INSTITUTE OF PHYSICS PUBLISHING METROLOGIA

Metrologia 41 (2004) S159S170 PII: S0026-1394(04)80313-X

Physical implications of Coulombs LawG Spavieri1, G T Gillies2 and M Rodriguez3

1 Centro de Astrofsica Teorica, Facultad de Ciencias, Universidad de Los Andes, Merida,5101 Venezuela2 Department of Mechanical and Aerospace Engineering, University of Virginia,PO Box 400746, Charlottesville, VA 22904, USA3 Departamento de Fsica, FACYT, Universidad de Carabobo, Valencia, 2001 Venezuela

E-mail: spavieri@ula.ve and gtg@virginia.edu

Received 3 March 2004Published 16 September 2004Online at stacks.iop.org/Met/41/S159doi:10.1088/0026-1394/41/5/S06

AbstractWe examine the theoretical and experimental foundations of CoulombsLaw and review the various roles it plays not only in electromagnetism andelectrodynamics, but also in quantum mechanics, cosmology, andthermodynamics. The many implications of Coulombs Law draw attentionto its fundamental importance within virtually all branches of physics andmake this elementary yet profound law one of the most useful of allscientific tools.

1. Introductory historical outlook

Few investigations in physics have enjoyed as sustained aninterest as have tests of Coulombs Law. As has beenthe case with most of the fundamental laws of physics, itwas discovered and elucidated through observations of basicphenomena. In his research, Coulomb was interested in themutual interactions between electric charges, a topic that hadbeen studied previously by Priestley [1], and in fact evenearlier, in 1755, during the experimental work of Franklin [2].

Franklin (17061790) was an American printer, writer,politician, diplomat, and scientist. He is credited with theinvention of such practical everyday items as bifocal eyeglassesand a free-standing, wood-burning heater called the Franklinstove. His principal connection to electrical experimentationcame via his investigations of the properties of Leyden Jars4.He is also commonly credited with giving the names positiveand 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(17331804)5, an English chemist and amateur natural4 In 1752, he flew a kite attached to a silk string in a thunderstorm, and showedthat 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 ifnear 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 (nitrousoxide, laughing gas), and dephlogisticated air (oxygen). His chemicalwritings 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. Hewas a politically involved Unitarian preacher and a sympathizerwith the French Revolution, and these aspects of his life forcedhim to move to America with his family in 1794. Priestleyis credited with the discovery of oxygen in 1774, which heproduced by focusing sunlight on mercuric oxide. During hisstudies of this dephlogisticated air, he noticed that it madehim light-headed and that it had a similar effect on animals.

The background studies underpinning Coulombs Lawbegan when Franklin took a small sphere made of corkand placed it inside a charged metallic cup (see figure 1)and observed that it did not move, suggesting that therewas no interaction between it and the cup. After Franklincommunicated his finding to Priestley, the Englishmanexplained the phenomenon in 1767, providing a line ofreasoning analogous to that used by Newton [3] to formulateand enunciate the law of universal gravitation.

Underlying Newtonian gravity was the observation thatthe gravitational field inside a spherical shell of homogeneousmaterial is null if the field is inversely proportional to thesquare of the distance r , i.e. if its intensity goes as r2.By approximating Franklins cup as a spherical shell, Priestleydeduced that the observed phenomenon should be physicallyon Different Kinds of Airs (17741777) and in the three-volume Experimentsand Observations Relating to Various Branches of Natural Philosophie(17791786). By dissolving fixed air in water, he invented carbonatedwater. He also noticed that the explosion of inflammable air with commonair produced dew. Lavoisier repeated this experiment and took credit forit. Priestley believed in the phlogiston theory, and was convinced that hisdiscovery of oxygen proved it to be correct.

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G Spavieri et al

Figure 1. Franklin placed a small sphere made of cork inside ametallic cup, and when this was charged he observed that the spheredid not move, suggesting that there was no interaction between itand the cup. Priestley explained the phenomenon in 1767, providinga line of reasoning analogous to that used by Newton for the law ofuniversal gravitation, implying that the field has an inverse squaredependence on distance. In the subsequent tests, instead of the corksphere inside a metallic cup, experimentalists considered a metallicshell enclosed within an outer charged shell or severalconcentric shells.

analogous to the gravitational case and he thus concluded thatthe electric force, like the gravitational force, must depend ondistance as r2. The lack of an observed ponderable forceon the cork sphere inside the charged spherical shell was thusevidence of an inverse square law behaviour in the electricforce.

Franklins work also served as inspiration for the effortsof Aepinus (17241802) [4], a German physicist6. Hemade experimental and theoretical contributions to the studyof 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 electricforce. His conjectures were made in analogy to what Newtonhad proposed in order to explain Keplers Law, the freefall of bodies near the Earths surface, and the outcomeof Cavendishs laboratory experiment that investigated thegravitational attraction between lead spheres [5].

All of these early qualitative phenomenological andtheoretical studies paved the way for the eventual quantitativeverification of the basic law describing the electrical force. Infact, the essay of Aepinus was read by Robison (17391805)[6], an English physician who in 1769 carried out experimentaltests of the inverse square law and used the results to surmisethat it was indeed correct. His determination was madesomewhat before that of Coulomb, but history has given thename of this interaction to the latter.

2. Early experimental verifications of CoulombsLaw

Robisons experiment was very straightforward. He measuredthe repulsive force between two charged masses, equilibrated6 He studied at Jena and Rostock and taught mathematics at Rostock from1747 to 1755. After a brief stay in Berlin he went to St Petersburg as professorof physics and academician, remaining there until 1798 and rising to a highposition as courtier to Catherine the Great.

by the force of gravity acting on them. By knowingtheir weight, and by repeating the measurements at differentdistances, it was possible to calculate the size of the electricalforce, evaluate its dependence on distance, and thus verify theexactness of the hypothesized 1/r2 law.

From his observations, Robison deduced that the law musthave the functional form

F 1r2

, (1)

where represents a measure of the precision to which the1/r2 behaviour is verified. He found an upper limit of0.06 for and thus could state that for masses in electricalrepulsion to each other, the force went as r2.06. However,for electrical attraction his limit was weaker, stated as rcwhere c < 2, but still essentially confirming the expectedr2 dependence. Unfortunately, Robison did not publishhis results until 1801, and by then Coulomb [7] had alreadypresented his. The parameter appearing in equation (1) andrelated to the precision of the 1/r2 behaviour, is not veryuseful from a theoretical point of view but is retained herebecause of its common historical use. Subsequent theoreticaldevelopments and improved understanding of the foundationsfor high precision tests of Coulombs Law have led to the useof the quantity = m c/h = 1C which, as considered below,is well motivated theoretically and represents the inverseCompton wavelength of a photon of mass m . CoulombsLaw is violated if = 0, i.e. if the photon mass is not zero.

Before discussing Coulombs experiment, we note thatCavendish, in addition to his celebrated measurement of themean density of the Earth, also carried out an early experimenton the physics of the electrical force. Inspired by the same ideathat motivated his predecessors, he too considered a metallicspherical shell enclosed within an outer shell consisting oftwo hemispheres that could be opened or closed. In the closedposition, the two hemispheres were connected electrically to anelectrostatic machine and charged while in ohmic contact withthe inner sphere. The hemispheres were then disconnectedfrom the inner sphere and opened, and it was verified that theyremained charged. At this point, an electrometer was used tocheck that the inner sphere was still uncharged, thus confirmingthe 1/r2 law but with an uncertainty that was smaller than thatof Robison (less than 1/60 of the charge moved to the innershell over the thin wire interconnecting the two spheres). Withreference to equation (1), Cavendish obtained 0.03 . Animproved version of the experiment was later performed byMaxwell [8], who increased the precision of the test and foundthat the exponent of r in Coulombs Law could differ from 2by no more than 5 105.

We now turn to the famous experiment of Coulomb of1788. Charles Augustin de Coulomb (17361806) was aFrench physicist and a pioneer in electrical theory. He wasborn in Angouleme. He served as a military engineer forFrance in the West Indies, but retired to Blois at the time ofthe French Revolution to continue his research on magnetism,friction, and electricity. In 1777, he invented the torsionbalance for the purpose of measuring the force of magneticand electrical attractions. With this device, Coulomb wasable to formulate the principle, now known as CoulombsLaw, governing the interaction between electric charges. In

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Physical implications of Coulombs Law

Light ray

Wire

Mirror d

Light source

q

qQ

Figure 2. The Coulomb torsion balance was similar in principle tothe torsion balance used by Cavendish for measurement of thegravitational attraction between masses. The interaction betweenthe charged spheres produces a measurable twist in the torsion fibre,which lets the apparatus rotate until equilibrium is reached. Byaccurately measuring the torsion angle, Coulomb confirmed the1/r2 law with a precision surpassing that of the previousexperiments of Robison and Cavendish.

1779, Coulomb published the treatise Theorie des machinessimples (Theory of Simple Machines), an analysis of frictionin machinery. After the war Coulomb came out of retirementand assisted the new government in devising a metric systemof weights and measures. The unit of quantity used to measureelectrical charges, the coulomb, was named for him.

His torsion balance, shown in figure 2, was similar inprinciple to that used by Cavendish for the measurement ofthe mean density of the Earth (now interpreted as a seminallaboratory test of Newtonian gravitation). The interactionbetween the charged spheres produces a torque that actson the torsion fibre, with the apparatus then rotating untilequilibrium is reached. By accurately measuring the torsionangle, Coulomb found a limiting value for in equation (1)of 0.01, thus surpassing the precision of the previousexperiments of Robison and Cavendish. The scalar expressionfor what has come to be known as Coulombs Law says that theforce F between two charges q1 and q2 separated by a distancer may be written in the simple form

F = k q1q2r2

, (2)

where the value of the constant of proportionality, k, will beconsidered in section 7.

The reasons why Coulomb achieved greater success andrecognition than did his predecessors are essentially two. First,he performed his tests with combinations of both negativeand positive charges. Cavendish used only charges of thesame sign, but Coulomb sought to measure both attractive andrepulsive forces. Second, he published his results immediately,while Robison did not make his findings available until 1801,thirteen years after Coulomb. Cavendish too delayed thedissemination of his work, and it thus garnered no attentionuntil nearly a century later when a citation to it was givenby Maxwell in his famous essay [8]. Coulombs prompt

publication of his results may signal that he was more awarethan his colleagues of how fundamental and important thiswork was.

3. Null tests of Coulombs Law: theory

The technique employed by Cavendish has been used in mostof the experimental work done since then, as it turned outto be potentially the most sensitive. It is intrinsically a nullexperiment, in the sense that the experimentalist seeks to verifywith great precision the absence of charge from the innersphere, rather than having to measure with less precision anon-null physical quantity, such as the twist in the fibre, as inthe torsion balance approach.

Following Robison and Maxwell and supposing that theexponent in Coulombs Law is not 2 but (2 + ), to firstorder in the electric potential at a point r due to the chargedensity distribution (r) is given by

V (r) =

d3r(r)

| r r |1+ . (3)

If the charge is uniformly distributed on a spherical shellof radius a > r , then (r , , ) = (r a) and wemay arbitrarily choose r to coincide with the z axis andexpression (3) becomes

V (r) =

sin d d

(r2 + a2 2ar cos )(1+)/2

= +1

1

d cos

(r2 + a2 2ar cos )(1+)/2

= f (r + a) f (r a)2ar

. (4)

In the case where Coulombs Law is perfectly valid, = 0 andV (r) = 2

+11

d(cos )(r2 + a2 2ar cos )1/2 =

1a

= const,(5)

so thatV (r)V (a) = 0 and the electric field inside the chargedspherical shell vanishes. Thus, in tests of Coulombs Law, weare interested in the potential induced on a sphere of radiusr by a charge distributed uniformly on a concentric sphere ofradius a > r , i.e.

V (r) V (a)V (a)

= ar

[f (a + r) f (a r)

f (2a)

] 1. (6)

To first order in , equation (6) yieldsV (r) V (a)

V (a)= M(a, r), (7)

where

M(a, r) = 12

[a

rln(

a + r

a r)

ln(

4a2

a2 r2)]

. (8)

Since M(a, r) turns out to be of order unity, is essentiallythe quotient [V (r) V (a)]/V (a) of the measured potentialdifference, V (r) V (a), and the applied voltage, V (a).

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As an alternative to equations (7) and (8), de Broglie [9]considered a simple generalization of Maxwells equationsinvolving a small non-zero rest mass of the photon. In thiscase, two charges will repel each other by a Yukawa forcederived from the potential

U(r) = er

r= e

r/C

r, (9)

where = m c/h = 1C is the inverse Compton wavelengthof the photon. In the limit a 1, U(r) = 1/r + 122rand equation (6) yields

V (r) V (a)V (a)

= 162(a2 r2). (10)

Although U(r) contains the term , what is testedexperimentally is the result of equation (10). Since the othertests of Coulombs Law are explicitly sensitive to 2 and not to, the quadratic dependence of V (r)V (a) on makes a testbased on this approach comparable to other tests. Thus, thepotential difference V (r)V (a) is not zero if Coulombs Lawis invalid or, equivalently, if the photon rest mass is non-zero.For direct tests of Coulombs Law that consist of measuringthe static potential difference of charged concentric shells, onemay use either equation (7) or equation (10). However, one canalso test Coulombs Law by determining with independent,indirect methods. In general, these would rely on either findingpossible variations due to the presence of the Yukawa potential(9) or on the standard fields of massless electrodynamics, suchas, e.g., measurements at either large distances or long times,where the percentage effect would be much higher. Typicalof these approaches are those that involve the magnetic fieldof the Earth. For example, one might consider (a) satelliteverification that the magnetic field of the Earth falls off as1/r3 out to distances at which the solar wind is appreciable[10], (b) observation of the propagation of hydromagneticwaves through the magnetosphere [11], (c) application of theSchrodinger external field method [12], or other methods suchas those described below. The three approaches outlined aboveshould all give roughly the same limit, 1011 cm1.

In the high-frequency (direct) null test of Coulombs Lawdescribed below, it is convenient to start from a relativisticallyinvariant linear generalization of Maxwells equations, namelythe Proca equations [13], which allow for a finite rest mass ofthe photon. Procas equations for a particle of spin 1 and massm are [14]

( + 2)A = 4c

J (11)and Gauss Law becomes

E = 4 2. (12)Equation (12) may be applied to two concentric,

conducting, spherical shells of radii r and a (a > r) with aninductor across (i.e. in parallel with) this spherical capacitor.If a potential V0eit is applied to the outer shell, the resultingelectric field is [15]

E(r) = (qr2 132V0eit r) r , (13)where q is the total charge on the inner shell.

The voltage across the inductor of capacity C is thengiven [15] by

V (r) V (a) = ar

E dl = qC

2 V0eit

6(a2 r2). (14)

Save for the standard term q/C (which is zero when thereis no charge on the inner shell), the term dependent on inequation (14) coincides with that of equation (10).

4. Direct tests of Coulombs Law

After the development of the phase-sensitive detectors suchas lock-in amplifiers, new and more sensitive attempts to testCoulombs Law were made such as the ones by Plimptonand Lawton [16], Cochran and Franken [17], and Bartlettand Phillips [18]. In this section, we consider Maxwellsderivation (equations (6) and (7)) applied to the simple caseof a conducting sphere containing a smaller concentric sphere.The potential of the outer sphere is raised to a value V andthe potential difference between them is measured. The actualshape of these conductors should not be relevant because theelectric field inside a cavity of any shape vanishes unlessCoulombs Law is violated. Thus, Cochran and Franken [17]could use conducting rectangular boxes in their experimentand set a limit of 0.9 1011.

The experiments of both Cavendish and Maxwell requiredconnecting the inner sphere to an electrometer. The accuracyof the experiment was thus limited by fluctuations in thecontact potentials while measuring the inner spheres voltage.Another problem was that of spontaneous ionization betweenthe spheres. These problems were overcome by Plimpton andLawton [16] by using alternating potentials. They developed aquasi-static method and charged the outer sphere with a slowlyalternating current. The potential difference between the innerand outer spheres was detected with a resonant frequencyelectrometer. It consisted of an undamped galvanometer withamplifier, placed within the globes, and with the input resistorof the amplifier forming a permanent link connecting them,so as to measure any variable potential difference. No effectwas observed when a harmonically alternating high potentialV (>3000 V), from a condenser generator operating at thelow resonance frequency of the galvanometer, was appliedto the outer globe. The sensitivity was such that a voltageof 106 V was easily observable above the small level ofbackground noise. With this technique they succeeded inreducing Maxwells limit to 2 109.

Another of the classic null experiments that tests theexactness of the electrostatic inverse square law was performedby Bartlett et al [19]. In this experiment, the outer shell of aspherical capacitor was raised to a potential V with respect toa distant ground and the potential difference V (r) V (a) ofequations (7) and (10) induced between the inner and outershells was measured. Five concentric spheres were used and apotential difference of 40 kV at 2500 Hz was imposed betweenthe two outer spheres. A lock-in detector with a sensitivityof about 0.2 nV measured the potential difference between theinner two spheres. Any deviation in Coulombs law should leadto a non-null result for V (r)V (a) proportional to as shownby equation (7). The result obtained by these authors was 1 1013. A comparable result was found even when

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Physical implications of Coulombs Law

the frequency was reduced to 250 Hz and the detector wassynchronized with the charging current rather than with thecharge itself.

The best result obtained so far through developmentsof the original Cavendish technique is still that from 1971by Williams et al [15], who improved an earlier experiment[20]. They used five concentric metallic shells in the formof icosahedra rather than spheres in order to reduce the errorsdue to charge dispersion. A high voltage and frequency signalwas applied to the external shells and a very sensitive detectorchecked for any trace of a signal related to variable chargingof the internal shell. The detector worked by amplifying thesignal of the internal shell and comparing it with an identicalreference signal, progressively out of phase at a rhythm of 360per half hour. Any signal from the detector would indicate aviolation of Coulombs Law. In order to avoid introducingunrelated fields, the reference signal and the detector outputsignal were transmitted by means of optical fibres. The outershell, of about 1.5 m diameter, was charged 10 kV peak-to-peakwith a 4 MHz sinusoidal voltage. Centred inside this chargedconducting shell is a smaller conducting shell. Any deviationfrom Coulombs Law is detected by measuring the line integralof the electric field between these two shells with a detectionsensitivity of about 1012 V peak-to-peak.

The null result of this experiment expressed in the form ofthe photon rest mass squared (equation (14) or equation (10))is 2 = 2.3 1019 cm2. Expressed as a deviation fromCoulombs Law in the form of equation (7), their result is = 6 1016, extending the validity of Coulombs Law bytwo 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 thevalidity of Coulombs Law are typically more precise thanthose that attempt to directly measure the interaction forcebetween charges. One of the problems arising when makingdirect measurements of the force between two macroscopiccharged bodies, as done when using a torsion balance, isthat the charges are distributed over conducting surfaces offinite size. In the ideal case, Coulombs Law describes theinteraction between two point charges separated by a preciselyknown distance. In any practical arrangement, even thecharge on a microscopically small conducting ball cannot beconsidered to be truly point-likeas if placed at the centrebut rather distributed over the balls surface. If the chargedball is interacting with another charged ball, the distributionon the surface is no longer uniform and has to be determinedusing the method of images. Saranin [21] has studied in detailthe departures from Coulombs Law that can occur when twoconducting 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 toattraction occurs in the general case of arbitrarily but similarlycharged spheres. The only exceptionand in it they alwaysrepel each otheris the case in which the charges on thespheres are related as the squares of their radii. The results ofSaranin help corroborate the idea that, even in principle, nullexperiments can be more precise than tests based on the direct

measurement of interaction forces between two macroscopiccharged bodies.

In view of the high levels of precision achieved in severalof these tests, it is interesting to consider what possiblecompeting effects gravity might introduce into them. Theresultf (a, r) in equation (5) was derived strictly from classicalelectrodynamics. In it, a uniform charge distribution isassumed and the effect of gravity is neglected. As notedby Plimpton and Lawton [16], if electrons have weightmeg the electron density on the conducting sphere must beasymmetrical, being greater at the bottom where the electronsare pulled by the force of gravity. For the experiment ofPlimpton and Lawton this effect is insignificant as it leads to amaximum potential difference over the globe of 1010 V, whichis far less than the minimum detectable voltage of 106 V.

Thus, while such a gravitational effect should be negligiblein the relatively low sensitivity experiment of Plimpton andLawton, it could conceivably be important in experimentaltests of higher sensitivity. According to this model, the overalleffect of gravity is to produce a distortion in what shouldotherwise be a uniform charge distribution. Of course, themore general problem is to account for effects of a non-uniform charge distribution regardless of the origin of thenon-uniformity. In equation (5), the null result comes fromthe assumption that Coulombs Law is valid and that thecharge is distributed uniformly on the sphere. However,Shaw [22] objected to the assumption that the charge willdistribute itself uniformly over a conducting spherical shell,even in the absence of any gravitational effect. In conventionalelectrostatics, the uniform charge distribution for Coulomb andYukawa potentials follows from the symmetry of the problemand the uniqueness of the solution. If these potentials are notvalid there is no guarantee of a uniform charge distributionand thus irregularities in the spherical surface would bias theconcept that the inner potential does not depend on the shapeof the outer sphere. However, considering that any violationof Coulombs Law is very small, departures from the expecteduniformity should give [19] only second-order corrections toequations (7) and (8).

6. Indirect tests of Coulombs Law

In addition to the tests discussed in the previous sections, therehave also been a number of indirect experimental verificationsof Coulombs Law, and these will be discussed briefly in whatfollows.

6.1. Geomagnetic and astronomical tests

A consequence of Coulombs Law is that the magnetic fieldproduced by a dipole goes as 1/r3 at distances from itscentre for which the dipole approximation is valid. For themagnetic field of a planet, this distance is equivalent to abouttwo planetary radii (at least). If the photon rest mass is notzerowhich is equivalent to a violation of Coulombs Lawa Yukawa factor er/C is introduced in the 1/r terms for theelectrostatic and magnetostatic potentials. In this case, themagnetic field produced by a dipole no longer goes as 1/r3but contains corrections related to the Compton wavelengthC = 1 = h/m c where m is the photon mass.

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G Spavieri et al

In terms of the Compton wavelength and photon mass,the sensitive explorations of the validity of Coulombs Lawreported by Williams et al [20] yielded an upper limit of1.05 1010 cm for C and 2 1047 g for m , while theirsubsequent direct test with concentric icosahedra [15] yieldedthe slightly better findings of 1.95 1010 cm for C and1.6 1047 g for m .

The first results obtained from orbital data were those ofGoldhaber and Nieto [14], who used satellite measurements ofthe Earths magnetic field to set limits of 5.5 1010 cm for Cand 4 1048 g for m . This corresponded to an equivalentvalue of 1.7 1016. Davis et al [23] verified the 1/r3behaviour of the magnetic field of Jupiter (much more intensethan that of the Earth) from observations of the Pioneer 10spacecraft, and were able to improve the precision of the limitsuch that C 3.14 1011, i.e. m 2.8 1011 cm1 =8 1049 g.

Further studies on the photon mass and planetary magneticfields were carried out by Bicknell [24]. He extendedSchrodingers [12] expression for a dipole field to a completespherical harmonic analysis of a static planetary field.Additional components of the magnetic field were identifiedwhich would arise from a non-zero photon mass. However, nonew limits were set on C because the most important termsin the expressions for the geomagnetic field were of the sameorder as Schrodingers apparent external field. An analysis ofEarths magnetic field, performed by Fischbach et al [25], ledto an upper limit value of m = 1 1048 g.

It is important to note that there is still another way ofcharacterizing deviations from Coulombs Law, based on anelectromagnetic analogue of a fifth force, and that this lineof thought leads to additional suggestions for experimentaltests. The relevant works in this area are those of Bartlett andLogl [26], Fischbach et al [25], and also Kloor et al [27].

An upper limit for the photon mass was also foundby Lowenthal [28], who used astronomical observationsof the gravitational deflection of electromagnetic radiation.Although this method does not lead to a better upper limitthan those mentioned above, the method is neverthelessinteresting and, because it is not related to the other techniques,it constitutes an independent approach to restricting themagnitude of the photon mass. The question posed byLowenthal was the following: if the general theory of relativitypredicts a deflection of starlight by the Sun of 1.75 s of arc, howis this deflection altered if the photon has a small rest mass (in units of c = h = 1)? Lowenthal showed that the deflectionvaries proportionally to 2 and he set the correction termequal to the difference between the measured deflection angleand the calculated deflection angle for photons of zero restmass. After taking into account the accuracy of the deflectionmeasurements, he obtained an upper limit of < 71040 g,which is not as stringent as the other limits, but still a usefulresult.

6.2. Lumped-circuit tests and the photon mass

Another indirect method to verify the 1/r2 form of CoulombsLaw involves observations of electromagnetic waves ofincreasing wavelength, . Underlying it is the concept thatif is small but =0, there is a max() beyond which there

cannot be electromagnetic wave propagation. With = 0 theequation for the scalar electromagnetic potential and vectorpotential A takes the form of a KleinGordon equation and thevelocity of propagation in vacuo of sinusoidal electromagneticwaves decreases as the wavelength increases, and it vanishesfor a value max() that depends on . The greater the observedmax(), the smaller is .

A very sensitive test of this concept using an LCR circuitwas carried out by Franken and Ampulski [29] in 1970. Theseauthors point out that the free-space phase velocity of lightmay be expressed as

(vc

)2=

2

2 2c, (15)

where is the angular frequency of the electromagnetic wavesin a resonant cavity and c = 2c/C. The latter quantity canbe rewritten as h c = mc2, which introduces a mass into theargument. In fact, the lowest resonant frequency of a cavity isc, irrespective of the size of the cavity. The phase velocitybecomes infinite at the critical frequency c while the groupvelocity approaches zero at this frequency, corresponding toa massive photon with zero momentum and hence an infiniteuncertainty in position, which, in turn, requires an infinitelylarge cavity for confinement. The results of the test [29]yield the values of 2 1012 cm for C and 1 1049 g form , corresponding to the equivalent value 4.3 108.A difficulty with this test is that so-called warped conductingcavities have been used in order to make up for the requirementof a very large cavity and there is no rigorous proof that thetheoretical considerations valid for a rectangular box apply alsoto warped cavities.

Several authors have discussed possible limitations of thetest by Franken and Ampulski. In one analysis of lumped-circuit tests of photon mass, Boulware [30] showed that theonly effect of a photon mass, if there is one, is to producesmall changes in the inductance and capacitance of the circuitas well as changes in its radiative half-life. The behaviour of alow-frequency, lumped LC circuit is essentially independentof the dynamics of the electromagnetic fields. Also Park andWilliams [31], and Kroll [32] have noted that in reducing thesize of their apparatus to table-top dimensions the authors havelowered the sensitivity of their experiment by the same ratio,thus making it difficult to set significant limits on the photonrest mass.

In examining the physical significance of the experimentby Franken and Ampulski, Goldhaber and Nieto [33] showthat in the massive-photon case the fields and currents of thesystem are changed only by order (D)2 from those of themassless-photon case (D is the dimension of the system). Thisimplies that the above-mentioned table-top experiment isonly weakly sensitive to small Yukawa-like deviations fromCoulombs Law and they argue that it is therefore unlikely thatthis test can improve present limits on the photon mass.

6.3. Cryogenic experiments

Modern theories that appeal to the concept of spontaneoussymmetry breaking assume that particles, which are masslessabove a certain critical temperature Tc, acquire mass belowthis temperature. Within this framework it is natural to

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Physical implications of Coulombs Law

Chargingcoaxcable

Copper pick-up coil

Silver surface (2) Inner surface (3)and mu-metal core

Superconductingsolenoid

20 FCapacitor C23

Twin Xto

lock-in

Silver groundsurface (1)Lead/aluminium outer shield

Figure 3. Schematic diagram of the cryogenic photon-mass experiment of Ryan et al. The system measures the current that flows betweentwo closed surfaces in response to an impressed voltage difference. The apparatus is kept at very low temperature (1.36 K). (Figure 1 fromRyan et al [34]. Copyright (1985) by the American Physical Society (with permission of Prof. Austin).)

speculate that the photon could also be massless above a criticaltemperature and acquire a rest mass below it.

A cryogenic photon-mass experiment was performed byRyan et al [34]. As shown diagrammatically in figure 3,it consisted essentially of a null experiment with concentriccontainers (closed surfaces), similar in a way to the previousdirect tests of Coulombs Law such as the one by Williamset al [15]. Unlike a standard Coulombs Law experiment, thismethod measured the current that flows between two closedsurfaces in response to an impressed voltage difference, not thevoltage difference itself. The apparatus was immersed in liquidhelium. The result of the experiment set a limit on the massof the photon of m (1.5 1.38) 1042 g at 1.36 K. Thesensitivity is lower than that of other previous tests. However,the result is still important because the thermodynamic validityof Coulombs Law is now extended from the standard terrestrial(room) temperatures to those typical of a planet far away fromthe Sun, or to those of the galactic environment.

6.4. The AharonovBohm effects and the mass of the photonAs discussed in the literature, several conjectures relatedto the AharonovBohm effect [35] have been developedassuming electromagnetic interaction of fields of infinite range,i.e. zero photon mass. The possibility that any associatedeffects, specifically the AharonovBohm effect itself, mightbe manifest within the context of finite-range electrodynamicshas been discussed by Boulware and Deser [36].

These authors consider the coupling of the photon massm , as predicted by the Proca equation in the form

F + m2A

= J (16)

and show that the component of the magnetic field for asolenoid of the type that might be used in a test of theAharonovBohm effect is

B = B0 + m2(), (17)

where the first term, B0, is the standard magnetic field forzero photon massthe field confined inside a long solenoidof radius a and carrying the current jand the second term

represents a correction due to the photon non-vanishing restmass m . The quantity (), which can be expressed in termsof the Bessel functions I0 and K0 that are regular at the originand infinity respectively, reads

() = j(a )[K0(m )

0

d I0(m )

+ I0(m )

a

d K0(m )]

j( a)K0(m ) a

0 d I0(m ). (18)

Thus, for the standard solenoid configuration, finite rangewould lead to a small B field leakage outside the solenoidrepresented by m2(). In principle, one could arrange forthe leakage flux addition to be an integer, in which case onlythe interior field would contribute, making the scenario just asnon-local as in the standard massless case.

Because of the extra mass-dependent term, some non-trivial limits on the range of the transverse photon might thusbe obtained from a table-top experiment. By comparing thetheoretical corrections to the flux through a small circularregion of radius with the flux for the massless case,Boulware and Deser were able to estimate the size of a possibleexperimental limit on the range m1 . For scale sizes of 10 cm, the observable ranges are m1 102 km.

It would also be interesting to consider the other non-localeffects of the AharonovBohm type, such as those associatedwith neutral particles that have an intrinsic magnetic or electricdipole moment [37]. The goal would be to see if they providesimilar correction terms that might be suitable for settingmore precise limits on the range of m1 . In the case ofthe standard AharonovCasher effect, this analysis has beenperformed by Fuchs [38] who points out that, unlike thecase for the AharonovBohm effect, a neutral particle witha magnetic dipole moment that couples to non-gauge fieldshas no classical acceleration regardless of the photon mass, sothat no observable corrections would be expected.

However, this may not be the case for a neutral particlewith an electric dipole moment. In fact, in the AharonovCasher effect a particle with magnetic dipole m acquires an

Metrologia, 41 (2004) S159S170 S165

G Spavieri et al

electromagnetic momentum c1E m because m coupleswith the electric field E, while a particle with an electricdipole de couples with the vector potential A [37]as happensin the case of the AharonovBohm effectthus acquiringan electromagnetic momentum c1(de )A. For field-freeeffects [37] on electric dipoles, a magnetic sheet is the sourceof A. Small leakages of the fieldconfined within the sheet inthe massless photon casemay lead to observable correctionsof the type predicted by Boulware and Deser, thus presentingan interesting possibility for future study.

6.5. Recent limits on the photon mass and the cosmicmagnetic vector potential

A novel experimental approach for measuring the photonmass m has been based on a toroidal Cavendish balanceused to evaluate the product 2A of the photon mass squared(1 = h/m c) and the ambient cosmic vector potential A.This approach, developed by Lakes [39], points out that theMaxwellProca equations modify the standard equations forthe curl of B to

B = 4c

j + 1c

E t

2A. (19)

Gauge invariance is lost if = 0, since in these equations thepotentials themselves have physical significance, in additionto that of the usual fields. If a toroid carries an electric current,or if it is permanently magnetized with a field B confinedwithin it, the corresponding magnetic vector potential maybe represented by a dipole field ad. If = 0, this dipolefield interacts with the ambient vector potential A to producea torque = ad A2 via the energy density of the vectorpotential. Thus, the method is based on the energy density ofA in the presence of m , not on measurement of the magneticfield.

The experiment does not yield a direct limit for , butrather on A2. The modified Cavendish balance used todetermine the product A2 consisted of a toroid of electricalsteel wound with many turns of wire that carried a current (seefigure 4). The apparatus was supported by water flotation [40].The experiment yielded A2 < 2 109 T m m2 and, ifthe ambient magnetic vector potential is A 1012 T m dueto cluster level fields, 1 > 2 1010 m. Even with moreconservative values for A, the limit set by this experimentimproves the precision attained with the Jovian magnetic field.

In the preceding section on lumped circuits we quoted theresult of Goldhaber and Nieto [33], which suggests that fortable-top experiments the sensitivity of any limit on 2 goesas 1/D2 where D is the dimension of the system. This lineof reasoning has become an often-used rule of thumb in thefield, and the inference of it is that table-top experiments willbe less sensitive than others. If so, it is then interesting toconsider why the experiment of Lakes [39] sets such a goodlimit. We believe that the difference is due mainly to the factthat in table-top tests such as those using lumped circuits, themodification of the vector potential A due to a massive photonpossesses a range that is logically limited to the dimensionsof the circuit or cavity where the electromagnetic waves areproduced and confined. However, in the Lakes experiment,the mass couples with a vector potential A that is of cosmic

+v DC supply

Amplifier

Computer

MagneticshieldAir

shield

18.5 mm145 mm

Flotation chamber

A

Ad

98 mm

Aambient

earth

SupportTungsten wire

Mirror

Sensor

Laser

Figure 4. Block diagram of the sophisticated Cavendish balanceused by Lakes to evaluate the product of the photon mass squaredand the ambient cosmic vector potential. The toroid carries anelectric current giving rise to a dipole field ad that, if m = 0,interacts with the ambient vector potential Aambient to produce atorque on the toroid. (Figure 1 from Lakes [39]. Copyright (1998)by the American Physical Society (with permission of Prof. Lakes).)

origin so that the dimensions of the sources of A correspondto astronomical or galactic distances as opposed to those oflaboratory size. Thus, experiments of that type are able tosidestep the implications of the argument of Goldhaber andNieto [33].

The experimental arrangement used by Lakes is such thatthe torque on the torsion balance will vary with the rotationof the Earth, making it possible to experimentally detect thevariations. However, if the cosmic ambient vector potentialwere to be fortuitously aligned with the Earths rotation axis,then this approach would fail. In order to avoid this possibility,Luo et al [41] performed an improved experiment by rotatingthe torsion balance to ensure the effectiveness of detection forall possible orientations of the vector potential. They werealso able to remove the influences of sidereal disturbances inthe environment by virtue of this method of modulation. Theexperimental result was A2 < 1.1 1011 T m m2. If theambient cosmic magnetic vector potential is A 1012 T m, ityields the new upper limit 1 > 1.661011 m and the photonmass is m < 1.2 1051 g.

Those latest results are certainly remarkable if oneconsiders that, according to the uncertainty principle, apurely theoretical estimate of the photon mass is given bym = h/(t)c2, which yields an order of magnitude numberof 1066 g, where the age of the universe is taken to beroughly 1010 years. Table 1 summarizes the results of thevarious experiments conducted since the time of Cavendish,and figure 5 presents a graphical comparison of the relatedprecisions.

In table 1, direct (d) and indirect (i) tests of CoulombsLaw are listed. The experiments performed up to 1971 directlymeasured the parameter that indicates the degree of precisionof Coulombs Law written in the form 1/r2+; most of thesuccessive experiments are instead tests of the photon restmass m (1 = h/m c) and determine its upper limit.For these latter experiments, the corresponding precision

S166 Metrologia, 41 (2004) S159S170

Physical implications of Coulombs Law

Table 1. Coulombs Law and photon mass experiments.

Author Date /cm1 m /g Cavendish 1773 d 3 102 1 102 2.8 1040Coulomb 1779 d 1 102 1 101 2.8 1039Robison 1801 d 6 102 6 102 2.8 1040Maxwell 1892 d 5 105 1 103 2.8 1041Plimpton and Lawton 1936 d 2 109 1 106 2.8 1044Cochran and Franken 1968 d 9 1012 9 108 2.6 1045Bartlett et al 1970 d 1 1013 1 108 2.8 1046Williams et al 1971 d 6 1016 5 1010 1.4 1047Goldhaber and Nieto 1968 i 1.7 1016 1.4 1010 4.0 1048Franken and Ampulski 1971 i 4.3 1018 3.6 1012 1.0 1049Lowenthal 1973 i 3.0 1008 2.5 102 7.0 1040Davis et al 1975 i 3.4 1017 2.8 1011 8.0 1049Crandall 1983 d 3.4 1016 2.8 1010 8.0 1048Ryan et al 1985 i 6.4 1011 5.3 105 1.5 1042Boulware and Deser 1989 i 1.2 1013 1 107 2.8 1045Chernikov et al 1992 i 3.6 1014 3 108 8.4 1046Fischbach et al 1994 i 4.3 1017 3.5 1011 1.0 1048Lakes 1998 i 6.8 1019 5.7 1013 1.6 1050Schaefer 1999 i 1.8 1012 1.5 106 4.2 1044Lou et al 2002 i 5.1 1020 4.3 1014 1.2 1045

1775 1800 1825 1850 1875 1900 1925 1950 1975 2000Year

10-20

10-15

10-10

10-5

100

Prec

ision

Figure 5. Graph showing the increasing precision by which Coulombs Law in the form 1/r2+ has been verified since the 1700s. Thelaw would be verified exactly when = 0. Although some of the recent experiments are relatively imprecise, they are neverthelessimportant because they extend the validity of Coulombs Law as a function of the physical conditions of the measurement.

is here evaluated indirectly as if, instead of a photon testmass, equivalent experiments of the type by Williams et al,maintaining the same proportion between and , wereperformed.

7. Coulombs Law and units of measurement

The various tests discussed above have confirmed the validityof Coulombs Law to a high degree of precision. We may writeit vectorially as

F = kq1q2 r21| r2 r1 |2 , (20)

where r1 and r2 are the position vectors of the charges q1 andq2, respectively. We are left with the task of determining theproportionality constant k, the value of which is strictly related

to the system of units chosen. Although arbitrary, in general,the choice has usually been made in such a way as to obtainunitary and dimensionless constants.

Three systems of units have historically been used themost. The first two have been superseded by the third:

(1) The non-rationalized Gaussian system, which coincideswith the old cgs electrostatic system.

(2) The rationalized Gaussian system.(3) The presently accepted International System (SI ).In case (1), the constant k is dimensionless and unitary, i.e.k = 1. From the expression of the force F above, we canimmediately establish the physical dimensions of the electriccharge [q] in terms of mechanical dimension such that

[q] = [r]

[F ], (21)

Metrologia, 41 (2004) S159S170 S167

G Spavieri et al

that is, the charge has the dimensions of length times the squareroot of a force. The corresponding unit was called the Franklin(or simply u.e.s) and was such that a force of 1 dyne wouldact between two unit charges separated by a distance of 1 cm(1 Franklin = 1 u.e.s = 1 cm3/2 g1/2 s1).

In the rationalized Gaussian system, k was alsodimensionless and had a numerical value of k = 1/4 . Theadded complexity of incorporating a numerical constant intothe statement of Coulombs Law is amply compensated by thedisappearance of the 4 in the integral relations (as in GaussLaw). In this system, the unit of the electric charge was derivedusing arguments similar to those of the first case, but the unitwas seldom used.

The accepted standard, of course, is the InternationalSystem or SI, within which the unit of electric charge is thecoulomb [C]. The constant k now takes the value (40)1where 0, which has the defined value 0 = 8.854.... 1012 F m1, is the electric constant, i.e. the permittivity offree space or the dielectric constant in vacuo. In this case,k is neither a pure number nor a dimensionless ratio, insteadhaving the dimensions of [0]1.

In spite of the ubiquity of the SI, use of the Gaussiansystem persists among many theoretical physicists becauseit offers a convenient form of k and velocities appear indimensionless form (instead of v the quotient v/c is used,where c is the speed of light in vacuo). Of course, the SI isused uniquely for any measurements and all modern electricalinstruments and devices are calibrated accordingly.

8. Modern interpretations and conclusions

In considering the various tests of Coulombs Law, or inapplications of it to diverse physical phenomena, it is importantto remember that this fundamental tenet of electrostatics is anidealization. Therefore, care must be taken when applying itand 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 thunderstormcharge distributions, as done by Stolzenburg and Marshall [42].They considered the charge distribution in a stratiform regionof a mesoscale convective system and determined that thegeneral vertical charge structure of that region is horizontallyextensive over at least 100 km. Charge distributions weremodelled in a three-dimensional domain using an approach thatcorrectly calculates the electric field due to all the point chargesand their corresponding images on the basis of CoulombsLaw. Atmospheric physics, planetary physics, astrophysics,and plasma physics are vast domains of application for effortsof this type, and this hints at the range of scales over which itis routinely applied.

Coulombs torsion balance was in essence the firstprecision mechanical detector of charge, i.e. the first high-precision electrometer. The modern analogues of suchinstruments are semiconductor-based field-effect devices, themost sensitive of which are cryogenically cooled transistorsthat function at the single-electrons level [43, 44]. Also,recently, a working nanometre-scale mechanical electrometerwas constructed by Cleland and Roukes [45], who state that,in principle, devices such as theirs should ultimately reach

sensitivities comparable with charge detection capabilities ofthe cryogenic single-electron transistors.

Coulombs Law is also at the heart of pedagogicalphysics. It is measured and tested by students in laboratoryclasses virtually every day. There are several types ofprecision apparatus that are commercially available for studentexperiments on Coulombs Law. In fact, a Coulomb nullexperiment that enables physics students to obtain rigorousupper bounds on the photon mass using an apparatus thatoperates with subnanovolt signals has been devised byCrandall [46]. It is inexpensive to assemble and use, and theexperimental arrangement can be adapted for use at severalcollege levels. Of course, interest in the pedagogic aspects ofCoulombs Law goes well beyond the obvious applicationsof it to electrostatics. By introducing the superpositionprinciple and the theory of special relativity, students are taughthow generalizations of Coulombs Law lead to Maxwellsequations [47].

In this review, we have presented a description of theongoing development of empirical studies of the forcesbetween charges, beginning with the very early work bypeople like Cavendish and others that established the natureof the inverse square law, and then proceeding to the modernexperiments that search for deviations from exactness of theinverse square law. Both Newtons Law of gravity andCoulombs Law possess the same spatial dependence, 1/r2,and it is fair to say that these statements of the inverse squarelaw have served as critical markers in guiding much of thedevelopment of theoretical physics, in parallel with the longhistory of improvement of the experimental situation. Withthe development of modern physics, Newtons Law is nowunderstood to be the weak-field limit of Einsteins generalrelativity, while Coulombs Law, as one statement of theelectromagnetic interaction, has been absorbed into the unifiedtheory of electro-weak interactions.

However, according to general relativity, matterintroduces curvature into spacetime and this is whatmodifies the law of gravitation as originally formulated byNewton. The result is that the theory is nonlinear in thesense that the principle of superposition of effects is notvalid. On the contrary, Coulombs Law is not altered in thedescription of physical phenomena provided by modern unifiedtheories. In the unified theory, the interaction (attractionor repulsion) between charges is no longer described interms of forces but rather in terms of an exchange of virtualparticles (photons) that yields the same effect as the classicalelectrostatic force. If the photon mass is zero, Coulombs Lawremains a fundamental law of electrodynamics, where linearityand 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, thesame result predicted by Coulombs Law, always supposingthat the photon mass is zero. All this can introduce the thoughtthat Coulombs Law is thus more fundamental than NewtonsLaw and that electromagnetic interactions are a more primarykind of fundamental interaction at the basis of any attempt ofunification of the forces and interactions of nature.

Finally, in discussing the physical implications ofCoulombs Law, it is tempting to consider the remarkably highdegree of precision with which the inverse square law holds,

S168 Metrologia, 41 (2004) S159S170

Physical implications of Coulombs Law

and thus speculate that its mathematical simplicity may be atthe basis of the successful incorporation of the electromagneticinteractions into unified theories. The precisions achieved bythe experimental tests are increasingly exact. In recent timesthey go from the value of = (2.7 3.1) 1016 from thedirect test of Williams et al [15] to the more recent limit on thephoton mass of m < 1.2 1051 g from the indirect test ofLuo et al [41].

This glimpse at several implications of Coulombs Lawreconfirms its importance and centrality in several areas ofphysics. Some far-reaching implications of the non-validity ofCoulombs Law or, equivalently, the existence of a non-zerophoton mass, include the wavelength dependence of the speedof light in free space [48], possible variability in the speed oflight [49], deviations from exactness in Ampe`re Law [50], theexistence of longitudinal electromagnetic waves [51], and anadditional Yukawa potential [25, 23] in magnetic dipole fields,as discussed earlier. The mass of the photon has been underscrutiny since the early days of quantum mechanics. However,the stringent limits imposed by the experiments discussedearlier seem to exclude the existence of a non-zero mass atleast in the normal conditions of the universe as we knowit. Nevertheless, in cosmological inflation scenarios whereelectrodynamics and gravitation are present in a unified way,the dynamical generation of a photon mass could be morelikely. In this context, the cosmologists Prokopec et al [52]point out some far-reaching implications of the non-validityof Coulombs Law. In describing the early stage of theuniverse, they find that quantum electrodynamics and curvedspace may do the trick: in a locally de Sitter inflationaryspacetime and in the presence of a light, minimally coupled,charged scalar field, the polarization of the vacuum induces aphoton mass that may result in the generation of intragalacticmagnetic fields responsible for seeding the galactic dynamomechanism [53]. Apart from indirect tests of potentiallyobservable effects of a dynamically generated photon massin inflation, a direct test would have been possible only inthe extreme conditions of the early universe described bycosmological inflation scenarios, hence it is unfortunatelydifficult to see how this far-reaching implication of the non-validity of Coulombs Law might be examined. Finally,stepping away from electrodynamics, quantum mechanics,and cosmology, and entering the arena of thermodynamics,still other studies suggest that a non-zero photon mass wouldimpact our understanding of blackbody radiation and otherfundamental aspects of thermodynamics [54].

Regarding the limits of validity of Coulombs Law,particularly as a function of distance r , the Jovianmeasurements show that it holds at least up to distances ofthe order of r 106 km. From the microscopic point of view,the inferior limits found from scattering experiments show thatit holds down at the atomic (r 108 cm) and nuclear scales(r 1013 cm), and of course the cryogenic test extends itsvalidity to very low temperatures. Coulombs Law providesa useful and exciting window on fundamental physics and itopens experimental doors to a number of diverse fields of study.With continued imagination and through careful laboratorytechnique, it promises to be a stimulating tool for still morefar-reaching tests of the limits of our knowledge of the physicalworld.

Acknowledgments

This work was supported in part by the CDCHT, ULA, Merida,Venezuela.

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