an inverse square dependence on distance. In the subsequent tests, instead of the cork sphere inside a metallic cup, experimentalists considered a metallic shell enclosed within an outer charged shell or several concentric shells. analogous to the gravitational case and he thus concluded that the electric force, like the gravitational force, must depend on distance as r−2. The lack of an observed ponderable force on the cork sphere inside the charged spherical shell was thus evidence of an inverse square law behaviour in the electric force. Franklin’s work also served as inspiration for the efforts of Aepinus (1724–1802) [4], a German physicist6. He made experimental and theoretical contributions to the study of electricity and in 1759 proposed in a theoretical essay, written in Latin, the existence of two types of electric charges (positive and negative) and a 1/r2 behaviour of the electric force. His conjectures were made in analogy to what Newton had proposed in order to explain Kepler’s Law, the free fall of bodies near the Earth’s surface, and the outcome of Cavendish’s laboratory experiment that investigated the gravitational attraction between lead spheres [5]. All of these early qualitative phenomenological and theoretical studies paved the way for the eventual quantitative verification of the basic law describing the electrical force. In fact, the essay of Aepinus was read by Robison (1739–1805) [6], an English physician who in 1769 carried out experimental tests of the inverse square law and used the results to surmise that it was indeed correct. His determination was made somewhat before that of Coulomb, but history has given the name of this interaction to the latter. 2. Early experimental verifications of Coulomb’s Law Robison’s experiment was very straightforward. He measured the repulsive force between two charged masses, equilibrated 6 He studied at Jena and Rostock and taught mathematics at Rostock from 1747 to 1755. After a brief stay in Berlin he went to St Petersburg as professor of physics and academician, remaining there until 1798 and rising to a high position as courtier to Catherine the Great. by the force of gravity acting on them. By knowing their weight, and by repeating the measurements at different distances, it was possible to calculate the size of the electrical force, evaluate its dependence on distance, and thus verify the exactness of the hypothesized 1/r2 law. From his observations, Robison deduced that the law must have the functional form F ∝ 1 r2±ε , (1) where ε represents a measure of the precision to which the 1/r2 behaviour is verified. He found an upper limit of 0.06 for ε and thus could state that for masses in electrical repulsion to each other, the force went as r−2.06. However, for electrical attraction his limit was weaker, stated as r−c where c < 2, but still essentially confirming the expected r−2 dependence. Unfortunately, Robison did not publish his results until 1801, and by then Coulomb [7] had already presented his. The parameter ε appearing in equation (1) and related to the precision of the 1/r2 behaviour, is not very useful from a theoretical point of view but is retained here because of its common historical use. Subsequent theoretical developments and improved understanding of the foundations for high precision tests of Coulomb’s Law have led to the use of the quantityµ = mγ c/h¯ = λ−1 C which, as considered below, is well motivated theoretically and represents the inverse Compton wavelength of a photon of mass mγ . Coulomb’s Law is violated if µ = 0, i.e. if the photon mass is not zero. Before discussing Coulomb’s experiment, we note that Cavendish, in addition to his celebrated measurement of the mean density of the Earth, also carried out an early experiment on the physics of the electrical force. Inspired by the same idea that motivated his predecessors, he too considered a metallic spherical shell enclosed within an outer shell consisting of two hemispheres that could be opened or closed. In the closed position, the two hemispheres were connected electrically to an electrostatic machine and charged while in ohmic contact with the inner sphere. The hemispheres were then disconnected from the inner sphere and opened, and it was verified that they remained charged. At this point, an electrometer was used to check that the inner sphere was still uncharged, thus confirming the 1/r2 law but with an uncertainty that was smaller than that of Robison (less than 1/60 of the charge moved to the inner shell over the thin wire interconnecting the two spheres). With reference to equation (1), Cavendish obtained ε 0.03 . An improved version of the experiment was later performed by Maxwell [8], who increased the precision of the test and found that the exponent of r in Coulomb’s Law could differ from 2 by no more than ε 5 × 10−5. We now turn to the famous experiment of Coulomb of 1788. Charles Augustin de Coulomb (1736–1806) was a French physicist and a pioneer in electrical theory. He was born in Angouleme. He served as a military engineer for ˆ France in the West Indies, but retired to Blois at the time of the French Revolution to continue his research on magnetism, friction, and electricity. In 1777, he invented the torsion balance for the purpose of measuring the force of magnetic and electrical attractions. With this device, Coulomb was able to formulate the principle, now known as Coulomb’s Law, governing the interaction between electric charges. In S160 Metrologia, 41 (2004) S159–S170
Physical implications of Coulomb's Law atttila publication of his results may signal that he was more aware than his colleagues of how fundamental and important this work was. Wire q 3.Null tests of Coulomb's Law:theory Mirror The technique employed by Cavendish has been used in most of the experimental work done since then,as it turned out to be potentially the most sensitive.It is intrinsically a null experiment,in the sense that the experimentalist seeks to verify with great precision the absence of charge from the inner sphere,rather than having to measure with less precision a Light ray non-null physical quantity,such as the twist in the fibre,as in the torsion balance approach. Following Robison and Maxwell and supposing that the Light source exponent in Coulomb's Law is not -2 but -(2+s),to first order in s the electric potential at a point r due to the charge Figure 2.The Coulomb torsion balance was similar in principle to the torsion balance used by Cavendish for measurement of the density distribution p(r')is given by gravitational attraction between masses.The interaction between the charged spheres produces a measurable twist in the torsion fibre, V(r)= dr p(r') Ir-r (3) which lets the apparatus rotate until equilibrium is reached.By accurately measuring the torsion angle,Coulomb confirmed the 1/r2 law with a precision surpassing that of the previous If the charge is uniformly distributed on a spherical shell experiments of Robison and Cavendish. of radius a r,then p(r',0',')08(r-a)and we may arbitrarily choose r to coincide with the z axis and 1779,Coulomb published the treatise Theorie des machines expression(3)becomes simples (Theory of Simple Machines),an analysis of friction o sine'de'do' in machinery.After the war Coulomb came out of retirement V(r)= and assisted the new government in devising a metric system (r2+a2 -2ar cos )(1+) of weights and measures.The unit of quantity used to measure *1 dcos0' electrical charges,the coulomb,was named for him. = 1(r2+a2 -2ar cos 0)(1+) His torsion balance,shown in figure 2,was similar in principle to that used by Cavendish for the measurement of =f(r+a)-f(r-a) (4) the mean density of the Earth(now interpreted as a seminal 2ar laboratory test of Newtonian gravitation).The interaction In the case where Coulomb's Law is perfectly valid, between the charged spheres produces a torque that acts s=0 and on the torsion fibre,with the apparatus then rotating until +1 d(cos0) equilibrium is reached.By accurately measuring the torsion V(r)=2πp a=const. angle,Coulomb found a limiting value for s in equation (1) (r2+a2 -2ar cos0= of g 0.01,thus surpassing the precision of the previous (5) experiments of Robison and Cavendish.The scalar expression so that V(r)-V(a)=0 and the electric field inside the charged for what has come to be known as Coulomb's Law says that the spherical shell vanishes.Thus,in tests of Coulomb's Law,we force F between two charges gi and q2 separated by a distance are interested in the potential induced on a sphere of radius r may be written in the simple form r by a charge distributed uniformly on a concentric sphere of radius a r,i.e. F=k9192 2 (2) V(r)-V(a)=a[f(atr)-f(a-r) (6) where the value of the constant of proportionality,k,will be V(a) f(2a) considered in section 7. To first order in,equation(6)yields The reasons why Coulomb achieved greater success and recognition than did his predecessors are essentially two.First, V(r)-V(a) =EM(a,r), (7) he performed his tests with combinations of both negative V(a) and positive charges.Cavendish used only charges of the same sign,but Coulomb sought to measure both attractive and where repulsive forces.Second,he published his results immediately, while Robison did not make his findings available until 1801, M(a,r)= (侣)-(】 (8) thirteen years after Coulomb.Cavendish too delayed the dissemination of his work,and it thus garnered no attention Since M(a,r)turns out to be of order unity,s is essentially until nearly a century later when a citation to it was given the quotient [V(r)-V(a)]/V(a)of the measured potential by Maxwell in his famous essay [8].Coulomb's prompt difference,V(r)-V(a),and the applied voltage,V(a). Metrologia,41(2004)s159-S170 S161
Physical implications of Coulomb’s Law Light ray Wire Mirror d Light source q θ q Q Figure 2. The Coulomb torsion balance was similar in principle to the torsion balance used by Cavendish for measurement of the gravitational attraction between masses. The interaction between the charged spheres produces a measurable twist in the torsion fibre, which lets the apparatus rotate until equilibrium is reached. By accurately measuring the torsion angle, Coulomb confirmed the 1/r2 law with a precision surpassing that of the previous experiments of Robison and Cavendish. 1779, Coulomb published the treatise Theorie des machines ´ simples (Theory of Simple Machines), an analysis of friction in machinery. After the war Coulomb came out of retirement and assisted the new government in devising a metric system of weights and measures. The unit of quantity used to measure electrical charges, the coulomb, was named for him. His torsion balance, shown in figure 2, was similar in principle to that used by Cavendish for the measurement of the mean density of the Earth (now interpreted as a seminal laboratory test of Newtonian gravitation). The interaction between the charged spheres produces a torque that acts on the torsion fibre, with the apparatus then rotating until equilibrium is reached. By accurately measuring the torsion angle, Coulomb found a limiting value for ε in equation (1) of ε 0.01, thus surpassing the precision of the previous experiments of Robison and Cavendish. The scalar expression for what has come to be known as Coulomb’s Law says that the force F between two charges q1 and q2 separated by a distance r may be written in the simple form F = k q1q2 r2 , (2) where the value of the constant of proportionality, k, will be considered in section 7. The reasons why Coulomb achieved greater success and recognition than did his predecessors are essentially two. First, he performed his tests with combinations of both negative and positive charges. Cavendish used only charges of the same sign, but Coulomb sought to measure both attractive and repulsive forces. Second, he published his results immediately, while Robison did not make his findings available until 1801, thirteen years after Coulomb. Cavendish too delayed the dissemination of his work, and it thus garnered no attention until nearly a century later when a citation to it was given by Maxwell in his famous essay [8]. Coulomb’s prompt publication of his results may signal that he was more aware than his colleagues of how fundamental and important this work was. 3. Null tests of Coulomb’s Law: theory The technique employed by Cavendish has been used in most of the experimental work done since then, as it turned out to be potentially the most sensitive. It is intrinsically a null experiment, in the sense that the experimentalist seeks to verify with great precision the absence of charge from the inner sphere, rather than having to measure with less precision a non-null physical quantity, such as the twist in the fibre, as in the torsion balance approach. Following Robison and Maxwell and supposing that the exponent in Coulomb’s Law is not −2 but −(2 + ε), to first order in ε the electric potential at a point r due to the charge density distribution ρ(r ) is given by V (r) = d3 r ρ(r ) | r − r | 1+ε . (3) If the charge is uniformly distributed on a spherical shell of radius a>r, then ρ(r , θ , ϕ ) = σ δ(r − a) and we may arbitrarily choose r to coincide with the z axis and expression (3) becomes V (r) = σ sin θ dθ dϕ (r2 + a2 − 2ar cos θ )(1+ε)/2 = +1 −1 d cos θ (r2 + a2 − 2ar cos θ )(1+ε)/2 = f (r + a) − f (r − a) 2ar . (4) In the case where Coulomb’s Law is perfectly valid, ε = 0 and V (r) = 2πρ +1 −1 d(cos θ ) (r2 + a2 − 2ar cos θ )1/2 = 1 a = const, (5) so thatV (r)−V (a) = 0 and the electric field inside the charged spherical shell vanishes. Thus, in tests of Coulomb’s Law, we are interested in the potential induced on a sphere of radius r by a charge distributed uniformly on a concentric sphere of radius a>r, i.e. V (r) − V (a) V (a) = a r f (a + r) − f (a − r) f (2a) − 1. (6) To first order in ε, equation (6) yields V (r) − V (a) V (a) = εM(a, r), (7) where M(a, r) = 1 2 a r ln a + r a − r − ln 4a2 a2 − r2 . (8) Since M(a, r) turns out to be of order unity, ε is essentially the quotient [V (r) − V (a)]/V (a) of the measured potential difference, V (r) − V (a), and the applied voltage, V (a). Metrologia, 41 (2004) S159–S170 S161
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-S170
G 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
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 S163
Physical 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 nonuniform 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