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