Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 9

I proceed to presentthe argument in the form which I believe may be close to what Helmholtz did.The gravitational potential energy of an outer spherical shell of radius r andmass dMr in the field of an inner shell of radius s and mass dMs is equal todE rsdM r dM sdM r dM spotrsrsE pot = −G, because −= −G=F2rdrr49P. Lenard: “Große Naturforscher’’. J.F. Lehmann Verlag München (1941).And yet, in 1921, when M.

Planck edited two of Helmholtz’s later papers onthermodynamics, he complained about the shear unbelievable number of calculationalerrors in Helmholtz’s papers. So, maybe Helmholtz might have profited, after all, fromsome formal mathematical education.51 H. Helmholtz: “Wissenschaftliche Abhandlungen.” Vol. I (1882), Vol. II (1883), Vol III(1895).50Electro-magnetic Energy29is the gravitational force on the outer shell. G is the gravitational constant.Therefore the potential energy of the outer shell in the field of all shells with s < ris equal toT' RQV)d/ TT/Tand the potential energy of the whole star is4' RQV) ³04/T/ T21 / 42 1)d/ T)dT .N³T2 42 0 T2by partial integrationThus Epot is determined by MR and R but also by the mass distribution Mr within thestar.

I believe that Helmholtz may have considered ȡ as homogeneous, equal toMR4 / 3π R3. In that case the calculation is very easy and one obtains2E pot-11 53 G M R . We calculate this value with G = 6.67·10Rm3kg s 2for the solar30mass MR = 2·10 kg and for the two cases when the sun has its present radius R =0.7·109m and when it has the radius R = 150·109 m of the earth’s orbit.

Thedifference is ǻEpot = 22.76 ·1040 J and, if we suppose that this energy is radiated offat the present rate, see above, we obtain ǻt = 20·10 6 years for the time needed forthe contraction. That is indeed close to the time given by Helmholtz.We shall recalculate Epot under a less sweeping assumption in Insert 7.6.Insert 2.2Helmholtz remained active until the last years of his life, and he took fulladvantage of what Clausius was to do. Later on – in Chap.

5 – we shallmention his concept of the free energy – Helmholtz free energy in Englishspeaking countries – in connection with chemical reactions.Electro-magnetic EnergyIt was not easy for a person to be a conscientious physicist in the midnineteenth century. He had to grapple with the ether or, actually, with up tofour types of ether, one each for the transmission of gravitation, magnetism,electricity and light. The ether – or ethers – did not seem to affect themotion of planets,52 so that matter moved through the ether without any52Actually Isaac Newton (1642–1727) conceived of a viscous interaction between the etherand the moon, and that idea led him to study shear flows in fluids.

Thus he discoveredNewton’s law of friction by which the shear stress in the fluid and the shear rate areproportional, with the viscosity as the factor of proportionality. Fluids that satisfy this law302 Energyinteraction, as if it were a vacuum. And yet, the ether could transmitgravitational forces. Its rest frame was supposed to define absolute space.The luminiferous ether – also assumed to be at rest in absolute space –carried light and that created its own problem. Indeed, light is a transversalwave and was known to propagate with the speed c = 3·105 kms .

One had toassume that the ether transmitted vibrations as a wave, like an elastic body.For the speed of propagation to be as big as it was, the theory of elasticityrequired a nearly rigid body. Therefore physicists had to be thinking ofsomething like a rigid vacuum.

Asimov remarks in his customaryflamboyant style that generations of mathematicians … managed to coverthe general inconceivability of a rigid vacuum with a glistening layer offast-talking plausibility.53And then there was electricity and magnetism, both exerting forces oncharges, currents, and magnets and that seemed to call for two more typesof ether. Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879) were, it seems, not unaffected by such thoughts. Maxwell developedelaborate analogies between electro-magnetic phenomena and vortices inincompressible fluids moving through a medium. It is true that Maxwellalways emphasized that he was thinking of analogies – rather than reality –when he set up his equations in terms of convergences in the medium, andof vortices.

However, Maxwell’s visualizations were incidental andHeinrich Rudolf Hertz (1857–1894), recognizing the fact, is on record ashaving said laconically that the theory of Maxwell is the system of Maxwellequations, cf. Fig. 2.8. Kelvin was among those who would have preferredsomething more concrete: a clear relation to a mechanical model.Maxwell’s equations, cf. Fig. 2.8, relate four vector fields54B – magnetic flux density E – electric fieldD – dielectric displacement H – magnetic field.J is the electric current and q is the electric charge density.

With all thesefields, the Maxwell equations are strongly underdetermined. But then thereare two additional relations, the so-called ether relations, which close thesystem, if q and J are known. The ether relations connect D to E andH to B. They readD = İ0 EandH = µ0 B ,AsVswhere İ0 = 8.85·10-12 Vcmand µ0 = 12.5·10-7 Acmare constants called thevacuum di-electricity and the vacuum permeability, respectively.– and there are many of them – are called Newtonian to this day. However, Newton couldnot detect any viscous effect between the ether and the moon.53I.

Asimov: “The rigid vacuum” in ‘‘Asimov on physics” Avon Books, New York (1976).54 Vectors are denoted by boldface letters, or by their Cartesian components. If the latternotation is used in formulae, summation over repeated indices is implied.Electro-magnetic Energy31In the vacuum there is neither current nor charge but the fields are there,and they propagate as waves. Indeed, if we apply the curl-operator to thefirst and third Maxwell equation and make use of the ether relations, weobtain˜ 2 Ei˜t 2˜ 2 Eiε 0 µ 0 ˜x j ˜x j10 and˜ 2 Bi˜t 2˜ 2 Biε 0 µ 0 ˜x j ˜x j10which are the well-known wave equations of mathematical physics.

Thespeed of propagation is H1P which happens to be equal to c, the speed of0 0light. (!!)Thus Maxwell was able to relate electro-magnetic wave propagation tolight. He says: The speed of the transversal waves in our hypotheticalmedium … is so exactly equal to the speed of light … that it is difficult torefuse the conclusion that light consists of the wave motion of the mediumthat is also the agent of electric and magnetic phenomena.55wBi curli Ewt0wBiwx i0wDi curli HwtJiwDiwx iqFig.

2.8. James Clerk Maxwell. Main system of Maxwell equationsAs a result, the magnetic and electric ether were cancelled out. Whatremained was the luminiferous ether – the rigid vacuum – and, perhaps,Newton’s ether that transmits gravitation. Actually Einstein threw out theluminiferous ether in 1905 as we shall see later, cf. Chap. 7. The gravitational ether is still an embarrassment to physicists today. Nobody believesthat it exists, but neither have gravitational waves convincingly been55Retranslated by myself from Giulio Peruzzi: ‘‘Maxwell, der Begründer derElektrodynamik” [Maxwell. The founder of electrodynamics] Spektrum derWissenschaften, German edition of Scientific American. Biografie 2 (2000).322 Energydiscovered – to the best of my knowledge – nor the particles that couldreplace them, the hypothetical gravitons.56This is all quite interesting but it distract us from the main subject in thischapter, which is energy or, here, electro-magnetic energy.

The Maxwellequations of Fig. 2.8, combined with the ether relations, imply – as acorollary – four equations which may be interpreted as equations of balanceof electro-magnetic momentum and energy, viz.w ( D u B) l w (( 12 E ˜ D 12 B ˜ H )G li Ei Dl Bi H l )wtwx iw ( 12 E ˜ D 12 B ˜ H ) w ( E u H ) iwx iwt qEl ( J u B) l J i Ei .In this interpretation we have( D – H )l momentum density( 12 E ¹ D 12 B ¹ H )įli Ei Dl Bi Hl pressure tensor12E ¹ D 12 B ¹ H energy density( E – H )i energy flux .The right-hand sides of the equations of balance represent – to withinsign – the density of the Lorentz force of an electro-magnetic fields oncharges and currents and the power density of the Lorentz force on a currentrespectively. If the current consists of a single moving charge e, the Lorentzforce becomes e( E ddxt u B) and the power equals e ddxt ˜ E .The trace of the pressure tensor is 3p, where p is the electro-magneticpressure.

Hence inspection of the balance equations shows that we haveelectro-magnetic pressure = 1/3 electro-magnetic energy density.This relation was to become important in Boltzmann’s investigation of radiationphenomena, cf. Chap. 7.That the Lorentz force on charged matter and its power should appear inan easily derived corollary – of balance type – of the Maxwell equationsplaces electro-magnetic energy firmly among the multifarious incarnationsof energy which altogether are conserved. Maxwell says: When I speak ofthe energy of the field, I wish to be understood literally. All energy isidentical to mechanical energy, irrespective of whether it appears in theform of motion or as elasticity or any other form.56You can still always make a learned physicist, who is happily expounding the properties ofblack holes, come to a full stop by asking a simple question.

Nothing can escape from ablack hole, not even light, which is why it is black. So, you must ask innocently: But thegravitons do come out, don´t they?Electro-magnetic Energy33Maxwell’s theory of electro-magnetism was created in three papers57between 1856 and 1865 and later summarized and extended in two books,58the latter of which appeared posthumously.The practical impact of Faraday and Maxwell was enormous, althoughnot immediate, and it was twofold: Telecommunication and energy transmission. It is true that electro-magnetic telecommunication by wirepreceded Maxwell’s work.

But, of course, wireless transmission was firmlybased on it after Hertz sent the first radio-signal – short for radiotelegraphic signal – from one side of his laboratory to the other one in 1888.Perhaps even more important is the electric generator which was inventedby Faraday in 1831 when he rotated a copper disk in a magnetic field, thusinducing a continuous electric current. The reversal of the process couldproduce – with the appropriate design – rotational motion of a shaft fromthe current fed into an electric motor.Generator and electric motor would eventually make it feasible toconcentrate steam power generation in some central plant in a city or thecountryside, rather than have each consumer set up his own steam engine.But that took time and the electrification of industry and transport – andhouseholds – was not complete until well into the 20th century.Faraday, however, was fully aware of the potential of his invention.There is a story about this, probably apocryphal: In 1844, when Faradaywas presented to Queen Victoria, she is supposed to have asked him whatone might do with his inventions.