27. DE BROGLIE'S PERPETUAL MOTION
 
   With the explanation of the photo-electrical effect the idea that light is dualistic in nature, particle (photons) and waves (electromagnetic wave) has become generally accepted. Also, from the relation and equivalence of mass and energy it results that every mass is accompanied by energy and energy is accompanied by mass . Consequently, every photon of energy has the mass
(27.1)

and also the momentum
(27.2)

from which it results that the wavelength of the photon is
(27.3)

where is the Planck constant.
   Therefore a beam of light possesses the momentum , which also results from electrodynamics. The existence of this momentum is considered to be proof that light has a particle nature as well as a wave nature. In contrast to this it is, ordinary particles and bodies in general are considered to posses an exclusively particle nature.
   Encouraged by the dual nature of light, de Broglie, in his doctoral thesis [L. de Broglie, Dissertation, Paris, 1924.; L. de Broglie, Phil. Mag., 47, 446, 1924.] of 1923, put forward the bold hypothesis that all particles have a wave nature as well. According to him, matter itself is dualistic in nature, not just light.
   De Broglie asserts that every particle of mass , moving at speed , is accompanied by a wave of wavelength
(27.4)

   Davisson and Germer allegedly confirmed the existence of de Broglie's wavelength experimentally in 1927, and discovered the diffraction of electrons. [C.J. Davisson and L.H. Germer, Nature 119, 558, 1927.]
   According to de Broglie's hypothesis, particles do not possess a wave nature when at rest. In order for the wave to accompany the particle, the particle must be set in motion. However, in order to set a particle in motion some energy must be expended. In order to set an electron in motion an acceleration voltage is used. In the electron microscope and X-ray devices, for example, the acceleration voltage is the anode voltage.
   The concept of the wave nature of the electron is employed in the electron microscope [23], [24]. The electron microscope is considered to be irrefutable proof that the electron has a wave nature. At this point X-ray devices, that existed before the electron microscope, are forgotten.
   In order to obtain de Broglie's wavelength of an electron we should find the momentum of the electron, which is dependent on the velocity, that is, on the kinetic energy of the electron.
   The equation for the kinetic energy of the electron, according to Eq. (23.38), in this case is given by
(27.5)

   From Eq. (27.5) we get that the mass of the electron accelerated by voltage is given by
(27.6)

where is the electrical charge of the electron.
   The relationship between mass and is given by Lorentz's Eq. (23.4) for transversal mass
(27.7)

Using Eqs. (27.6) and (27.7) we find that the momentum of the electron is given by
(27.8)

   So, de Broglie's wavelength is given by
(27.9)

   In case of non-relativistic speeds of electrons, when , de Broglie's wavelength is
(27.10)

   The wavelengths calculated according to Eq. (27.9) are a little different from the wavelengths calculated according to Eq. (27.10). For example, if the accelerating voltage is 60000 V, then the wavelength calculated by use of Eq. (27.9) is = 0.04866·10-10 m and calculated by use of Eq. (27.10) is = 0.05101·10-10 m.
   When we know de Broglie's wavelength we are able, using Planck's equation for the energy of the wave , to calculate the energy contained in the wave of that wavelength, and then to compare it with the energy expended to generate that wave. Such information can be used to estimate the correctness of de Broglie's hypothesis. Of course, this is correct only if de Broglie's wave has electromagnetic nature.
   For example, the energy of de Broglie's wave of wavelength = 0.04866·10-10 m is

and the energy spent, in order to generate that wave is

   Comparing these two results we find that the energy of the wave is 4.247 times greater that the energy expended to generate it The alleged gain in energy, that is the gain coefficient, decreases with increases in the anode voltage. For example, at an anode voltage of 100 V the gain coefficient is 101.1. Such a result is surprising, and runs counter to the law on the conservation of energy. If de Broglie were right, and if an electromagnetic wave were involved, this would mean that the long awaited secret of perpetual motion had been uncovered. Unfortunately, this is impossible. De Broglie's hypothesis is, in fact, the result of a tendency to find symmetry in nature even where it does not exist.
   We have seen above how the electron in motion generates an electromagnetic field, with which it joins and from which it breaks away on the decrease of the velocity of motion. The relation between the mass of an electron and the field generated appears between the mass of the electron and the energy of the generated field as well. The faster the electron moves, the greater the energy of the field generated and the greater the mass of the electron. As a result, the increase in mass of the electron in motion, as we saw before, must be called electromagnetic mass.
   The phenomenon of the generation of an electromagnetic field by the motion of an electron is as well known as X-ray radiation. However, the wavelength of X-ray radiation is considerably larger than de Broglie's wavelength at the same acceleration voltage and corresponds to the energy spent in its generation. Therefore, some kind of dualism of electrified particles in motion really does exist, even without de Broglie's wave. That kind of dualism, in distinction from de Broglie's, has sound basis in proved fact. De Broglie starts from the symmetry in which it is understood that light has a particle nature, and as a result it brings that assertion into doubt as well.
   Neutral particles in motion do not generate an electromagnetic field. As a result they do not behave as if they were waves, that is they are not accompanied by a wave as is an electron, or some other electrified particle, in motion.
 
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