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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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On approximate formulasfor theelectrostatic force between two conducting spheres

Facultad deCiencias FõÂsico-MatemaÂticas, Universidad AutoÂnoma de Puebla, Apartado Postal 1152,

72000 Puebla, Pue.,MeÂxico

RauÂl A. Brito-Orta

Instituto de FõÂsica ``Luis Rivera Terrazas,'' Universidad AutoÂnoma de Puebla, Apartado Postal J-48,

72570 Puebla, Pue., MeÂxico

~Received21 March 1996; accepted 23 October 1997! A series expression for the electrostatic force between two charged conducting spheres having equal

radii and charges is derived using the method of electrical images. This expression is a special case

of that for two spheres with arbitrary charges and radii, found by Maxwell using zonal harmonics. Keeping in mind the use of approximate formulas for the interpretation of classroom measurements of the electrostatic force between spheres, we comment on two incorrect approximate formulas and examine the contribution of the ®rst few non-Coulomb terms of the correct formula by comparing with values obtained using a computational approach. ©

1998 American Association of Physics Teachers.

I. INTRODUCTION

Coulomb's law is demonstrated in classroom experiments in different ways, using either a kind of torsion balance 1 or some other tool for electrostatic force measurement.

2±4

Al- though there are still some doubts about the actual experi- mental determination of the inverse square law by Coulomb himself, 5 nowadays we see classroom measurements that have suf®cient precision even to detect deviations from the simple inverse square relationship between force and distance

6±8

valid for point charges. That such deviations, due to the redistribution of charge caused by the mutual electrostatic in¯uence, must have oc- curred in Coulomb-like experiments was mentioned by

Maxwell.

9

He also admitted that a quantitative account of

this effect requires an ``intricate investigation,'' which, for the case of two spheres, was ®rst carried out ``in extremely able manner'' by Poisson and later ``greatly simpli®ed by Sir W. Thomson in his Theory of Electrical Images.'' A possible role of induction effects in Coulomb's original experiment was recently discussed by Soules 10 using numerical methods forthecalculation ofthe force.

Many authors

11±13

give, as an illustration of the applica-

bility of the method of image charges, a general formula forthe force between two spheres which takes into account the

effects of charge redistribution. Nevertheless, the derivation of an approximate formula, suitable to obtain the theoretical insight needed to deal with situations met in classroom mea- surements related to Coulomb's law, is commonly left as a homework exercise for interested readers. Maxwell's com- ment cited above and the examples given below show that this isnotatrivial task. In Sec. II we present such a formula, derived by the method of electrical images. Two incorrect approximate for- mulas, published in this journal, are discussed in Sec. III. In Sec. IV we consider the accuracy of the correct formula, keeping in mind its application in classroom measurements.

Conclusions are giveninSec.V.

II. A CORRECT APPROXIMATE FORMULA FOR

THE ELECTROSTATIC FORCE BETWEEN

TWO SPHERES

Calculations for the general case of spheres with arbitrary radii and charges are very complicated. They become sim- pler if the spheres have equal radii and charges. Consider the case of two perfectly conducting spheres with radiusaand charge1qwhose center-to-center distance isd.

352352Am. J. Phys.66~4!, April 1998 © 1998 American Associationof Physics Teachers This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

157.92.4.75 On: Wed, 23 Mar 2016 03:06:38

We replacethe spheres bytwo in®nitesets of image charges,q n andq n

8, located on the line of centers between

the spheres at positions, with respect to the center of one of the spheres, given byx n andx n

8, wheren51,2,3,... . Since

the spheres are identical in size and charge, we have 6 q n 5q n 8,x n 85d2x
n ,n51,2,3,...,~1! with q n 52aq
n21 d2x n21 ,n.1,~2! and x n 5a 2 d2x n21 ,n.1,~3! withx 1

50. Using these recurrence relations, we can ®nd

expressions for all the charges in terms ofq 1 . For example, the ®rst ®ve chargesare givenby q 1 5q 1 q 2 52bq
1 q 3 5b 2 q 1 /~12b 2 !,~4! q 4 5b 3 q 1 /~2112b 2 q 5 5b 4 q 1 /~123b 2 1b 4 andthey are located,respectively, at x 1 50,
x 2 5ab, x 3

5ab/~12b

2 !,~5! x 4

5ab~12b

2 !/~122b 2 x 5

5ab~122b

2 !/~123b 2 1b 4 where b5a/d.~6!

Notice that if we writeq

n as a series in powers ofbthe leading term will be proportional to b n21 . It may be seen from Eq.~4!that the total charge in each sphere, q5q 1 1q 2 1q 3

1{{{,~7!

is proportional toq 1 , so thatq 1 can be written asqtimes a seriesinpowers of b. The magnitude of the force between the spheres is given by the derivative of the potential energyW5qq 1 /(4pe 0 a) with respect tod. We substitute forq 1 its series expansion and take the derivative to ®nd the following expression for the repulsive electrostatic force between two conducting spheres: F5F C ~124b 3 26b
5 114b
6 28b
7 154b
8 250b
9 1154b
10 2264b
11 1494b
12

21092b

13

11830b

14

24192b

15

17140b

16

215894b

17

128234b

18

260320b

19

1112056b

20

2230032b

21

1{{{!,~8!

where F C 5q 2 /~4pe 0 d 2 !.~9! This formula coincides with one which can be derived from a general formula found by Maxwell using zonal harmonics. 14

We give this series to order 21 because it is the

same order as obtained from Maxwell's formula. The result can be extended to any number of terms using a program thatcan carry out symbolic calculations, like

MATHEMATICAor

MAPLE.

To compare approximate and ``exact'' numerical compu- tations, notice two things about Eq.~8!as an approximation. First, Eq.~8!was obtained using a ®nite number of charges. To ®nd a correct expression to orderNin b,itis necessary to take into account the contributions ofN12 im- age charges within each sphere. This is clear from Eq.~2!, where it can be seen that the chargeq N13 contributes terms of orderN12 or higher in bin the series expansion ofq 1 After taking the derivative to ®nd the force, these terms will be of orderN11 or higher. These considerations can be important also when comparing Eq.~8!to other series expan- sions found by analytical approaches. Second, Eq.~8!can be considered a series expression for the total force between two sets of 23 charges located at certain positions. For this case, we would obtain better re- sults by expanding the series to higher order. This may be important when comparing with numerical approaches using a ®nite number of image charges and adding the forces be- tween every pair of image charges~one charge of the pair within the ®rst spherical surface and the other within the second!. No series expansion is needed. For distances that are large in comparison with the radius, the series expression for the force converges very quickly because b!1. This is not the case when the spheres are very close. Then, as will be shown later, the required number of charges is very large and analytical approaches turn out to be very impractical. For small distances one must make a nu- merical calculation.

III. COMMENTS ON TWO INCORRECT

APPROXIMATE FORMULAS

Larson and Goss,

6 using an amazingly simple measuring tool, obtained results which could not be ®tted using Cou- lomb's law. Not knowing a formula which could be used for bringing together experiment and theory, they attempted to derive one. Using the method of images, Larson and Goss found the approximate formula: F LG 5F C ~124bquotesdbs_dbs41.pdfusesText_41

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