VI. SUMMARY The N (Macmillan, New I have argued that Newton's first two laws of motion are n, Princi- ns of force but rather contain strong 966)and was further s of Space-Time Theories of a four- A. Weyl, multaneity d New York, first given rie de la Suppl. 40, relativity by re these objects eorie de la 41,1-25 la gravi- ACKNOWLEDGMENT I wish to thank James Supplee for several helpful discus- n with Frank- sions and suggestions on improving the presentation of this lin Polloo paper. A. Einstein, "Zur electro nn. Phys. Leip- zig17,891-921(1905) J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Berlin, Sitzber. 778-786, 799-801 (1915); English translation available Systems(Harcourt, San Diego, 1988), 3rd ed., p. 45. in A. Einstein, The Principles of Relativity (Dover, New York, 1952). Precise calculation of the electrostatic force between charged spheres including induction effects Jack A. Soules Department of Physics, Cleveland State University, Cleveland, Ohio 44115 (Received 12 January 1989; accepted for publication February 1990) method of images is applied iteratively to compute precisely the electrostatic force between show that Coulomb overlooked induction effects revealed in his data. Most textbooks intr It requires qua ectrodynamics to calculate those ef- lomb's experiment and his results which ed by F~1/d2." This h fects when two electrons are really close. After perorming was a stu- the calculations for the force between charged conducting dent. When the sim nough to spheres (see Appendix for details), we decided to compare produce detectable fo to move the charg ental forces measured by d con- producinga ective d ha charged the descriptic pith balls een mathematical n not given but from course, the standard form to point charges that are i ly about 18 lines (1 in. ) (See n the centers of the balls was 1195 Am. J. Phys. 58(12), December 1990 1990 American Association of Physics Teachers 1195
d=2a sin 0/2, which Coulomb treated as approxi&9 a0. He found that when the angle was decreased from 36 18, the force quadrupled, within an error of less than 1% However, to quadruple again the angle had to be reduced, not to 9, but to 8. Coulomb noticed this error but made no further comment on it. Evidently he did not expect his apparatus to give a more precise result even though it was capable of it. Of course, we know that the experiment was eporting the force accurately. It was the theory that was in error! At or 9, the parameter d /a is somewhere between 3.7 and 5.5, where we see from Fig. 1 that the force is about 5%0 less than 1/d2 Coulomb was so eager to prove the 1/d2 law that he overlooked the experimental observation of po- larization Because Coulomb did not specify the exact diameters of his pith balls, we also calculated the repulsive force curve for balls with diameters in the ratio 3: 2. The resulting force at the distance d/(a, +a,)=2 or more is only slightly CENTER TO CENTER DESTANCE, d less(1%)than the force shown in Fig. l, leaving our conclusion intact. However, at close approach, the repul ion between dissimilar balls falls significantly (-50%) Fig. 1. Behavior of the electric force between identical charged spheres as below the curve for identical balls shown in Fig. 1 a function of center-to-center distance It is too much to expect Coulomb to have understood the concept of equipotential surfaces in 1785. However, he did know that the electrical"fluid"was mobile on the pith balls since if it were not so he could not have charged them. He failed to recognize, however, that this same mobility would ause the charges to move on the balls in the presence of the electrical force, thus spoiling his effort to identify the cen- ter of a ball with the center of the Coulombs torsion balance was and is an extremely sen sitive and precise device for physical measurements. How ever, it is a small miscarriage of justice to name the electro- static force law in his honor. Most scientists of his time expected the force law to be 1/d- and others had mea sured it to be 1/(d2+e), for example. It was Priestleywho recognized that if cork balls inside a charged cup were un affected by the charge on the cup then F-1/d2 exactly.He drew this conclusion from Newton's work on the force of gravity within a hollow sphere. This theoretical insight is ar more satisfacte tory in establishing F-1/d2 than any ex periment, which, as can be worked out from Table I, will give a force law more like Frep -1/D-1/D+-2/D etc. where D=d/a When Coulomb attempted to measure the attractive force between unlike charges he met a serious problem. The force increases so rapidly at small distances that his torsion lance for which the force was linear with distance, could not compensate. To measure the attractive force he used a suspension with so little restoring torque that its natural period of oscillation was very long. He then bro charged sphere close to a suspended disk and measured the period under conditions in which the restoring force was, for all practical purposes, the electrical force. See Fig ere in this experiment was 6 in. in rad Is and the small disk was about 2 in. in radius. The separa- tion in various trials ranged from 9 in.(center-to-center)to 18 in Coulomb tells us he charged the disk by induction when it was“ some inches”away There are two ways of estimating the charge on tI sna disk. a disk carrying charge g has a potential plified diagram of Coulomb's apparatus for measuring electro- 1.5708(=m /2)times the potential of a sphere of the same radius.If we compute the potential at the center of the disk 1196 Am J Phys., Vol 58, No 12, December 1990 Jack A. Soule 1196
Table l calculated values of Fas a function of center-to-center distance. We know that even in the event disk =0 there is an attrac- tive force! We return then to our iterative procedure to Distance D compute the attractive force between a large sphere as a multiple of a attraction Coulomb repulsion F*(alarge=1)and a small sphere(asmall =32)carrying @o and -00240@o, respectively 100 009960.009990 0124150.01234600122770.0121190 Table II compares the results of the force computation 0.015 751 0.015 625 0.015 501 0.015 373 including induction with the pure 1/d2 results. We have 00206560.02040800201650.019975 suppressed the 1/4Eo factor and the ratio of inches to 6.0 0.028 324 0.027778 0.027250 0.026963 meters in both columns. It is clear that the attractive force 5.0 0041404004 00366800038 is much stronger than 1/d2 for d/a<1.5 0.067 0.0625 00584570058106 Coulomb,'s torsional oscillator should have a period in- 0. 134819 0.111 111 0.094 4370.096022 versely proportional to F/, therefore, directly propor 2.4 0.285 277 0.1736 0.127 089 0.133 005 tional to dif F-1/d2. He presents his results in the form of 02406470.16 0.1210910.126208 a table 0.3525140.1890360.1333130.139797 04702690.2066120.1397940.146283 The distances are as the numbers: 3 6 8 0.7597390.2267570.1465830.152019 The times of the same numbers of oscillations are 1.225380.2379540.150120.154385 3.76450.2475190.154 By theory they ought to have been: 20 40 54 0.154 0.15625 He then explains the small discrepancy at large distance by claiming that the charge had probably drained away a bit 1/D2-1/D4-2/D=d/a; F*isa simple function that (Note: He evidently did not reverse the order of the experi closely approximated Repulsion ment to see if that was really the casel) From our enlightened position, we choose to present his ata differently, making the period at large separation the due to the large sphere and place a charge on the disk of most reliable. We then have the followina(4 in.) 6(18 opposite sign so as to reduce its potential to zero, we get The distances are as the numbers 0.014 920hn when the disk is 8 in from th )3(9in.) ter of the large sphere. This calculation ignores induction By theory (1/d2)the times should be: 60 22.5 effects on the large sphere. Alternatively we can replace the By theory (corrected for induction)the time should be disk with a small sphere of equal capacitance, in which case 521.5 the small sphere will have a radius of By experiment the times are: 60 4 2/m×1×ags=0.0309 a,arge. Using the image charge it Clearly, induction effects can account for a substantial eration technique described in the Appendix, we find that part of the discrepancy he small sphere is at zero potential when it is charged to The intellectual climate of the 18th century was such about -0.0240sphere about 50% more charge that Coulomb could not hope to gain much respect from with these figures in hand, we can proceed to calculate inventing a clever and sensitive apparatus for measuring what Coulomb should have observed for the attractive forces. Proving that electrical charges obeyed a force law force. Of course, the actual charge on the disk could be identical to the gravitational force law, on the other hand, considerably different from our estimated value since we would have great philosophical significance. We should have no way of knowing what Coulomb meant by"some not be rised, therefore that Coulomb "stretched his inches. "In the absence of induction effects, the charge on data to"prove"the 1/d2 law rather than to respect the the small disk would not matter in determining the force precision of his instrument enough to find an entirely new law. However, in the presence of induction it clearly does. electrical effect Since our calculation technique was so convenient we roceeded to examine the induced force in two additiona cases:()when an uncharged sphere is brought near to a charged one; and (2)when two equally charged spheres are of different radii Table Il. Representative forces in simulation of Coulomb's attractive e as a multiple of a Force with induction Force, I/D2 0.111 0.161 Fig 3. Simplified diagram of Coulomb,'s apparatus for measuring electro- 1.812 0850 1197 Am J Phys., Vol 58, No 12, December 1990 Jack A. soules
ion of any electrostatic problem involving two spheres, need for the solution of a differential equation either by saying that the gradient of the field is too smal ver the diameter of the small sphere or by noting that the image charges are small and highly localized on both APPENDIX spheres. When the small sphere is charged, however, and ught near the uncharged large sphere the resulting force Consider two similar charges, o, a distance d apart. If very large. Essentially, the small sphere sees its negative we surround one charge with a conducting sphere of radius image close by and is strongly attrached to it. Or, in a differ- a and require that the sphere be an equipotential surface, ent description, the potential of the small sphere is large we find that an image charge of magnitude and the field gradient over the diameter of the large sphere q1 (A1) is also large must be placed a distance a'/d from the center of the When the two spheres are of different radii a surprising sphere to produce a spherical equipotential at the radius a2 esult occurs. If a,> 1.24a2 there exists a center-to-center he original charge go at the center of the sphere must be spacing where the force between equally charged spheres increased to go 1q1 to maintain the total charge equal to becomes attractive.Of course, in the laboratory we seldom go. A second conducting sphere of radius a around the deal with equal charges on dissimilar spheres. It is more second charge go will be an equipotential if two new image common to charge the spheres to same potential charges are placed inside, corresponding to the images of In Fig 4 is shown the position of the null point, at which 1 and of o' lg l. Each image charge is smaller than its the force between equally charged spheres changes sign object charge and of opposite sign. By repeated iteration of from repulsion to attraction. Distances in Fig. 4 are ex- this scheme, a task well suited to a personal computer,we pressed as d/(a,+ a2)and the abscissa is the"eccentric- found that a string of charges of alternating sign and rapid ly decreasing magnitude can replace the charged spheres These results should not surprise us. If the larger sphere and produce the appropriate field. Figure 5 shows the loca were very much larger than the smaller one its charge tions of the image charges when d=2.1a would create only a small repulsive force on the small The computer program calculates all of the anand sphere. The small sphere, on the other hand would be at tracted to its own oppositely charged image by a much from larger force. Our calculations show that the onset of this D,=d-xn+1 effect occurs at about a,= 1. 24a2 D I=l/Dn, The program we have written is convenient for the solu- -gn/Dn The resulting charges are in the right proportions but no longer sum up to @o so their sum S is calculated and a new o= 1/S is chosen, The process is repeated until IS-1<10-. To calculate the total force between the spheres we simply sum all of the forces between the sets of charges, a task that the computer does very quickly. In ractice, when the distance between the spheres d is greater than 4a, as few as 10 charges suffice for six-digit accuracy On the other hand, when the spheres touch, d= 2a, and more than 100 charges on each side are needed to compute 2*419 q。+1428 46" Fig. 4. Unequal spheres, equally charged; center-to-center distance where force changes sign. Fig. 5. Distribution of image charges when D=2. 1a, repulsive case 1198 Am J Phys., Vol 58, No 12, December 1990 Jack a. soule
the force. Because the charges alternate in sign, the series between conducting spheres of arbitrary radius and spac. The problem of the attractive force between unlike g does converge for all d> 2a ges is tackled in the same way. The charges q? appear at the same positions and with the same magnitudes but David nick, Fundamentals of Physics(Wiley, this time are all of one sign, negative on one side and posi- New York, 1986), 2nd ed, p. 457. Most elementary texts have a similar statement. tive on the other The resulting total force between the 2 Charles-Augustin Coulomb, "Law of electric force, "Mem.Acad.R charge systems is slightly larger than a-/d but, unlike the repulsive case, begins to diverge rapidly as d-2a In Fig. I 'William francis Magie, A Source Book in Physics(McGrawHill, New are plotted the logarithms of the repulsive, the a/d2, and York, 1935), p.408 the attractive forces as a function of the logarithm of the Coulombs many papers were reprinted in 1884 by Gauthier-Villars parameter /a Ofcourse, the a/d graph is a straight line. under the auspices of the Societe Francaise de Physique. This is the The distribution of image charges in a typical example is version that is most commonly available in American libraries. In the shown in Fig. 5 reprinted version of the 1785 papers on the electrostatic force there are The calculation of electric potentials, fields, and charge insertions in which the various assumptions of Coulomb are ana- distributions by the method of images is described in many yzed mathematically. In the first of these, the annotator calls attention undergraduate and graduate electricity texts. Smythe's to the small error due to coulomb s failure to account for inductive text is particularly useful because it was written after the GAbriel G. Luther and william r. Towler,"Redetermination of th possibly tedious calculations feasible. Smythe, for example,(1982 shows that the image charges for two conducting spheres Cf, for example, the discussion in Franklin Miller, college Physics(Har- satisfy a difference equation that can be solved in closed court Brace Jovanovich, New York, 1972), 3rd ed, p. 372 form: 7 Reference 6 q,=R,R2 sinh a/R, sinh na +R, sinh(n-1 sCf. Wolfgang K. H. Panofsky and Melba Phillips, Classical Electricity and Magnetism(Addison-Wesley, Reading, MA, 1955), p. 85 with mundT. Whittaker and George N. Watson, A Course of Moder (Cambridge U P, London, 1963), p. 382 Panofsky and Phill nian hs a=cosh-(d2-Ri-R2/2R, R 2 that the potential of the disk is We have chosen to present our calculations in computer v(a)=mEa x Jo,(x)sin x dx. iterative form simply because the interesting questions of physics and of mathematics(convergence, divergence)are okisl whittaker and Watson show that S.x- Jo(xrsin x dx=r/2,for more transparent in that form. Either method can be used William R. Smythe, Static and Dynamic Electricity(McGraw-Hill,New to calculate the potentials, forces, and capacitances York, 1968), 3rd ed, pp.128-131 1199 Am J Phys., Vol 58, No. 12, December 1990 Jack A Soules 1199