Wilhelm Weber: On the Energy of Interaction (translated and edited by A. K. T. Assis)
aa r X i v : . [ phy s i c s . h i s t - ph ] S e p Wilhelm Weber: On the Energy ofInteraction (translated and edited byA. K. T. Assis)
Wilhelm Weber
Editor’s Note: An English translation of Wilhelm Weber’s 1878 paper“Ueber die Energie der Wechselwirkung”, [Web78b]. This work is an ex-cerpt from Weber’s seventh major Memoir on Electrodynamic Measurements,“Elektrodynamische Maassbestimmungen”, [Web78a] with English translationin [Web20].
Posted in September 2020 at , , (Excerpt by the author from the Treatise on Elektrodynamische Maass-bestimmungen in Volume XVIII of the
K¨onigl. S¨achs. Gesellschaft der Wis-senschaften .) , ,
6. A Particle Driven by both an Electric and a Non-Electric Forcewhile Enclosed in an Electrified Spherical Shell
Regarding the applications of the fundamental electric law, in order toshow that none of the “inconsistent and absurd” consequences occur, throughwhich Helmholtz wished to refute this fundamental law, we will only considerhere the application to the motion of a mass point µ (with an electric quan-tum ε ) enclosed in an electric spherical shell , when acted on by both an electric force and a non-electric constant force a . From this fundamental law, Helmholtz deduced in Borchardt’s
Journal ,[Volume] LXXV, the equation of the vis viva for this mass point µ withelectric quantum ε , [inside] a spherical shell of radius R uniformly covered [Web78b], related to [Web78a] with English translation in [Web20]. Translated and edited by A. K. T. Assis, . I thank F.D. Tombe for relevant suggestions. The Notes by H. Weber, the Editor of Volume 4 of Weber’s
Werke , are represented by[Note by HW:]; while the Notes by A. K. T. Assis are represented by [Note by AKTA:]. [Note by HW:] Annalen der Physik und Chemie , edited by G. Wiedemann, Vol. 4,Leipzig, 1878, pp. 343-373. [Note by HW:] As § § §
6, has been printed here. [Note by AKTA:] Pp. 343-365 of [Web78b] coincide with pp. 645-664 line 12 fromabove of the Abhandlungen der mathematisch-physischen Classe der K¨oniglich S¨achsischenGesellschaft der Wissenschaften (Leipzig) , [Web78a], and with pp. 364-382 line 10 fromabove of Volume 4 of Weber’s
Werke , [Web94]. [Note by AKTA:] Weber is referring here to his electrodynamic force law which hepresented in 1846, [Web46] with partial French translation in [Web87] and a completeEnglish translation in [Web07].Weber studies in this paper of 1878 the motion of a particle with mass µ and electriccharge ε moving inside a uniformly electrified spherical shell. He considers two forcesacting on this particle, namely, the electric force exerted by the shell and a non-electricconstant force a . He considers this constant force a to be the weight of the particle near thesurface of the Earth, namely, a = µg . He is replying to Helmholtz’s criticisms presentedin 1873, [Hel73], see also [Hel72a] with English translation in [Hel72b]. [Note by AKTA:] [Hel73]; see also [Hel72a] with English translation in [Hel72b]. with electricity, which appears as follows: (cid:18) µ − π c · Rεε ′ (cid:19) q − V + C = 0 , where ε ′ denotes the quantum of electricity per unit area on the surface ofthe spherical shell, q the velocity of the mass point µ and V the potential ofthe non-electric force. From this equation it has been concluded that when, with an existing difference between the potential V of the non-electric force and the constant C , ε ′ would have increased from 0 to [8 π/ c ] Rε · ε ′ = µ , then the vis viva ofthe point mass µ would have increased from µq = V − C up to µq = ∞ ,which would be an infinitely large work output . The removal of this ob-jection can now be obtained from the complete presentation of the wholeprocess of motion in its context, as indicated earlier in these
Annalen , [Vol- [Note by AKTA:] [Hel73, Section 12, pp. 48-54], see also [Neu74, §§ viv viva ( living force in English or lebendige Kraft in German) wascoined by G. W. Leibniz (1646-1716).Originally the vis viva of a body of mass m moving with velocity v relative to an inertialframe of reference was defined as mv , that is, twice the modern kinetic energy. However,during the XIXth century many authors like Weber and Helmholtz defined the vis viva as mv /
2, that is, the modern kinetic energy.Weber, for instance, in his paper of 1871 on the conservation of energy discussed twoelectrified particles of charges e and e ′ separated by a distance r . He then said the following,[Web71, Footnote 1, pp. 256-257 of Weber’s Werke ] with English translation in [Web72,p. 9]: If ε and ε ′ denote the masses of the particles e and e ′ , and α and β thevelocities of ε in the direction of r and at right angles thereto, and α ′ and β ′ the same velocities for ε ′ , so that α − α ′ = dr/dt is the relative velocity ofthe two particles, then 12 ε (cid:0) α + β (cid:1) + 12 ε ′ (cid:0) α ′ + β ′ (cid:1) is the total vis viva of the two particles.In 1872 Helmholtz expressed himself as follows, [Hel72b, p. 533]:If we, as has always hitherto been done, name vis viva or actual energy thesum of the moved inert masses multiplied each by half the square of itsvelocity, then, [...] [Note by AKTA:] ε ′ is the surface charge density. The total charge over the wholesurface of the spherical shell of radius R is then given by 4 πR ε ′ . [Note by AKTA:] Arbeitsleistung in the original. This expression can also be translatedas “work performed”. ume] XLVI, p. 29. , Let us denote by η that charge ε ′ on the unit area of the spherical shell forwhich the velocity q of the mass µ would be infinite, then set η = [3 c µ/ πRε ],and assume that ε has a certain constant value, while ε ′ grows uniformlyfrom 0 at time t = − ϑ up to η at time t = 0, the latter value being graduallyattained. Furthermore, to simplify the analysis, take the center of the sphereas the starting point of the path s where the particle µ at time t = − ϑ (where ε ′ = 0) is at rest, that is, with ε ′ = 0 we have s = 0 and q = 0.Then with the help of the values ε ′ = η (cid:18) tϑ (cid:19) , µ = 8 π c · Rεη and dVds = a , (see Article 12 of the Abhandlung ) , the following equation is obtained: dq = − aϑµ · dtt . The integral of this equation can be written as: q = − aϑ µ · log C t , [Note by HW:] Wilhelm Weber’s Werke , Vol. IV, p. 333. [Note by AKTA:] [Web75, p. 29 of the Annalen der Physik und Chemie and p. 333 ofWeber’s
Werke ]. [Note by AKTA:] Weber will consider the motion of the particle along a straight linebeginning at the center of the shell. We can represent this motion as taking place alongthe x axis, with x = 0 at the center of the shell, so that the path or trajectory s = x mighthave positive or negative values. When s = ± R the particle would reach the sphericalshell of radius R . [Note by AKTA:] Due to a misprint this expression appeared in the original as ε = 0. [Note by HW:] Wilhelm Weber’s Werke , Vol. IV, p. 333. [Note by AKTA:] [Web75, p. 29 of the Annalen der Physik und Chemie and p. 333 ofWeber’s
Werke ]. [Note by AKTA:] What Weber writes here as log of a magnitude θ should be under-stood as the natural logarithm of θ to the base of Euler’s constant e = 2 . ... , namely,log θ = log e θ = ln θ . His integration can be expressed as follows: Z qq =0 dq = − aϑµ Z tt = − ϑ dtt = − aϑµ [ln | t | ] tt = − ϑ = − aϑµ ln r t ϑ , such that q = − aϑ µ ln t ϑ . in which C = 1 /ϑ , because q = 0 should take place for t = − ϑ . Therefore,as q = ds/dt : ds = − aϑ µ · log t ϑ · dt . From this it follows through integration: s = aϑµ −
12 log t ϑ ! · t + C ′ . Since now s = 0 for t = − ϑ , it results C ′ = aϑ /µ , therefore: s = aϑ µ tϑ −
12 log t ϑ !! . When we set the non-electric force acting on µ as a = gµ , with q ′ beingthe ratio of the velocity q to gϑ , and with s ′ being the ratio of s [the path]to gϑ , then these formulas can be written as: dq ′ dt = − t ,q ′ = −
12 log t ϑ ,s ′ = 1 + tϑ −
12 log t ϑ ! . Now they can be used for the construction of all motions of the particle µ with an uniformly growing charge ε ′ and can be represented in a tabularoverview, where e is the base of the natural logarithm: [Note by AKTA:] Weber is assuming here that the constant force a is the weightof the particle of mass µ near the surface of the Earth, namely, a = µg . Moreover, heis defining the dimensionless displacement s ′ = s/ ( gϑ ) and the dimensionless velocity q ′ = q/ ( gϑ ) = ( ds/dt ) / ( gϑ ). [Note by AKTA:] Instead of dq ′ /dt , the expression in the fourth column of the firstline in the next Table should be the dimensionless acceleration given by1 g d sdt = 1 g dqdt = ϑ dq ′ dt = − ϑt . tϑ s ′ q ′ dq ′ dt ε ′ η − − e − − e − e − e − − e − − e − e − e − − e − − e − e − e − ... ... ... ... ...0 1 ∞ ±∞ e − e − − e e − + e − e − − e e − + e − e − − e e − +1 2 0 − e − − e − e + e − e − − e − e The curve
ABCDEF GH in the next Figure represents, according to thisinformation, the dependence of the velocity q ′ as a function of the path length s ′ , namely, s ′ as abscissa and q ′ as ordinate. This curve goes from the center A of the sphere as the starting point of the coordinates out to B , C andapproaches asymptotically the ordinate for s ′ = 1, then returning from thereto D , E , F , where it intersects the axis of abscissas at the point s ′ = 2, andthen goes on to G and H , where s becomes = − R and µ hits the sphericalshell. One can see from this overview that the particle µ , which would havecovered the distance gϑ in the time ϑ due to the acceleration g comingfrom the non-electric force, covers twice this path under the joint action ofthe electric force; moreover, while it had reached the velocity gϑ without theelectric force, it now reaches an infinitely large velocity with [the joint actionof] the electric force.However, with this attained infinitely large velocity , the particle µ doesnot cover the smallest finite path element , due to the fact that at the same mo-ment the acceleration dq/dt , which became equally infinitely large, suddenlyjumps from + ∞ to −∞ , that is, changes to an infinitely large deceleration ,causing the velocities to become equal long before and after this moment.For instance, the velocity q at time t = + ϑ (that is, after the time interval [Note by AKTA:] When the ordinate q ′ = 0 the letters from left to right along theabscissa s ′ should read as follows: H ◦ , A , K , F ′ and F . Due to a misprint the first point H ◦ was printed as H . When the ordinate q ′ is equal to −
1, the letters along the abscissafrom left to right are G and G ′ . Close to q ′ = − . s ′ = − H , while closeto q ′ = − . s ′ = − . H ′ . Figure 1: [Ordinate q ′ as function of abscissa s ′ .]2 ϑ calculated from the beginning of the motion) is equal to the velocity inthe beginning, at time t = − ϑ , namely q = 0, where the path s , when thespherical shell is large enough for s to still have room inside it, would havegrown again by gϑ , so that s would become = 2 gϑ . The charge ε ′ wouldthereby have grown up to 2 η . From now on, however, with time and charge[of the spherical shell] continuing to increase, the displacement of the particle µ from the center of the shell would decrease quickly up to s = 0, and thenbecome negative up to s = − R , where the particle µ would hit the sphericalshell at time t , which can be determined through the equation − R = gϑ " tϑ −
12 log t ϑ ! , and with the velocity q which, after t has been determined, is found from theequation q = [ gϑ/
2] log[ t /ϑ ].It has been assumed up to now, that the radius R of the sphere is largerthan the largest value which s has reached at time t = + ϑ , namely, 2 gϑ .If R were smaller, then it is evident that the particle µ would have collidedearlier against the spherical shell, namely, at the moment in which s wouldbecome = R , which can be determined from the equation R = gϑ " tϑ t ϑ ! . Now, finally, when there is no continuous increase in the electric charge ε ′ ,as previously assumed, but instead of this the charge ε ′ remains constant afterit reaches the value η and surpasses it by any assumed arbitrarily small value,then let us designate this constant charge as η (1 + e − n ), and consequentlythe time at which this occurred as t = + e − n ϑ , the velocity of the particle µ at this moment as q = ngϑ , and the distance of the particle from the centerof the sphere as s = (1 + (1 + n ) e − n ) gϑ . This results in the differentialequation: dq = − ae n µ · dt , and from it through integration: q = − ae n µ t + C .
Now if the time is calculated from the moment in which the charge [onthe spherical shell] has become constant, where the velocity q = ngϑ , thusyielding C = ngϑ , therefore, as [the constant force] a has been set = gµ , [weobtain]: q = dsdt = − ge n · t + ngϑ . From this one obtains through a second integration: s = ngϑt − ge n · t + C ′ , and, as has already been mentioned, for t = 0 we have the value from s =(1 + (1 + n ) e − n ) gϑ , yielding consequently: C ′ = (cid:16) n ) e − n (cid:17) gϑ , therefore: s = ngϑ · t − ge n · t + (cid:16) n ) e − n (cid:17) gϑ . This formula for the displacement s and the obtained formula for the velocity,namely: q = − ge n · t + ngϑ are now used, for a constant remaining charge ε ′ , to determine all motions ofthe particle µ . They can be represented in a tabular overview, for instancein the following Table for the case in which n = 2, when s/ ( gϑ ) = s ′ and q/ ( gϑ ) = q ′ are set as above: tϑ s ′ q ′ ε ′ η e e e e e e e e − e e − e e − e − e − t = 2 ϑ/e onwards, the displacement of the particle µ from the center of theshell decreases and very soon becomes negative, until finally the particle µ ,when s becomes = − R , collides against the spherical shell, at time t andwith the velocity q , which can be determined from the two equations: − R = (cid:18) e (cid:19) gϑ + 2 gϑ · t − e g · t , q = 2 gϑ − e g · t . One can see from this presentation of the whole process in its context ,that none of the “inconsistent or absurd” consequences, by which Helmholtzwanted to refute the established fundamental law, actually occur.The curve
ABCDE on page 7 represents the dependence of the velocity q as a function of the displacement s of the particle µ from the center ofthe sphere, with a uniformly increasing charge ε ′ , up to the moment whenthis charge becomes greater than η , namely, = η (1 + [1 /e ]). This curvecan now be continued in two ways, either for a charge [on the sphericalshell] continuing to grow uniformly as before, which is represented by thecurve EF GH and which has already been considered, or for a charge ε ′ = η (1 + [1 /e ]) which remains constant from now on, which is related to thedeterminations in the Table mentioned above, after which the curve EF ′ G ′ H ′ forms the continuation of curve ABCDE .In both cases the particle µ moves in a continuous path, namely, in thefirst case along a straight line from A up to F and from there back to A andfurther to H ◦ , where the particle hits the spherical shell; in the second casealong a straight line from A up to F ′ and from there back to A and H ◦ .Also the velocity of the particle along its path changes always contin-uously, except at one point K , in the middle of the path AF , where thevelocity of the particle becomes infinitely large, and at the same time withit the work performed from the beginning of the motion onwards. But if werepresent this performed work as positive , this is immediately followed by a negative case which is also infinitely large.Each of these two performed works can be divided into two parts, namely,the first or positive case of the work performed along the path from A to apoint at a distance = [( n + 1) /e n ] · gϑ before K , and in the work performedalong this last distance before K = [( n + 1) /e n ] gϑ ; the latter or negative case of the work performed on the way through the distance after K =[( n + 1) /e n ] gϑ , and on the rest of the way up to F or F ′ .Of these four performed works, the two on the path = [( n + 1) /e n ] gϑ before and after K are infinitely large , but oppositely equal , while the othertwo are also oppositely equal, but have finite values. Since n can now beconsidered so large, that the time [interval] of the first two, infinitely largeperformed works, namely, 2 ϑ/e n , can be regarded as negligible, one has twoinfinitely large, but oppositely equal performed works taking place in an [Note by AKTA:] See Footnote 21 on page 6. n was = 2, one can choose anotherexample, where n is much larger, so that the difference of the charge ε ′ , whichbecame constant, from η becomes vanishingly small; no substantial changeis brought about by this and one can see from the presentation of the wholeprocess in context, that none of the “inconsistent and absurd” consequences,by which Helmholtz wanted to refute the established fundamental law, everreally take place.2 References [Hel72a] H. Helmholtz. Ueber die Theorie der Elektrodynamik.
Monats-berichte der Berliner Akademie der Wissenschaften , pages 247–256, 1872. Reprinted in H. Helmholtz, Wissenschaftliche Abhand-lungen (Johann Ambrosius Barth, Leipzig, 1882), Vol. 1, Article34, pp. 636-646.[Hel72b] H. von Helmholtz. On the theory of electrodynamics.
PhilosophicalMagazine , 44:530–537, 1872.[Hel73] H. v. Helmholtz. Ueber die Theorie der Elektrodynamik. ZweiteAbhandlung. Kritisches.
Journal f¨ur die reine und angewandteMathematik , 75:35–66, 1873. Reprinted in H. Helmholtz, Wis-senschaftliche Abhandlungen (Johann Ambrosius Barth, Leipzig,1882), Vol. 1, Article 35, pp. 647-683; with additional materialfrom 1881 on pp. 684-687.[Neu74] C. Neumann. Ueber das von Weber f¨ur die elektrischen Kr¨afteaufgestellte Gesetz.
Abhandlungen der mathematisch-physischenClasse der K¨oniglich S¨achsischen Gesellschaft der Wissenschaften(Leipzig) , 11:77–200, 1874.[Web46] W. Weber. Elektrodynamische Maassbestimmungen — ¨Uber einallgemeines Grundgesetz der elektrischen Wirkung.
Abhandlun-gen bei Begr¨undung der K¨oniglich S¨achsischen Gesellschaft derWissenschaften am Tage der zweihundertj¨ahrigen GeburtstagfeierLeibnizen’s herausgegeben von der F¨urstlich Jablonowskischen Ge-sellschaft (Leipzig) , pages 211–378, 1846. Reprinted in WilhelmWeber’s
Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893),pp. 25-214.[Web71] W. Weber. Elektrodynamische Maassbestimmungen insbesonde-re ¨uber das Princip der Erhaltung der Energie.
Abhandlungender K¨oniglich S¨achsischen Gesellschaft der Wissenschaften, ma-thematisch-physische Klasse (Leipzig) , 10:1–61, 1871. Reprintedin Wilhelm Weber’s
Werke , Vol. 4, H. Weber (ed.), (Springer,Berlin, 1894), pp. 247-299.[Web72] W. Weber. Electrodynamic measurements — Sixth memoir, re-lating specially to the principle of the conservation of energy.
Philosophical Magazine , 43:1–20 and 119–149, 1872. Translatedby Professor G. C. Foster, F.R.S., from the
Abhandlungen der mathem.-phys. Classe der K¨oniglich S¨achsischen Gesellschaft derWissenschaften , vol. x (January 1871).[Web75] W. Weber. Ueber die Bewegung der Elektricit¨at in K¨orpern vonmolekularer Konstitution. Annalen der Physik und Chemie , 156:1–61, 1875. Reprinted in Wilhelm Weber’s
Werke , Vol. 4, H. Weber(ed.), (Springer, Berlin, 1894), pp. 312-357.[Web78a] W. Weber. Elektrodynamische Maassbestimmungen insbeson-dere ¨uber die Energie der Wechselwirkung.
Abhandlungender mathematisch-physischen Classe der K¨oniglich S¨achsischenGesellschaft der Wissenschaften (Leipzig) , 11:641–696, 1878.Reprinted in Wilhelm Weber’s
Werke , Vol. 4, H. Weber (ed.),(Springer, Berlin, 1894), pp. 361-412.[Web78b] W. Weber. Ueber die Energie der Wechselwirkung.
Annalen derPhysik und Chemie , 4:343–373, 1878. Reprinted in Wilhelm We-ber’s
Werke , Vol. 4, H. Weber (ed.), (Springer, Berlin, 1894), pp.413-419.[Web87] W. Weber. Mesures ´electrodynamiques. In J. Joubert, editor,
Col-lection de M´emoires relatifs a la Physique,
Vol. III:
M´emoires surl’ ´Electrodynamique , pages 289–402. Gauthier-Villars, Paris, 1887.[Web94] W. Weber.
Wilhelm Weber’s Werke,
H. Weber, (ed.), volume 4,
Galvanismus und Elektrodynamik , second part. Springer, Berlin,1894.[Web07] W. Weber, 2007. Determinations of electrodynamic mea-sure: concerning a universal law of electrical action, 21st Cen-tury Science & Technology, posted March 2007, translatedby S. P. Johnson, edited by L. Hecht and A. K. T. Assis.Available at http://21sci-tech.com/translation.html and .[Web20] W. Weber, 2020. Electrodynamic measurements, especially on theenergy of interaction. Second version posted in August 2020 at