12. The Electron theory of Lorentz

   "It is the theory of H.A. Lorentz (proposed in
1892) that signi-fied the climax and the final step of the physics of the material ether.
    It is a one-fluid theory of electricity that had been developed atomistically, and it is this feature which, as we shall presently see, determines the part allocated to the ether.
    The fact that electric charges have an atomic structure, that is,  

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occur in very small indivisible quantities, was first stated by Helm-holtz (in 1881) in order to make intelligible Faradays laws of eloctro-lysis (p.157)  Actually, it was only necessary to assume that every atom in an electrolytic solution enters into a sort of chemical bond with an atom of electricity or an electron in order to make intelligible the fact that a definite amount of electricity always separates out equivalent amounts of substances.
    The atomic structure of electricity proved of particular value for explaining the phenomena which are observed in the passage of the electric current through a rarefied gas. Here it was first discovered that positive and negative electricity behave quite differently. If two metal electrodes are introduced into a glass tube and if a current is made to pass between them
(Fig. 104). Very complicated pheno-mena are produced so long as gas is still present at an appreciable
Fig. 104   A tube to produce cathode rays.   K  cathode, A  anode.
Pressure in the tube. But if the gas is pumped out more and more, the phenomena becomes increasingly simple. When the vacuum is very high, the negative electrode, the cathode K, emits rays which pass through a hole in the positive pole, the anode A, and are observed behind   A   through fluorescence produced on a screen (as known from any television set). These rays are called cathode rays. It was shown that they could be deflected by a magnet in the manner of a stream of negative electricity. The greatest share in investigating the nature of cathode rays was taken by Sir J. J. Thomson and P.L. Lenard. The negative charge of the rays could also be directly demonstrated by collecting it in a hollow conductor. Furthermore, the rays are deflected by an electric field applied per  

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pendicular to their path, and this deflection is opposite to the direc-tion of the field, which again proves the charge to be negative.
     The conviction that the nature of cathode rays is corpuscular became a certainty when physicists succeeded in deducing quantita-tive conclusions concerning their velocity and their charge.
     If we picture the cathode ray as a stream of small particles of mass m
el, then clearly it will be the less deflected by a definite elec-tric or magnetic field the greater its velocity - just as the trajectory of a rifle bullet is straighter the greater its velocity. Now it is possible to produce cathode rays that can be bly deflected - that is, slow cathode rays - by using a small difference of potential between cathode and anode. If one chooses a b difference of potential between the two poles, the rays are bly accelerated by the field from K to A, which may be calculated from the fundamental equation of mechanics    
el b = K = e E,
where e is the charge and E is the field strength. We are here clearly dealing with a case analogous to that of "falling" bodies, in which the acceleration is not equal to that of gravity g but to e  E.  If the ratio e were known, the velocity v could be found                                                                                     m
el                  m el

from the laws for falling bodies. But there are two unknowns, e and v, and hence another measurement is necessary if they                                                                                      m

are to be determined. This is obtained by applying a lateral magnetic force. In discussing Hertz's theory (V,11, 1b, p. 194)
we saw that a magnetic field H sets up in a body moving perpendicular to H an electrical field E= v H,
which is perpendicular both to H and to v. Hence a deflecting force eE = e v H  will act on every cathode ray particle so that

there will be an acceleration b = e    v H perpendicular to the original motion. This may be found by measuring the lateral
                                     Mel  c
deflection of the ray.  

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Hence we have a second equation for determining the two unknowns  e and  v.
   The determinations carried out by this or a similar method have led to the result that, for velocities that are not two great   e
el and v. has a definite constant value:              e  = 5.31 x 10 17  electrostatic units per gm.
      (66)                mel
On the other hand, in dealing with electrolysis (V, 2, formula (48),p.159), we stated that hydrogen carries an amount of electricity Co = 2.90 x 10
14  electrostatic units per gm. If we now make the readily suggested assumption that the charge of a particle is in each case the same, namely an atom of electricity or an electron, we must conclude that the mass of the cathode ray particle mel must bear the following ratio to that of the hydrogen atom  mH:
el   =   e   :   e   =  2.90 x 10 14   =    1  .
H      mH   mel      5.31 x 10 17      1830
Thus, the cathode ray particles are nearly 2000 times lighter than hydrogen atoms, which are the lightest of all chemical atoms. This leads us to conclude that cathode rays are a current of pure atoms of electricity.
   This view has stood the test of innumerable researches Negative electricity consists of freely moving electrons, but positive electricity is bound to matter and never occurs without it. Thus recent experi-mental researches have confirmed and given a precise form to the old hypothesis of the one-fluid theory. The amount of the charge e of the individual electron has also been successfully determined. The first experiments of this type were carried out by Sir J. J. Thomson (
1898). The underlying idea is : Little drops of oil or water, or tiny spheres of metal of microscopic or submicroscopic dimensions, which are produced by condensation of a vapour or by spraying of a liquid in air, fall with constant velocity, since the fric-tion of the air prevents acceleration. By measuring the rate of fall the size of the particle can be determined, and then their mass M is  

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obtained by multiplying their size by the density. The weight of such a particle is then Mg, where g = 981 cm./sec.2 is the acceleration due to gravity. Now such particles may be charged electrically by sub-jecting the air to the action of x-rays or the rays of radioactive sub-stances. If an electric field E, which is directed vertically upwards, is then applied, a sphere carrying the positive charge e is pulled upwards by it, and if the electric force  eE is equal to the weight Mg,
The sphere will remain poised in the air. The charge e may then be calculated from the equation  eE = Mg. Millikan (
Who carried out the most accurate experiments of this sort, found that the charge of the small drops is always an exact multiple of a definite minimum charge. Thus we shall call this the elementary electrical quantum Its value is
e = 4.77 x 10-
10 electrostatic units. (67)
    The absolute value of the elementary charge plays no essential part in Lorentz's theory of electrons. We shall now describe the physical world as suggested by Lorentz.
     The material atoms are the carriers of positive electricity, which is indissolubly connected with them. In addition they also contain a number of negative electrons, so that they appear to be electrically neutral with respect to their surroundings.
In nonconductors the electrons are tightly bound to the atoms; they may only be displaced slightly out of the positions of equilibrium so that the atom becomes a dipole. In electrolytes and conducting gases it may occur that an atom has one or more electrons too many or two few; it is then called an ion or a carrier, and it wanders in the electric field carrying elec-tricity and matter simultaneously. In metals the electrons move about freely and experience resistance only when they collide with the atoms of the substance. Magnetism comes about when the electrons in certain atoms move in closed orbits and hence represent Amperian molecular currents.
     The electrons and the positive atomic charges swim about in the sea of ether in which an electromagnetic field exists in accordance with Maxwell's equations. But we must set
* = 1, u = 1 in them, and, in place of the density of the conduction current, we have the convec-tion current pv of the electrons. The equations thus become  

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div E = 4
pi p;         curl H - 1   E  = 4 p- ,          ]                                                                             (68)
pi         c             ]
                                  div H = 0;                curl e + 1   H  =                    
pi                        ]
and include the laws of Coulomb, Biot and Savert, and Faraday in the usual way.
     Thus all electromagnetic events consist fundamentally of the motions of electrons and of the fields accompanying them.
All matter is an electrical phenomenon. The various properties of matter depend on the various possibilities of motion of the electrons with respect to atoms, in the manner just described. The problem of the theory of electrons is to derive the ordinary equations of Maxwell from the fundamental laws (68) for the individual, invisible electrons and atoms, that is, to show that material bodies appear to have, according to their nature, respectively, a conductivity
o, a dielectric constant *, and a permeability u.
     Lorentz has solved this problem and has shown that the theory of electrons not only gives Maxwell's laws in the simplest case, but, more than this, also explains numerous facts which were inexplicable for the descriptive theory or could be accounted for only with the aid of artificial hypotheses. These facts comprise, above all, the more refined phenomena of optics, color dispersion, the magnetic rotation of the plane of polarization (p. 184) discovered by Faraday, and similar interactions between light waves and electric or magnetic fields. We shall not enter further into this extensive and mathe-matically complicated theory, but shall restrict ourselves to the ques-tion which is of primary interest to us : What part does the ether play in this concept of matter ?      Lorentz proclaimed the very radical thesis which had never before been asserted with such definiteness:
     The ether is at rest in absolute space
     In principle this identifies the ether with absolute space. Absolute space is no vacuum, but something with definite properties whose state is described with the help of two directed quantities, the elec-trical field E and the magnetic field H, and, as such, it is called the ether.
      This assumption goes still farther than the theory of Fresnel. In  

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the latter the ether of astronomic space was at rest in a special inertial system, which we might regard as being absolute rest. But the ether inside material bodies is partly carried along by them.
Lorentz dispenses with even this partial convection and arrives at practically the same result. To see this, we consider the phenomenon that occurs in a dielectric between the plates of a condenser. When the latter is charged, a field perpendicular to the plate arises (Fig.
89), and this displaces the electrons in the atoms of the dielectric sub-stance and transforms them into dipoles, as we explained earlier (pp. 197 and 198). The dielectric displacement in Maxwell's sense *E, but only a part of it is due to the actual displacement of the
Fig. 105  A light ray with its electric vibration E and its magnetic vibration H travelling in an insulator with velocity c1. The insulator moves with velocity v.
Electrons. For a vacuum has the dielectric constant
* = 1, and hence the displacement E; consequently the true value of the electronic displacement is *E - E = (* - 1) E. Now we have seen that the experiments of Rontgen and Wilson on the phenomenona in moving insulators affirm that actually only this part of the displacement takes part in the motion. Thus Lorentz's theory gives a correct account of electromagnetic facts without having recourse to hypo-theses about the ether carried along by matter.
     The fact that the convection of light comes out in exact agreement with Fresnel's formula (44), (p.134) is made plausible by the following argument:  

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As in Wilson's experiment, we consider a dielectric body which moves in the x-direction with the velocity v and in which a light ray travels in the same direction (Fig. 105). Let this ray consist of an electrical vibration E parallel to the y-axis and a magnetic vibration parrallel to the z-axis. Now we know from Wilson's experiment that such a magnetic field in the moving body produces a corres-ponding displacement of the value (*- 1) v  H in the  y-direction                                                                                                                                                 c
From this we get a superposed electrical field if we divide by *. Thus the total electrical field is      E + *
- 1  v  H.
                                                                                                                                                                *    c
     If the convection were complete, as is assumed in Hertz's theory. We should have only * in place of *
- 1, thus the total field would have the value   E +  v - H.  We see that in our formula v is replaced by       * - 1 v.
            c                                                                                         *          
Therefore, this value should correspond to the absolute velocity of the ether within matter according to Fresnel's theory, that is, to the convection coefficient called   in optics (compare formula (44)). This is precisely the case, for, according to Maxwell's electromagnetic theory (p.188), the dielectric constant is equal to the square of the index of refraction n,
i.e., * = n2. If we insert this value, we get                                                
                                                                         * -  1 v = n2
- 1 v = ( 1 -    1  ) v =
                                                                            *          n2                         n2
in agreement with formula (44).
     We recall that Fresnel's theory encountered difficulties through color dispersion; for if the refractive index  n depends
On the fre-quency (color) of the light, so also will the convection coefficient  . But the ether can be carried along in only one definite way, not differently for each color. This difficulty vanishes for the theory of electrons, since the ether remains at rest and it is the electrons situ-ated in the matter that are carried along: color dispersion is due to their being forced into vibration by light and reacting, in turn, on the velocity of light.  

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We cannot enter further into the details of this theory and its many ramifications, but we shall summarize the results as follows: Lorentz's theory presupposes the existence of an ether that is absolutely at rest. It then proves that, in spite of this, all electromagnetic and optical phenomena depend only on the relative motions of translation of material bodies so far as terms of the first order in come into account. Hence it accounts for all known phenomena, above all for the fact that the absolute motion of the earth through the ether cannot be demonstrated by experiments on the earth involving only quantities of the first order (this is the optica, or rather the electro-magnetic principle of relativity).      There is, however, one experiment of the first order which can no more be explained by Lorentz's theory than by any of the other theories previously discussed: this would be a failure of the experiment to find an absolute motion of the whole solar system with Romer's method (see pp. 91 and 129).       The deciding point for Lorentz's theory is whether it stands the test of experiments that allow quantities of the second order in to be measured. For they should make it possible to establish the absolute motion of the earth through the ether. Before we enter into this question, we have yet to discuss an achievement of Lorentz's theory of electrons through which its range became greatly extended, namely, the electrodynamic interpretation of inertia."
For pages
91 and 129 see original whatsit.
13. Electromagnetic Mass
   " The reader will have remarked that from the moment when we left the elasic ether and turned our attention to the electrodynamic ether, we havehad little to say about mechanics. Mechanical and electrodynamic phenomena each form a realm for themselves. The former take place in absolute Newtonian space, which is defined by the law of inertia and which betrays its existence through centrifugal forces; the latter are states of the ether which is at rest in absolute space. Acomprehensive theory, such as Lorentz's aims at being, cannot allow these two realms to exist side by side unassociated. We have seen that physicists in spite of incredible effort and ingenuity were not able to reduce electrodynamics to terms of  

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mechanics. The converse then boldly suggests itself: can mechanics be reduced to terms of electrodynamics ?
     If this could be carried out successfully the absolute abstract space of Newton would be transformed into the concrete ether. The iner-tial resistances and centrifugal forces would appear as physical actions of the ether, say, as electromagnetic fields of particular form, but the principle of relativity of mechanics would lose its strict validity and would be true, like that of electrodynamics, only approxi-mately, for quantities of the first order in   = v  
Science has not hesitated to take this step, which entirely reverses the order of rank of concepts. And, although the doctrine of an
Fig. 106  A circuit with condenser K, coil S, and spark gap F used to demon-strate the oscillations of electricity.
Ether absolutely at rest later had to be dropped, this revolution, which ejected mechanics from its throne and raised electrodynamics to sovereign power in physics was not in vain Its results have retained their validity in a somewhat altered form.
   We saw already (p. 185) that the propagation of electromagnetic waves comes about through the mutual action of electrical and magnetic fields producing an effect analogous to that of  mechanical inertia. An electromagnetic field has a power of persistence quite similar to that of matter. To generate the field, work must be per-formed, and when it is destroyed, this work again appears.  This is  

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observed in all phenomena that are connected with electromagnetic vibrations - for example, in the various forms of wireless trans-mitters.  An old-fashioned wireless transmitter of the Marconi type contains an electric oscillator, consisting essentially (Fig. 106) of a spark gap F, a coil S, and a condenser K (two metal plates that are
Seperated from each other) connected by wires to form an "open" circuit. The condenser is charged until a spark jumps across the gap at F. This causes the condenser to become discharged and the quantities of electricity that have been stored flow away. They do not simply neutralize each other, but shoot beyond the state of equilibrium and again become collected on the condenser plates, but with reversed signs, just as a pendulum swings past the position of equilibrium to the opposite side. When the condenser has thus
Fig.107 The   electric   field around a charge at rest.   Fig. 108  The electric field around a charge is supplemented by a                                                                                                                           magnetic field when it is moved.
Been charged up afresh, the electricity again flows back causing another spark to jump the gap, and thus the system oscillates to and fro until its energy has been used up in warming the conducting wires or in being passed on to other parts of the apparatus, for example, the emitting antenna. Thus the oscillation of the electricity proves the inertial property of the field, which exactly corresponds to the inertial property of the field, which exactly corresponds to the inertia of mass of the pendulum bob. Maxwell's theory represents this fact correctly in all details. The electromagnetic vibrations that  

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occur in a definite apparatus can be predicted by calculation from the equations of the field.
     This led J.J.Thompson to infer that the inertia of a body must be increased by an electric charge which is imparted to it. Let us con-sider a charged sphere first at rest and then moving with the velocity v. The stationary sphere has an electrostatic field with lines of force directed radially outwards (Fig.
107); the moving sphere has, in addition, a magnetic field with circular lines and forces that encircles the path of the sphere  (Fig. 108);  the moving sphere has, in addition, a magnetic field with circular and forces that encircle the path of the sphere  (Fig. 108).  For a moving charge is a convec-tion current (combined with a displacement current) and produces a magnetic field in accordance with Biot and Savert's law. Both states have the inertial property above described. The one can be transformed into the other only by the application of work. The force that is necessary to set the stationary sphere into motion is thus greater for the charged than for the uncharged sphere.
To accelerate the moving charged sphere still further, the magnetic field H must be clearly be strengthened. Thus, again, an increase of force is necessary.
    We remember that a force K that acts for a short time
t represents an impulse J = Kt  which produces a change of velocity    
w in a mass m in accordance with the formula (
7) (11. 9, p. 34):                                                                                                                     mw = J
If the mass carries a charge, a definite impulse J will produce a smaller change of velocity and the remainder J' will be used to change the magnetic field. Thus we have  
                                                                                                                    mw = J - J'.
Now, calculation gives the rather obvious result that the impulse J' necessary to increase the magnetic field is greater, the greater the change of velocity w, and, indeed, it is approximately proportional to this change of velocity. Thus we may set
J' = m'w, where m' is a factor of proportionality which, moreover, may depend on the state, that is, the velocity v, before the change of velocity occurs. We then have          mw = J - m' w
                     (m + m' ) w = J.

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Thus, it is as if the mass m were augmented by an amount m' , which is to be calculated from the electromagnetic field equations and which may be depende on the velocityv. The exact value of m' for any velocity v may be calculated only if assumptions are made about the distribution of the electric charge over the moving body. But the limiting value for velocities that are small compared with that of light c, that is, for small values of  , is obtained independently of such assumptions as                               m' = 4   S ,                                                 (69)                                        3   c2
where S is the electrostatic energy of the charges on the body.
     We have seen that the massof the electron is about 2000 times smaller than that of the hydrogen atom. Hence the idea occurs that the electron has, perhaps, no "ordinary" mass at all, but is nothing other than an "atom of electricity," and that its mass is entirely electromagnetic in origin. Is such an assumption reconcil-able with the knowledge that we have of the size, charge, and mass of the electron?
     Since the electrons are to be the structural elements of atoms, they must at any rate be small compared with the size of atoms. Now we know from atomic physics that the radius of  atoms is of the order 10-
8 cm. Thus the radius of the electron must be smaller than 10-8 cm. If we imagine the electron as a sphere of radius a with charge e distributed over its surface, then, as may be derived from Coulomb' law, the electrostatic energy is   S = 1  e2. Hence, by (69),                                                                                                                                                  2  a
the electromagnetic mass becomes                        
el = 4  S  =  2  e2 .
                                                                                   3  c2    3  ac 2
From this we can calculate the radius a:
                                                                                       a = 2  e      e
                                                                                             3  c2  mel
On the right-hand side we know all the quantities,  e  from the deflection of the cathode rays (formula (66), p. 202), e from Milikan's  

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measurements (formula (67), p. 203, and c the velocity of light. If we insert the values given, we get

a = 1.88 x 10-13 cm.

A length which is about 100,000 times smaller than the radius of an atom.
     Thus the hypothesis that the mass of the electron is electromagnetic in origin does not conflict with the known facts. But this does not prove the hypothesis.
     At this stage the theory found b support in refined observa-tions of cathode rays and of the -rays of radioactive substances, which are also ejected electrons. We explained above how electric and magnetic action on these rays allows us to determine the ratio of the charge to mass,   e  , and also their velocity v, and that at first a definite valuefor   e   was obtained,
el                                                                                                  mel
which was independent of   v.  But, on proceeding to higher velocities, a decrease of  e  was found.  This effect was particularly clear and could be measured quantitatively in the case of the -rays of radium, which are only slightly slower than light.  The assumption that an electric charge should depend on the velocity is incompatible with the ideas of the electron theory. But that the mass should depend on the velocity was certainly to be expected if the mass was to be electromagnetic in origin. To arrive at a quantitative theory, it is true, definite assumptions had to be made about the form of the electron and the distribution of the charge on it. M.
Abraham (1903) regarded the electron as a rigid sphere, with a charge distributed on the one hand, uniformly over the interior, or, on the other, over the surface,and he showed that both assumptions lead to the same dependence of the electromagnetic mass on the velocity, namely, to an increase of mass with increasing velocity. The faster the electron travels, the more the electromagnetic field resists a further increase of velocity. The increase of  mel  explains the observed decrease of  e  , and Abraham's theory agrees quantitatively very well with the                                                                           mel

results of measurement  

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of Kaufmann (1901) if it is assumed that there is no "ordinary" mass present.
    Thus, the object of tracing the inertia of electrons back to electro-magnetic fields in the ether was attained. At the same time a further perspective presented itself. Since atoms are the carriers of positive electricity, and also contain numerous electrons, perhaps their mass is also electromagnetic in origin? In that case, mass as the measure of the inertial persistence would no longer be a primary phenomenon, as it is in elementary mechanics, but a secondary consequence of the ether. Therefore, Newton's absolute space, which is defined only by the mechanical law of inertia, becomes super-fluous; its part is taken over by the ether whose electromagnetic properties are well known.
     We shall see (V, 15, P. 221) that new facts contradict this view. But the relationship between mass and electromagnetic energy, which was first discovered in this way, constitutes a fundamental discovery the deep significance of which was brought into prominence only when Einstein proposed his theory of relativity".
The Zed Aliz Zed made a nine from the anagram   EI
NSTEIN     "We have yet to add that, besides Abraham's theory of the rigid electron, other hypothesi were set up and worked out mathematically. The most important is that of Lorentz (1904) which is closely connected with the theory of relativity. Lorentz assumed that every moving electron contracts in the direction of motion, so that from a sphere it becomes a flattened    
spheroid of revolu-tion, the amount of flattening depending in a definite way on this velocity. This hypothesis seems at first sight strange. It certainly gives a simpler formula for the way electromagnetic mass depends on velocity than does Abraham's theory, but this in itself does not justify it. The actual confirmation came from the course Lorentz's theory of electrons took when it had to consider quantities of the second order in the discussion of experimental researches, to which we shall presently direct our attention. Lorentz's formula then turned out to have a universal significance in the theory of relativity. The experimental decision between it and Abraham's theory will be discussed later (VI, 7, p 278).     At the beginning of the new century, after the theory of electrons had reached the stage above described, the possibility of forming a uniform physical picture of the world seemed at hand - a picture  

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which would reduce all forms of energy, including mechanical inertia, to the same root, to the electromagnetic field in the ether. Only one form of energy - gravitation - seemed still to remain outside the system; yet it could be hoped that that, too would allow itself to be interpreted as an action of the ether.




    14. Michelson and Morley's Experiment

     Twenty years before this period, however, the base of the whole structure had already cracked and, while the building was going on above, the foundations needed repairing and strengthening.
     We have several times emphasized that any decisive experiment in regard to the theory of the stationary ether had to be precise enough to determine quantities of the second order in . Only then could it be ascertained whether or not a
fast- moving body is swept by an ether wind which blows away the light waves as is demanded by theory.
     Michelson and Morley (
1881) were the first sucecessfully to carry out the most important experiment of this type They used Michel-son's interferometer (IV, 4, p. 102) which they had refined to a precision instrument of unheard-of efficiency.
     In investigating the influence of the earth's motion on the velocity of light (IV,
9, P. 130), it has been found that the time taken by a ray of light to pass back and forth along a distance l parallel to the earth's motion differs only by a quantity of the second order from the value it has when the earth is at rest. We found earlier that this time was                                                                     t1= l (     1     +     1   ) =    21c  ,
               c + v       c - v      c2 - v2
for which we may also write
1 = 21      1  .  
                                                                                            c    1-
   If this time could be so accurately measured that the fraction   1    could be distinguished from 1 in spite of the extremely
                                                                                1- 2
small value of the quantity 2, we should have means of proving the exis-tence of an ether wind.  

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It is by no means possible, however, to measure the short time taken by a light ray to traverse a certain distance.  Interferometric methods give us rather only differences of the times taken by light to traverse different routes between two given points. But they give these with amazing accuracy.
     For this reason Michelson and Morly caused a second ray of light to traverse a path AB of the same length l backwards and forwards, but perpendicular to the earth's orbit (Fig, 109). While

Fig. 109  The path of the light in Michelson's experiment.

the light passes from Ato B, the earth moves a short distance forward so that the point B arrives at the point B' of the ether. Thus the true path of the light in the ether is AB', and if it takes a time t to cover this distance, then AB, = ct. During the same time A has moved on to the point A' with the velocity v, thus AA' = vt. If we now apply Pythagoras' theorem to the right-angled triangle  AA'B, we get  
2t2 =l 2 + v2t2
2 (c2- v2) =l2,       t2   =    l2        =    l2        1    ,                                                                                                  c2 - v2          c2     1- 2
                                                                                      t  =  l        1    .
                                                                                             c   /  1
- 2                                                              

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The light requires exactly the same time to make the return journey, for the earth shifts by the same amount, so that the initial point  A moves from  A' to A ".
      Thus the light takes the following time for the journey backwards and forwards:
                                                                                        t  =  2l        1    .
                                                                                                c    /  1
- 2
The difference between the times taken to cover the same distance parallel and perpendicular to the earth's motion is thus
1- t2  = 21  (      1   .  -           1     ).
                                                                                               c      1-
2           / 1- 2

Now, by neglecting terms of higher order than the second in   (similar to what was done on p. 126) we may approximate by replacing     1     by    1 +
2, and        1       by 1 +  2*  .
                                        /1 -
2                                    2
Hence we may write to a sufficient degree of approximation  t
1- t2  = 2l      [ ( 1 + 2) - ( 1 + 2 ) ] = 21   2  = 1  2 .
                                                                                                           c                               2           c    2      c
     The retardation of the one light wave compared with the other is thus a quantity of the second order.
     This retardation may be measured with the help of Michelson's interferometer (Fig 110).  In this the light coming from the source Q is divided at the half silvered-plate P into two rays which run in perpendicular directions to the mirrors S
1 and S2,                                                                                                                        
At which they are reflected and sent back to the plate  P. From P onwards they run parallel into the telescope F where they interfere. If the distances S
1P and S2P are equal and if one arm of the apparatus is placed in the direction of the earth's motion, one duplicates the    /                                                                                                                  
* To show the approximate validity of the equation       1     = 1 + 2    write it   1 = (1-  2) ( 1 + 2) - 4,
- 2
which is correct if  
4 is neglected. In the same way squaring        1      = 1 +  2 , one obtains      1     =  1 + 2 + 4;  if the                                                                                          / 1- 2                2                        1- 2                    4
last term is neglected one has the same formula as above.
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case just discussed. Thus the two rays reach the field of vision with a difference of time   l  2.  Hence the                                                                                                                                                                  c
interference fringes are not situated precisely where they would be if the earth were at rest. But if we now turn the apparatus through
90º until the other arm is paral-lel to the direction of the earth's motion, the interference fringes will be displaced by the same amount but in the opposite direction.

Fig. 110   Michelsons's interfero-meter.

Fig.  111   Two waves of the same wave length , one shifted against the other  by 2  l2.  
Hence if we observe the position of the interference fringes while the apparatus is being rotated, a displacement should be measured which corresponds to the double retardation 2  1
                                                                                2.                                                                                             c
    If T is the period of vibration of the light used, the ratio of the retardation to the period is 2l  
2 and

since by formula (35)  (p.99) the wave length = Ct, we may write this ratio as 2  1  2.
     Hence, when the apparatus is rotated, the two interfering trains of waves experience a relative displacement whose ratio to the wave length is given by   2l
2  (Fig. 111). The interference fringes them-selves arise because the rays which leave the                                                  
source in different  

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directions have to traverse somewhat different paths. The distance between two fringes corresponds to a path difference of one wave length, hence the observable displacement of the fringes is the fraction   2l 2 of the width of the fringe.                                                                                                                                                  
    Now Michelson, in a repetition of his experiment with Morley (1887) carried out on a larger scale, extended the length of the path traversed by the light by means of several reflections forward and back to 11m. = 1.1 x 10  
3 cm. The wave length of the light used was about = 5.9 x 10-5 cm. We know that   is approximately equal to 10 - 4, and hence 2 = 10 - 8.
So we get
2  = 2 x 1.1 x 103 x 10 - 8  = 0.37,                                                                                                 5.9 x 10 -5
that is, the interference fringes must be displaced by more than one-third of their distance apart when the apparatus is turned through
90º. Michelson was certain that the one-hundredth part of this displacement would still be observable.
      When the experiment was carried out, however, not the slightest sign of the expected displacement manifested itself, and later repe-titions with still more refined means led to no other result. From this we must conclude that the ether wind does not exist. The velocity of light is not influenced by the motion of the earth even to the extent involving quantities of the second order.