Lecture 1
Planetary systems and the Bohr atom model.
Central and centrifugal forces in astronomy
Newton force plus centrifugal force analogous to that of Lorentz
Bohr planetary model of the hydrogen atom
Conclusion
Bibliography
It is shown in each of the following three cases: 1) Earth revolving around the Sun, 2) Moon revolving around the Earth and 3) electron revolving around the hydrogen nucleus; that the corresponding attraction force (gravitational or electric) is balanced by the centrifugal mass force. In astronomy such a balance results in an approximated version of the third Kepler law and together, with the requirement of angular momentum conservation, results in the second Kepler law. An analogy between the centrifugal mass force and the magnetic component of the Lorentz force is demonstrated. It is shown that the general formula for the electric intensity field generated by a moving electron, in the particular case of an electron revolving around the nucleus, gives no electromagnetic radiation when the orbit is stable, and there is non zero radiation when the electron jumps from the higher orbit to the lower one. As the result it is claimed that Bohr planetary model of hydrogen atom is fully consistent with the electromagnetic theory.
Central and centrifugal forces in astronomy
Planetary orbits have thus far not been determined based on the condition that gravitational attraction force has to be balanced by the centrifugal force. They are calculated using the experimental Kepler laws. The centrifugal mass force is often treated as the apparent one (Feynman, Vol. I, part 1, sec.12-5). And, in the published text of D.L. & I. R. Goldstein, especially devoted to the problem of planetary orbits., it is not considered. Knowing the gravitational constant, orbital radius, angular velocity of revolution and corresponding masses, one can easily calculate gravitational force and the centrifugal force caused by revolution around a nucleus. Absolute values of these forces are the same as can be seen in examples A and B.Example A. Earth revolving around the Sun. Gravitational attraction of Sun balanced by centrifugal force affecting the orbiting Earth.
(1) 
Earth mass 
Earth angular velocity 
mean orbital radius 
gravitational constant 
Sun mass Example B. Moon revolving around the Earth. Gravitational attraction of Earth balanced by centrifugal force affecting the orbiting Moon.
(2) 
Moon mass 
Moon angular velocity 
mean orbital radius 
gravitational constant 
Earth mass Equations (1) and (2) resemble the third Kepler law. Assuming that the angular momentum of the revolving body is constant
(3)and combining equation (1) or equation (2) with equation (3) we obtain the equation equivalent to the second Kepler law.
(4) In a particular case when the angular velocity and the orbit radius are a constant, the equation (4) means a circular orbit. In the general case, the second Kepler law means an ellipse (the first Kepler law), which is a generalisation of a circle.
Newton force plus centrifugal force analogous to that of Lorentz
Knowing the expression for the Lorentz force, Newton force and centrifugal force, one can see an analogy between the sum of the Newton force
and centrifugal force 
on the one hand and the Lorentz force 
on the other hand.
(5)
(6)Feynman (Vol. II, part 1,sec. 13-1, Polish translation) writes: "Force affecting an electric charge not only depends upon its location but also upon its velocity. Every point in space is characterised by two vectors, which determine force acting on any charge. For the first there is an electric force (here
), which constitutes a component independent from charge velocity. For the second there is an additional force called the magnetic one (here 
), which does depend upon charge velocity". In the case of a revolving charge, the vector product of the charge velocity v and the magnetic induction vector B (caused by that velocity) also must be oriented along the orbit radius (see Fig.2),
Fig.1 Revolving mass generates vector
, which results in centrifugal force 
Fig.2 Revolving charge generates magnetic induction B, which results in centrifugal magnetic force
.If for a comparison sake we assume that kg/As = 1, then a unit of B [kg/As²] would be the same as a unit of angular velocity
[1/s], and a unit of E [kgm/As³] would be the same as a unit of the acceleration a [m/s²]. In the case where the vector of magnetic induction is vertical, the corresponding vector of angular velocity is vertical also.Problem of proper sign and the possible coefficient 2 for the angular velocity will be discussed in the next lecture.
Bohr planetary model of the hydrogen atom
Postulated here analogy between a revolving mass object and a revolving electric charge (open for a discussion) suggests that while considering electrons revolving around the nucleus we have to take into account two central forces (Newton's and Coulomb's) and the two centrifugal forces (that generated by an orbiting mass and the magnetic force generated by revolving charge).Knowing gravitational constant, mass of proton and having other data shown in example C below, one can calculate two mass forces and two forces acting on electron charge. The gravitational force and the magnetic centrifugal force are much weaker than the two others. One can easily check that the absolute values of the Coulomb central force and the mass centrifugal force in example C are the same. The corresponding balance equation can be found in Greiner (example 21.2, page 443).
Although verified by many experiments, the Bohr model has been rejected because of theoretical reasons stating that the revolving electron must radiate an electromagnetic energy and as a result must be drawn into the nucleus. Let us check it. According to Feynman (Vol. II, part 1, sec.21-1,) it has been accepted that the Coulomb law is not general enough and in the case of a moving charge has to be changed. The proper expression for the electric field E in the case of a moving electron should read
(7)where:
q - charge of an orbiting electron,
R - distance between the electron and the nucleus in time t,
R' - distance between the electron and the nucleus in the retarded time t',
- unit vector in direction R' In the case of an electron revolving around the nucleus, the nucleus is considered as fixed and - as in example C - only its electric field is of interest.
Example C. Electron revolving around the nucleus. The Coulomb attraction balanced by centrifugal mass force. The magnetic component of the Lorentz force is much smaller than the centrifugal (mass) force.
(8)
(9)
electron mass 
angular electron velocity 
mean orbital radius
electron charge 
dielectric constant 
magnetic induction Would there exist a "moon of an electron", the electric field of an electron would be interesting also. As this is not the case we are not especially interested in the electric field of the revolving electron given by the two first terms in the equation (7). It is the third term, responsible for radiation of electromagnetic energy from a revolving electron, which we are interested in.
However in the case of an electron revolving around the nucleus with constant angular velocity, the second time derivative of the unit vector "e" is zero (unit vector's direction changes linearly with time). So while the angular velocity is constant no radiation should be expected.
As a matter of fact, only then, when the angular velocity (and the orbit) of a revolving electron does change (electron jumping on lower orbit) there is a corresponding radiation of electron energy (photon emission). When the energy is increasing (electron jumping on higher orbit) there is a photon absorption.
Thesis of the lecture are as follows:
| 1 | Balance of gravitational force, on the one hand, and the centrifugal (mass) force, on the other hand, determines orbit stability for one mass object revolving around another mass object. Such a condition constitutes the theoretical background of the (experimental) third Kepler law. See examples A and B. |
| 2 | Taking into account point 1, and requiring conservation of the angular momentum, we obtain the (experimental) second Kepler law. The experimental first Kepler law is a consequence of the second Kepler law. |
| 3 | Apart from obvious analogy between Coulomb's and the Newton's attraction laws, there exists some analogy between the magnetic component of the Lorentz force, on the one hand, and the centrifugal mass force, on the other hand |
| 4 |
Vector of magnetic induction B generated by rotating charge results in the magnetic (centrifugal) force acting on that charge itself in a way similar to the action of the angular velocity on the revolving mass object.
|
| 5 | Gravitational attraction between a proton and an electron is much smaller than the corresponding Coulomb force, however the centrifugal mass force is much greater then the corresponding centrifugal magnetic force. Because of that, in the case of atoms, balance of the Coulomb force and the mass centrifugal force constitutes the condition for a stable orbit of the revolving elctron. |
| 6 | If the stability is disturbed (electron jumps from one orbit to the other) there is a radiation of energy predicted by the equation (7). There should not be and there is no electromagnetic radiation when the electron does not jump from one orbit to the other. |
V. Acosta, C.L. Covan, B.J. Graham, 1973, Essentials of Modern Physics, Harper & Row, Publishers, New York.
R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, 1963, Addison - Wesley Publishing Co. Inc., Reading, Massachusetts, USA.
D.L. & I.R. Goldstain, 1996, Feynman Lost Lectures. The Motion of Planets around the Sun, W. W. Norton & Company.
W. Greiner, 1998, Classical Electrodynamics, Springer-Verlag, New York.
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